Global Finite-Energy Solutions of the Compressible Euler–Poisson Equations for General Pressure Laws with Large Initial Data of Spherical Symmetry

Abstract

We are concerned with global finite-energy solutions of the three-dimensional compressible Euler–Poisson equations with gravitational potential and general pressure law, especially including the constitutive equation of white dwarf stars. In this paper, we construct global finite-energy solutions of the Cauchy problem for the Euler–Poisson equations with large initial data of spherical symmetry as the inviscid limit of the solutions of the corresponding Cauchy problem for the compressible Navier–Stokes–Poisson equations. The strong convergence of the vanishing viscosity solutions is achieved through entropy analysis, uniform estimates in $L^p$, and a more general compensated compactness framework via several new ingredients. A key estimate is first established for the integrability of the density over unbounded domains independent of the vanishing viscosity coefficient. Then a special entropy pair is carefully designed via solving a Goursat problem for the entropy equation such that a higher integrability of the velocity is established, which is a crucial step. Moreover, the weak entropy kernel for the general pressure law and its fractional derivatives of the required order near vacuum ($\rho =0$) and far-field ($\rho =\infty$) are carefully analyzed. Owing to the generality of the pressure law, only the $W^{-1,p}_{\textrm{loc}}$-compactness of weak entropy dissipation measures with $p\in [1,2)$ can be obtained; this is rescued by the equi-integrability of weak entropy pairs which can be established by the estimates obtained above, so that the div-curl lemma still applies. Finally, based on the above analysis of weak entropy pairs, the $L^p$ compensated compactness framework for the compressible Euler equations with general pressure law is established. This new compensated compactness framework and the techniques developed in this paper should be useful for solving further nonlinear problems with similar features.

1 Introduction

We are concerned with global finite-energy solutions of the three-dimensional (3-D) compressible Euler–Poisson equations (CEPEs) that take the form:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}\rho + \nabla \cdot \mathcal {M}=0,\\&\partial _{t}\mathcal {M}+ \nabla \cdot \Big (\frac{\mathcal {M}\otimes \mathcal {M}}{\rho }\Big )+\nabla P+\rho \nabla \Phi =0,\\&\Delta \Phi ={k_{g}\rho }, \end{aligned} \right. \end{aligned}$$

(1.1)

for $(t,\textbf{x}):=(t,x_1,x_2,x_3)\in \mathbb {R}_+^4:=\mathbb {R}_{+}\times \mathbb {R}^{3}=(0,\infty )\times \mathbb {R}^3$. System (1.1) is used to model the motion of compressible gaseous stars under a self-consistent gravitational field (cf. [7]), where $\rho$ is the density, $P=P(\rho )$ is the pressure, $\mathcal {M}\in \mathbb {R}^{3}$ is the momentum, $\Phi$ represents the gravitational potential of gaseous stars as ${k_{g}}>0$, $\nabla =(\partial _{x_1}, \partial _{x_2}, \partial _{x_3})$, and $\Delta =\partial _{x_1x_1}+\partial _{x_2x_2}+\partial _{x_3x_3}$. Without loss of generality, by scaling, we take ${k_{g}}=1$ throughout this paper.

The constitutive pressure-density relation $P(\rho )$ depends on the types of gaseous stars. The class of polytropic gases, i.e.,

$$\begin{aligned} P(\rho )=\kappa \rho ^{\gamma } \qquad \text { for }\kappa >0\;\text { and}\; \gamma \in (1,3), \end{aligned}$$

(1.2)

has been widely investigated in mathematics. From the point view of astronomy, the constitutive pressure $P(\rho )$ for certain gaseous stars is not of the polytropic form. For example, the pressure law of a white dwarf star takes the following form (cf. [7, 66]):

$$\begin{aligned} P(\rho )=\mathcal {C}_{1}\int _{0}^{\mathcal {C}_{2}\rho ^{\frac{1}{3}}}\frac{s^4}{\sqrt{\mathcal {C}_{3}+s^2}}\,\textrm{d}s \qquad \, \text { for}\, \rho >0, \end{aligned}$$

(1.3)

where $\mathcal {C}_{1}, \mathcal {C}_{2}$, and $\mathcal {C}_{3}$ are positive constants. It can be checked that $P(\rho )\cong \kappa _1\rho ^\frac{5}{3}$ as $\rho \rightarrow 0$ and $P(\rho )\cong \kappa _2\rho ^\frac{4}{3}$ as $\rho \rightarrow \infty$ for some positive constants $\kappa _1$ and $\kappa _2$.

In this paper, we consider a general pressure law in which any pressure function $P(\rho )$ satisfies the following conditions:

• (i) The pressure function $P(\rho )$ is in $C^1([0,\infty ))\cap C^4(\mathbb {R}_+)$ and satisfies the hyperbolic and genuinely nonlinear conditions:

  $$\begin{aligned} P'(\rho )>0,\quad 2P'(\rho )+\rho P''(\rho )>0\qquad \,\, \text {for }\rho >0. \end{aligned}$$

  (1.4)

• (ii) There exists a constant $\rho _{*}>0$ such that

  $$\begin{aligned} P(\rho )=\kappa _{1} \rho ^{\gamma _1}\big (1+\mathcal {P}_1(\rho )\big ) \qquad \text { for } \rho \in [0, \rho _{*}), \end{aligned}$$

  (1.5)

  with some constants $\gamma _1\in (1,3)$ and $\kappa _1>0$, and a function $\mathcal {P}_1(\rho )\in C^4(\mathbb {R}_+)$ satisfying that $|\mathcal {P}_1^{(j)}(\rho )|\le C_{*}\rho ^{\gamma _1-1-j}$ for $\rho \in (0,\rho _{*})$ and $j=0,\cdots ,4$, where $C_{*}>0$ is a constant depending only on $\rho _{*}$.

• (iii) There exists a constant $\rho ^{*}> \rho _{*}>0$ such that

  $$\begin{aligned} P(\rho )=\kappa _2\rho ^{\gamma _2}\big (1+\mathcal {P}_2(\rho )\big ) \qquad \text { for } \rho \in [\rho ^{*},\infty ), \end{aligned}$$

  (1.6)

  with some constants $\gamma _2\in (\frac{6}{5},\gamma _{1}]$ and $\kappa _{2}>0$, and a function $\mathcal {P}_2(\rho )\in C^{4}(\mathbb {R}_{+})$ satisfying that $|\mathcal {P}_{2}^{(j)}(\rho )|\le C^{*}\rho ^{-\epsilon -j}$ for $\rho \in [\rho ^{*},\infty )$ and $j=0,\cdots ,4$, where $\epsilon >0$, and $C^{*}>0$ is a constant depending only on $\rho ^{*}$.

It is direct to see that the polytropic gases in (1.2) satisfy assumptions (1.4)–(1.6). Moreover, the white dwarf star (1.3) is also included with

$$\begin{aligned} \gamma _1=\frac{5}{3},\quad \kappa _{1}=\frac{1}{5\sqrt{\mathcal {C}_3}}\mathcal {C}_1\mathcal {C}_2^5, \quad \gamma _2=\frac{4}{3},\quad \kappa _{2}=\frac{1}{4}\mathcal {C}_1\mathcal {C}_2^4,\quad \epsilon =\frac{2}{3}. \end{aligned}$$

(1.7)

The restriction: $\gamma _2>\frac{6}{5}$ is necessary to ensure the global existence of finite-energy solutions with finite total mass. Such a condition is also needed for the existence of the Lane–Emden solutions; see [7, 47].

We consider the Cauchy problem of (1.1) with the initial data:

$$\begin{aligned} (\rho ,\mathcal {M})(0,\textbf{x})=(\rho _0,\mathcal {M}_0)(\textbf{x})\,\rightarrow \, (0,\textbf{0})\qquad \,\, \text { as }\,|\textbf{x}|\rightarrow \infty , \end{aligned}$$

(1.8)

subject to the far field condition:

$$\begin{aligned} \Phi (t,\textbf{x})\,\rightarrow \, 0\qquad \,\, \text { as}\, |\textbf{x}|\rightarrow \infty . \end{aligned}$$

(1.9)

The global existence of solutions of the Cauchy problem (1.1) and (1.8)–(1.9) is a longstanding open problem. Many efforts have been made for the polytropic gas case (1.2). Considerable progress has been made on the smooth or special solutions under some restrictions on the initial data. Among the most famous solutions of CEPEs (1.1) are the Lane–Emden steady solutions (cf. [47]), which describe spherically symmetric gaseous stars in equilibrium and minimize the energy among all possible configurations (cf. [46]). There exist expanding solutions for the non-steady CEPEs (1.1). Hadzić–Jang [34] proved the nonlinear stability of the affine solution (which is linearly expanding) under small spherically symmetric perturbations for $\gamma =\frac{4}{3}$, while the stability problem for $\gamma \ne \frac{4}{3}$ is still widely open. A class of linearly expanding solutions for $\gamma =1+\frac{1}{k}$ with $k\in \mathbb {N}\backslash \{1\}$, or $\gamma \in (1,\frac{14}{13})$, was further constructed in [35]. For $1< \gamma <\frac{4}{3}$, the concentration (collapse) phenomena may happen. Indeed, as $\gamma =\frac{4}{3}$, there exists an homologous concentration solution; see [28, 30, 55]. More recently, Guo–Hadzić–Jang [31] observed a continued concentration solution for $1< \gamma <\frac{4}{3}$; see also [37]. A kind of smooth radially symmetric self-similar solutions exhibiting gravitational collapse for $1\le \gamma <\frac{4}{3}$ can be found in [32, 33]. We refer to [51, 54] for the local well-posedness of smooth solutions.

Owing to the strong nonlinearity and hyperbolicity, the smooth solutions of (1.1) with (1.2) may break down in a finite time, especially when the initial data are large (cf. [16, 55]). Therefore, weak solutions have to be considered for large initial data. For gaseous stars surrounding a solid ball, Makino [56] obtained the local existence of weak solutions for $\gamma \in (1,\frac{5}{3}]$ with spherical symmetry; also see Xiao [69] for global weak solutions with a class of initial data. For this case, the possible singularity at the origin is prevented since the domain was considered outside a ball. Luo–Smoller [50] proved the conditional stability of rotating and non-rotating white dwarfs and rotating supermassive stars; see also Rein [61] for the conditional nonlinear stability of the Lane–Emden steady solutions.

Another fundamental question is whether global solutions can be constructed via the vanishing viscosity limit of the solutions of the compressible Navier–Stokes–Poisson equations (CNSPEs):

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}\rho + \nabla \cdot \mathcal {M}=0,\\&\partial _{t}\mathcal {M}+ \nabla \cdot \Big (\frac{\mathcal {M}\otimes \mathcal {M}}{\rho }\Big ) +\nabla P+\rho \nabla \Phi =\varepsilon \nabla \cdot \Big (\mu (\rho )D\big (\frac{\mathcal {M}}{\rho }\big )\Big ) +\varepsilon \nabla \Big (\lambda (\rho ) \nabla \cdot \big (\frac{\mathcal {M}}{\rho }\big )\Big ),\\&\Delta \Phi =\rho , \end{aligned} \right. \end{aligned}$$

(1.10)

where $D(\frac{\mathcal {M}}{\rho })=\frac{1}{2}\big (\nabla (\frac{\mathcal {M}}{\rho })+(\nabla (\frac{\mathcal {M}}{\rho }))^{\bot }\big )$ is the stress tensor, the Lamé (shear and bulk) viscosity coefficients $\mu (\rho )$ and $\lambda (\rho )$ depend on the density (that may vanish on the vacuum) and satisfy

$$\begin{aligned} \mu (\rho )\ge 0,\quad \mu (\rho )+3\lambda (\rho )\ge 0\qquad \,\,\, \text {for}\, \rho \ge 0, \end{aligned}$$

(1.11)

and parameter $\varepsilon >0$ is the inverse of the Reynolds number. Formally, as $\varepsilon \rightarrow 0$, the sequence of the solutions of CNSPEs (1.10) converges to a corresponding solution of CEPEs (1.1). However, the rigorous proof has been one of the most challenging problems in mathematical fluid dynamics; see Chen-Feldman [9] and Dafermos [21].

The limit problem with vanishing physical viscosity dates back to the pioneering paper by Stokes [65]. Most of the known results were around the inviscid limit from the compressible Navier–Stokes to the Euler equations for the polytropic gas case (1.2). The first rigorous proof of the vanishing viscosity limit from the Navier–Stokes to the Euler equations was provided by Gilbarg [29], in which he established the existence and inviscid limit of the Navier–Stokes shock layers. For the case of large data, due to the lack of $L^{\infty }$ uniform estimate, the $L^{\infty }$ compensated compactness framework [22,23,24, 36, 48, 49] fails to work directly in the inviscid limit of the compressible Navier–Stokes equations. An $L^{p}$ compensated compactness framework was first studied by LeFloch–Westdickenberg [43] for the isentropic Euler equations for the case $\gamma \in (1,\frac{5}{3})$ in (1.2), and was further developed by Chen–Perepelitsa [13] to all $\gamma >1$ for (1.2) with a simplified proof; see also [17] for spherically symmetric solutions of the M-D isentropic Euler equations. We also refer to [63, 64] for the 1-D case of asymptotically isothermal gas, i.e., $\gamma _2=1$ in (1.6). More recently, Chen–He–Wang–Yuan [10] established both the strong inviscid limit of CNSPEs (1.10) and the global existence of spherically symmetric solutions of CEPEs (1.1) with large data for polytropic gases (1.2).

The main purpose of this paper is to establish the global existence of spherically symmetric finite-energy solutions of (1.1) with general pressure law (1.4)–(1.6):

$$\begin{aligned} \rho (t,\textbf{x})=\rho (t,r),\quad \mathcal {M}(t,\textbf{x})=m(t,r)\frac{\textbf{x}}{r},\quad \Phi (t,\textbf{x})=\Phi (t,r) \qquad \,\, \text {for }r=|\textbf{x}|, \end{aligned}$$

(1.12)

subject to the initial condition:

$$\begin{aligned} (\rho ,\mathcal {M})(0,\textbf{x})=(\rho _0,\mathcal {M}_{0})(\textbf{x})=(\rho _{0}(r),m_0(r)\frac{\textbf{x}}{r})\,\rightarrow \, (0,\textbf{0}) \qquad \,\, \text {as }r\rightarrow \infty , \end{aligned}$$

(1.13)

and the asymptotic boundary condition:

$$\begin{aligned} \Phi (t,\textbf{x})=\Phi (t,r)\,\rightarrow \, 0\qquad \,\, \text {as }r\rightarrow \infty . \end{aligned}$$

(1.14)

Systems (1.1) and (1.10) for spherically symmetric solutions take the following respective forms:

$$\begin{aligned} \left\{ \begin{aligned}&\rho _{t}+m_{r}+\frac{2}{r}m=0,\\&m_{t}+\Big (\frac{m^2}{\rho }+P(\rho )\Big )_{r}+\frac{2}{r}\frac{m^2}{\rho }+\rho \Phi _{r}=0,\\&\Phi _{rr}+\frac{2}{r}\Phi _{r}=\rho , \end{aligned} \right. \end{aligned}$$

(1.15)

and

$$\begin{aligned} \left\{ \begin{aligned}&\rho _{t}+m_{r}+\frac{2}{r}m=0,\\&m_{t}+\Big (\frac{m^2}{\rho }+P(\rho )\Big )_{r}+\frac{2}{r}\frac{m^2}{\rho }+\rho \Phi _{r}\\&\qquad =\varepsilon \Big ((\mu (\rho )+\lambda (\rho ))\big (\big (\frac{m}{\rho }\big )_r+\frac{2}{r}\frac{m}{\rho }\big )\Big )_{r} -\frac{2\varepsilon }{r}\mu (\rho )_{r}\frac{m}{\rho },\\&\Phi _{rr}+\frac{2}{r}\Phi _{r}=\rho . \end{aligned}\right. \end{aligned}$$

(1.16)

The study of spherically symmetric solutions is motivated by many important physical problems such as stellar dynamics including gaseous stars and supernovae formation [7, 60, 68]. An important question is how the waves behave as they move radially inward near the origin, especially under the self-gravitational force for gaseous stars. The spherically symmetric solutions of the compressible Euler equations may blow up near the origin [20, 44, 57, 68] at certain time in some situations. Considering the effect of gravitation, a fundamental problem for CEPEs (1.1) is whether a concentration (delta-measure) is formed at the origin. This problem was answered in [10] for polytropic gases in (1.2) when the initial total-energy is finite that no delta-measure is formed for the density at the origin for the two cases: (i) $\gamma >\frac{6}{5}$; (ii) $\gamma \in (\frac{6}{5}, \frac{4}{3}]$ and the initial total-energy is finite and the total mass is less than a critical mass.

In this paper, we establish the global existence of finite-energy solutions of the Cauchy problem (1.1) and (1.13)–(1.14) with spherical symmetry as the inviscid limits of global weak solutions of CNSPEs (1.10) with general pressure law (1.4)–(1.6), especially including the white dwarf star (1.3). The $L^p$ compensated compactness framework for the general pressure is also established. Moreover, it is proved that no delta-measure is formed for the density at the origin in the limit, and the critical mass for the white dwarf star is the same as the Chandrasekhar limit for the polytropic gas (1.2) with $\gamma =\frac{4}{3}$. The precise statements of the main results are given in Sect. 2.

To achieve these, the main strategy is to develop entropy analysis, uniform estimates in $L^p$, and a more general compensated compactness framework to prove that there exists a strongly convergent subsequence of solutions of CNSPEs (1.10) and show that the limit is the finite-energy weak solution of CEPEs (1.1) with general pressure law. This consists of the following three steps:

• Establish the uniform $L^p$ estimates of the solutions of CNSPEs (1.10) independent of $\varepsilon$ for some $p>1$;

• Show the compactness for weak entropy dissipation measures;

• Prove that the associated Young measure $\nu _{(t,r)}$ is the delta measure almost everywhere which leads to a subsequence of solutions of CNSPEs (1.10) strongly converging to the global finite-energy solution of CEPEs (1.1).

The generality of pressure $P(\rho )$ causes essential difficulties in the analysis for all of the above steps. We now describe these difficulties and show how they can be overcome:

(i) The crucial step in the $L^p$ estimates is to show that $\rho |u|^3$ ($u:=\frac{m}{\rho }$ is the velocity) is uniformly bounded in $L^1_\textrm{loc}$. This estimate might be obtained through constructing appropriate entropy $\hat{\eta }$, which is a solution of $(\rho ,u)$ to the entropy equation:

$$\begin{aligned} \eta _{\rho \rho }-\frac{P'(\rho )}{\rho ^{2}}\eta _{uu}=0, \end{aligned}$$

(1.17)

with corresponding entropy flux $\hat{q}$. If $(\rho ,u)$ is the solution of (1.16), any entropy-entropy flux pair (entropy pair, for short) $(\hat{\eta },\hat{q})$ satisfies

$$\begin{aligned}&(\hat{\eta } r^{2})_{t}+(\hat{q}r^{2})_{r} +2r\, (-\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_m)\\&\quad =\varepsilon \,r^{2}\big ((\rho u_{r})_{r}+2\rho \big (\frac{u}{r}\big )_{r}\big )\hat{\eta }_{m} -\rho \int _{a}^{r}\rho \, z^{2}\textrm{d}z\,\hat{\eta }_{m}; \end{aligned}$$

see (5.68) below. For the polytropic gas case (1.2), there is an explicit formula of the entropy kernel $\chi (\rho ,u)$ so that $\chi * \psi$ is the entropy, where $*$ denotes the convolution and $\psi (s)$ is any smooth function. By choosing $\psi (s)=\frac{1}{2}s|s|$ as in [10], the corresponding entropy flux $\hat{q}$ satisfies that $\hat{q}\ge c_0\rho |u|^3$ and $-\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_m\le 0$. Then the uniform bound of $\rho |u|^3r^2$ in $L^1_\textrm{loc}$ follows (cf. [10]).

However, there is no explicit formula of the entropy kernel $\chi$ for the general pressure satisfying (1.4)–(1.6). Even for the special entropy pair generated by $\psi (s)=\frac{1}{2}s|s|$, it is difficult to prove that $\hat{q}\ge c_0\rho |u|^3$ and $-\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_m\le 0$, due to the lack of explicit formula of the entropy kernel $\chi$. Hence, the above approach does not apply directly, so we have to seek a new method to establish the uniform local integrability of $\rho |u|^3$. One of the novelties of this paper is that a special entropy $\hat{\eta }$ is constructed by solving a Goursat problem of the entropy equation (1.17) in the domain: $|u|\le k(\rho ):=\int _{0}^{\rho }\sqrt{P'(y)}/{y}\,\textrm{d} y$, so that $\hat{\eta }$ is chosen as the mechanical energy $\eta ^*$ (see (2.13)) when $u\ge k(\rho )$, $-\eta ^*$ when $u\le -k(\rho )$, and the boundary condition for the Goursat problem is given on the characteristics curves: $u\pm k(\rho )=0$. One advantage of such a special entropy pair $(\hat{\eta },\hat{q})$ is that $\hat{q}\ge c_0\rho |u|^3$ as $|u|\ge k(\rho )$, and $|\hat{q}|\le C\rho ^{\gamma _2+1}$ for large $\rho$ as $|u|\le k(\rho )$ via careful analysis for the Goursat problem; see Lemma 5.8 for details. Moreover, $-\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_m$ vanishes as $|u|\ge k(\rho )$. Similarly, $|-\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_m|\le C\rho ^{\gamma _2+1}$ for large $\rho$ as $|u|\le k(\rho )$.

To show $\rho |u|^3$ is uniformly bounded in $L^1_\textrm{loc}$, it remains to prove that

$$\begin{aligned} \int _{0}^{T}\int _{d}^{\infty }\rho ^{\gamma _2+1}\,r\textrm{d}r\textrm{d}t \end{aligned}$$

(1.18)

is uniformly bounded for any $T>0$ and $d>0$. It should be noted that the local integrability $\int _{0}^{T}\int _{d}^{D}\rho ^{\gamma _2+1}\,\textrm{d}r\textrm{d}t\le C$ was obtained in [10], but it is not enough yet to obtain the uniform $L^1_\textrm{loc}$ estimate for $\rho |u|^3$. Fortunately, we can obtain even stronger estimate than (1.18), i.e.,

$$\begin{aligned} \int _{0}^{T}\int _{d}^{\infty }\rho ^{\gamma _2+1}\,r^2\textrm{d}r\textrm{d}t\le C, \end{aligned}$$

(1.19)

by an elaborate analysis; see Lemma 5.6 and Corollary 5.7 for details.

(ii) For the polytropic gas case in (1.2), Chen–Perepelitsa [13, 14] and Chen–He–Wang–Yuan [10] proved the $H_\textrm{loc}^{-1}$–compactness for weak entropy dissipation measures via the explicit formula of the weak entropy kernel $\chi$ by convolution with any test function of compact support, which also implies that the entropy pair $(\eta , q)$ is in $L^r_\textrm{loc}, r>2$. However, it is not clear how the $H_\textrm{loc}^{-1}$–compactness for the general pressure satisfying (1.4)–(1.6) can be shown by using the expansions of the weak entropy kernel established in [11, 12]. Motivated by [64], we instead show the $W_{\textrm{loc}}^{-1,p}$–compactness for $1\le p<2$, so that an improved div-curl lemma (cf. [19]) applies, which leads to the commutation identity for the entropy pairs. In fact, we can show that the entropy flux function q is bounded by $\rho ^{\frac{\gamma _2+1}{2}}$ (see (4.81)) as $\rho$ is large by careful analysis on the expansion of the entropy pair so that $q\in L^2_\textrm{loc}$. Then the interpolation compactness yields the $W^{-1,p}$ compactness for $1\le p<2$; see Lemma 7.1 for details.

(iii) The argument for the reduction of the associated Young measure $\nu _{(t,r)}(\rho , u)$, introduced in [10, 13, 14], for the polytropic gas case in (1.2), can be roughly stated as follows: Show first that every connected subset of the support of the Young measure is a bounded interval; then use the $L^\infty$ reduction technique introduced in [8, 22, 24, 48] for a bounded supported Young measure to show that the Young measure is either a delta measure or supported on the vacuum line. This method essentially relies on the explicit formula of the weak entropy kernel $\chi$. For the general pressure law satisfying (1.4)–(1.6), the above method does not apply directly, since it is difficult to show that every connected subset of the support of the Young measure is a bounded interval. Motivated by [11, 12, 48, 63, 64], we carefully analyze the singularities of $\partial ^{\lambda _1+1} \chi$ with $\lambda _1=\frac{3-\gamma _1}{2(\gamma _1-1)}$ for large $\rho$ and fully exploit the property: $(\rho ^{\gamma _2+1},\rho |u|^3)\in L^1(\textrm{d}\nu _{(t,r)})$ so that the $\partial ^{\lambda _1+1}-$derivatives can be operated in the commutation relation; see Lemmas 4.11–4.14 for details. Then we prove that the Young measure is either a delta measure or supported on the vacuum line by similar arguments as in [11, 12, 48, 64]. This new compensated compactness framework and the techniques developed in this paper should be useful for solving further nonlinear problems with similar features.

Finally, we remark that there are some related results on CNSPEs (1.10) and the compressible Euler equations. For weak solutions of CNSPEs (1.10), we refer to [26, 38, 40, 41] with constant viscosity, and [25, 27, 70] with density-dependent viscosity. Recently, Luo-Xin-Zeng [51,52,53] proved the large-time stability of the Lane–Emden solution for $\gamma \in (\frac{4}{3},2)$. We also refer to the BD entropy developed in [2,3,4,5], which provides a new estimate for the gradient of the density. For the compressible Euler equations, we refer to [8, 15, 39, 44, 62] and the references cited therein.

The rest of this paper is organized as follows: In Sect. 2, the finite-energy solutions of the Cauchy problem (1.1) and (1.8)–(1.9) for CEPEs are introduced, and the main theorems of this paper are given. In Sect. 3, some elementary quantities and basic properties about the pressure and related internal energy are provided, and then some remarks on $M_\textrm{c}$ are also given. The entropy analysis for weak entropy pairs for the general pressure satisfying (1.4)–(1.6) is presented in Sect. 4, especially a special entropy pair is constructed by solving a Goursat problem for the entropy equation (2.14). In Sect. 5, a free boundary problem (5.1)–(5.6) for (1.16) is analyzed, and some uniform estimates of solutions are derived, including the basic energy estimate, the BD-type entropy estimate, and the higher integrabilities of the density and the velocity. In Sect. 6, the global existence of weak solutions of CNSPEs (1.10) is established, and some uniform $L^p$ estimates in Theorem 2.1 are also obtained. In Sect. 7, we prove the $W_{\textrm{loc}}^{-1,p}$–compactness of the entropy dissipation measures for the weak solutions of (1.16) and complete the proof of Theorem 2.1. In Sect. 8, the $L^p$–compensated compactness framework for the general pressure law (1.4)–(1.6) (Theorem 2.2) is established, which leads to the proof of Theorem 2.3 by taking the inviscid limit of weak solutions of CNSPEs (1.10) in Sect. 9. Appendix A is devoted to the presentation of both the sharp Sobolev inequality that is used in Sect. 5 and some variants of Grönwall’s inequality which are used in the proof of several estimates in Sect. 4.

Notations: Throughout this paper, we denote $C^{\alpha } (\Omega ), L^{p}(\Omega ), W^{k, p}(\Omega )$, and $H^{k}(\Omega )$ as the standard Hölder space, and the corresponding Sobolev spaces, respectively, on domain $\Omega$ for $\alpha \in (0,1)$ and $p\in [1, \infty ]$. $C_{0}^{k}(\Omega )$ represents the space of continuously differentiable functions up to the kth order with compact support over $\Omega$, and $\mathcal {D}(\Omega ):=C_{0}^{\infty }(\Omega )$. We also use $L^{p}(I; r^{2}\textrm{d}r)$ or $L^{p}([0, T) \times I; r^{2}\textrm{d} r\textrm{d} t)$ for an open interval $I \subset \mathbb {R}_{+}$ with measure $r^{2}\textrm{d}r$ or $r^{2}\textrm{d}r \textrm{d}t$ correspondingly, and $L_{\textrm{loc}}^{p}([0, \infty ); r^{2}\textrm{d}r)$ to represent $L^{p}([0, R]; r^{2}\textrm{d} r)$ for any fixed $R>0$.

2 Mathematical Problem and Main Theorems

The spherically symmetric initial data function $(\rho _{0},\mathcal {M}_{0})(\textbf{x})$ given in (1.13) is assumed to be of both finite initial total-energy:

$$\begin{aligned} E_{0}&:=\int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}_{0}}{\sqrt{\rho _{0}}}\Big |^{2} +\rho _{0} e(\rho _{0})\Big )(\textbf{x}) \,\textrm{d}\textbf{x}=\omega _3\int _0^\infty \Big (\frac{1}{2}\frac{m^2_{0}}{\rho _{0}} +\rho _{0} e(\rho _{0})\Big )(r)\, r^2\textrm{d}r<\infty , \end{aligned}$$

(2.1)

and initial total-mass:

$$\begin{aligned} M:=\int _{\mathbb {R}^3}\rho _0(\textbf{x})\,\textrm{d} \textbf{x}=\omega _3\int _{0}^{\infty }\rho _0(r)\,r^{2}\textrm{d} r<\infty , \end{aligned}$$

(2.2)

where the internal energy $e(\rho )$ is related to the pressure by

$$\begin{aligned} e'(\rho )= \frac{P(\rho )}{\rho ^2}, \qquad e(0)=0, \end{aligned}$$

(2.3)

and $\omega _n:=\frac{2\pi ^{\frac{n}{2}}}{\Gamma (\frac{n}{2})}$ denotes the surface area of the unit sphere in $\mathbb {R}^n$. The initial potential $\Phi _0(\textbf{x})$ is determined by

$$\begin{aligned} \Delta \Phi _0(\textbf{x})=\rho _0(\textbf{x}), \qquad \lim _{|\textbf{x}|\rightarrow \infty }\Phi _0(\textbf{x})=0. \end{aligned}$$

(2.4)

For $\gamma _{2}\in (\frac{6}{5},\frac{4}{3}]$, we define the critical mass $M_\textrm{c}$ as follows:

(i) When $\gamma _2=\frac{4}{3}$,

$$\begin{aligned} M_\textrm{c}:=M_\textrm{ch}, \end{aligned}$$

(2.5)

where $M_\textrm{ch}$ is the Chandrasekhar limit that is the total mass of the Lane–Emden steady solution $(\rho _{s}(|\textbf{x}|),0)$ for $P(\rho )=\kappa _2\rho ^{\frac{4}{3}}$:  $\rho _{s}(|\textbf{x}|)$ has compact support and is determined by the equations:

$$\begin{aligned} \nabla _{\textbf{x}}P(\rho _{s}(|\textbf{x}|))+\rho _{s}(|\textbf{x}|)\nabla _{\textbf{x}}\Phi (\textbf{x})=0,\quad \Delta _{\textbf{x}}\Phi (\textbf{x})=\rho _{s}(|\textbf{x}|), \quad P(\rho _{s}|\textbf{x}|)=\kappa _2(\rho _{s}(|\textbf{x}|))^{\frac{4}{3}}, \end{aligned}$$

with the center density $\rho _{s}(0)=\varrho$. It is well-known that $M_\textrm{ch}$ is a uniform constant with respect to the center density $\varrho$ (cf. [7]).

(ii) When $\gamma _2\in (\frac{6}{5},\frac{4}{3})$,

$$\begin{aligned} M_\textrm{c}:=\sup _{\beta >0}M_\textrm{c}(\beta ) \end{aligned}$$

(2.6)

with

$$\begin{aligned} (4-3\gamma _2)\Big (\frac{B_{\beta }}{3(\gamma _2-1)}\Big )^{-\frac{3(\gamma _2-1)}{4-3\gamma _2}}M_\textrm{c}(\beta )^{-\frac{5\gamma _2-6}{4-3\gamma _2}} -\omega _{3}^{-1}\beta M_\textrm{c}(\beta )=\frac{E_{0}}{\omega _{3}}, \end{aligned}$$

(2.7)

and

$$\begin{aligned} \begin{aligned}&B_{\beta }:= \frac{2}{3} \omega _{4}^{-\frac{2}{3}} \omega _{3}^{\frac{4-3 \gamma _2}{3(\gamma _2-1)}} (C_{\max }(\beta ))^{\frac{5\gamma _2-6}{3(\gamma _2-1)}},\\&C_{\max }(\beta ):=\sup _{\rho \ge 0}\big (\rho ^{\gamma _2-1}(\beta +e(\rho ))^{-1}\big )^{\frac{1}{5\gamma _2-6}}>0. \end{aligned} \end{aligned}$$

(2.8)

It is clear in (2.6)–(2.8) that $M_\textrm{c}(\beta )$ is well determined for $\beta >0$ and $\gamma _2\in (\frac{6}{5},\frac{4}{3})$. Some useful properties of $M_\textrm{c}:=\sup _{\beta >0}M_\textrm{c}(\beta )$ will be presented in Proposition 3.3 below. We also point out that $M_\textrm{c}$ in (2.5) is strictly larger than the one obtained in [10, (2.8)] for $\gamma _2=\frac{4}{3}$ (cf. [18]).

For the spherically symmetric initial data $(\rho _{0}, m_{0},\Phi _{0})(r)$ imposed in (1.12)–(1.14) satisfying (2.1)–(2.2), using similar arguments as in [10, Appendix A], we can construct a sequence of approximate initial data functions $(\rho _{0}^{\varepsilon },m_{0}^{\varepsilon }, \Phi _{0}^{\varepsilon })(r)$ satisfying

$$\begin{aligned} \begin{aligned}&\int _{0}^{\infty } \rho _{0}^{\varepsilon }(r)\, r^{2}\textrm{d} r=\frac{M}{\omega _3}, \qquad \Phi _{0r}^{\varepsilon }=\frac{1}{r^2}\int _{0}^{r}\rho _{0}^{\varepsilon }(z)\,z^2\textrm{d}z,\\&\,E_{0}^{\varepsilon }:=\omega _{3} \int _{0}^{\infty } \Big (\frac{1}{2}\Big |\frac{m_{0}^{\varepsilon }}{\sqrt{\rho _{0}^{\varepsilon }}}\Big |^{2} +\rho _{0}^{\varepsilon } e(\rho _{0}^{\varepsilon })\Big )\,r^{2}\textrm{d}r\le C(E_{0}+1)<\infty ,\\&\,E_{1}^{\varepsilon }:=\varepsilon ^{2}\omega _{3} \int _{0}^{\infty }\big |\partial _{r} \sqrt{\rho _{0}^{\varepsilon }(r)}\big |^{2} \,r^{2}\textrm{d}r \le C \varepsilon (M+1)<\infty . \end{aligned} \end{aligned}$$

(2.9)

Moreover, as $\varepsilon \rightarrow 0$, $(E_{0}^{\varepsilon }, E_{1}^{\varepsilon }) \rightarrow (E_{0}, 0)$ and

$$\begin{aligned}&(\rho _{0}^{\varepsilon }, \rho _{0}^{\varepsilon }u_{0}^{\varepsilon })(r) \rightarrow (\rho _{0}, \rho _{0}u_{0})(r) \qquad \text{ in } L^{\tilde{q}}([0, \infty ); r^{2}\text {d} r) \times L^{1}([0, \infty ); r^{2}\text {d} r),\\&\Phi _{0r}^{\varepsilon }\rightarrow \Phi _{0r}\qquad \text{ in } L^2([0,\infty );r^{2}\text {d}r), \end{aligned}$$

where $\tilde{q}\in \{1,\gamma _{2}\}$. Furthermore, there exists $\varepsilon _0\in (0,1]$ such that, for any $\varepsilon \in (0,\varepsilon _0]$,

$$\begin{aligned} M<M_\textrm{c}^{\varepsilon }\qquad \text {for }\gamma _2\in (\frac{6}{5},\frac{4}{3}], \end{aligned}$$

(2.10)

where $M_\textrm{c}^{\varepsilon }$ is defined in (2.5)–(2.8) by replacing $E_0$ with $E_0^{\varepsilon }$.

Now we introduce the weak entropy pairs of the 1-D isentropic Euler system (cf. [11, 42]):

$$\begin{aligned} \left\{ \begin{aligned}&\rho _{t}+m_{r}=0,\\&m_{t}+\big (\frac{m^2}{\rho }+P(\rho )\big )_{r}=0. \end{aligned} \right. \end{aligned}$$

(2.11)

A pair of functions $(\eta (\rho ,m),q(\rho ,m))$ is called an entropy pair of (2.11) if

$$\begin{aligned} \nabla q(\rho ,m)=\nabla \eta (\rho ,m)\nabla \Big (\begin{matrix}m\\ \frac{m^2}{\rho }+P(\rho )\end{matrix}\Big ). \end{aligned}$$

(2.12)

Moreover, $\eta (\rho ,m)$ is called a weak entropy if $\eta (\rho ,m)\vert _{\rho =0}=0$, and a convex entropy if $\nabla ^2\eta (\rho , m)\ge 0$. The mechanical energy and energy flux pair is defined as

$$\begin{aligned} \eta ^{*}(\rho ,m)=\frac{1}{2}\frac{m^2}{\rho }+\rho e(\rho ),\qquad q^{*}(\rho ,m)=\frac{1}{2}\frac{m^3}{\rho }+m(\rho e(\rho ))', \end{aligned}$$

(2.13)

which is a convex weak entropy pair. From (2.12), any entropy satisfies

$$\begin{aligned} \eta _{\rho \rho }-\frac{P'(\rho )}{\rho ^{2}}\eta _{uu}=0 \end{aligned}$$

(2.14)

with $u=\frac{m}{\rho }$. It is known in [11, 12, 48, 49] that any regular weak entropy can be generated by the convolution of a smooth function $\psi (x)$ with the fundamental solution $\chi (\rho ,u,s)$ of the entropy equation (2.14), i.e.,

$$\begin{aligned} \eta ^{\psi }(\rho ,u)=\int _{\mathbb {R}}\chi (\rho ,u,s)\psi (s)\,\textrm{d}s. \end{aligned}$$

(2.15)

The corresponding entropy flux is generated from the flux kernel $\sigma (\rho ,u,s)$ (see (4.56)), i.e.,

$$\begin{aligned} q^{\psi }(\rho ,u)=\int _{\mathbb {R}}\sigma (\rho ,u,s)\psi (s)\,\textrm{d}s. \end{aligned}$$

(2.16)

We first consider the Cauchy problem of CNSPEs (1.10) with approximate initial data:

$$\begin{aligned} (\rho ,\mathcal {M},\Phi )\vert _{t=0}=(\rho _0^{\varepsilon },\mathcal {M}_{0}^{\varepsilon },\Phi _{0}^{\varepsilon })(\textbf{x}):=(\rho _{0}^{\varepsilon }(r), m_{0}^{\varepsilon }(r)\frac{\textbf{x}}{r}, \Phi _{0}^{\varepsilon }(r)), \end{aligned}$$

(2.17)

subject to the far field condition:

$$\begin{aligned} \Phi ^{\varepsilon }(t,\textbf{x})\longrightarrow 0\qquad \text {as }|\textbf{x}|\rightarrow \infty . \end{aligned}$$

(2.18)

For concreteness, we take $\varepsilon \in (0,1]$ and the viscosity coefficients $(\mu (\rho ),\lambda (\rho ))=(\rho ,0)$ in (1.10).

Definition 2.1

A triple $(\rho ^{\varepsilon }, \mathcal {M}^{\varepsilon },\Phi ^{\varepsilon })(t,\textbf{x})$ is said to be a weak solution of the Cauchy problem (1.10) and (2.17) if

• (i) $\rho ^{\varepsilon }(t, \textbf{x}) \ge 0$, and $(\mathcal {M}^{\varepsilon }, \frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}})(t, \textbf{x})=\textbf{0}\,$ a.e. on $\{(t, \textbf{x})\,:\,\rho ^{\varepsilon }(t, \textbf{x})=0\}\,$(vacuum),

  $$\begin{aligned} \begin{aligned}&\rho ^{\varepsilon } \in L^{\infty }(0, T ; L^{\gamma _2}(\mathbb {R}^{3})), \quad \nabla \sqrt{\rho ^{\varepsilon }} \in L^{\infty }(0, T ; L^{2}(\mathbb {R}^{3})), \\ {}&\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} \in L^{\infty }(0, T ; L^{2}(\mathbb {R}^{3})),\quad \Phi ^{\varepsilon }\in L^{\infty }(0,T;L^{6}(\mathbb {R}^3)),\quad \nabla \Phi ^{\varepsilon }\in L^{\infty }(0,T;L^2(\mathbb {R}^3)). \end{aligned} \end{aligned}$$

• (ii) For any $t_{2} \ge t_{1} \ge 0$ and any $\zeta (t, \textbf{x}) \in C_{0}^{1}([0, \infty ) \times \mathbb {R}^{3})$, the mass equation (1.10)$_{1}$ holds in the sense:

  $$\begin{aligned} \int _{\mathbb {R}^{3}}(\rho ^{\varepsilon } \zeta )(t_{2}, \textbf{x})\,\textrm{d} \textbf{x} -\int _{\mathbb {R}^{3}}(\rho ^{\varepsilon } \zeta )(t_{1}, \textbf{x})\,\textrm{d} \textbf{x} =\int _{t_{1}}^{t_{2}} \int _{\mathbb {R}^{3}}(\rho ^{\varepsilon } \zeta _{t}+\mathcal {M}^{\varepsilon } \cdot \nabla \zeta )(t, \textbf{x}) \,\textrm{d} \textbf{x} \textrm{d} t. \end{aligned}$$

• (iii) For any $\Psi =(\Psi _{1}, \Psi _{2}, \Psi _{3})(t,\textbf{x}) \in (C_{0}^{2}([0, \infty ) \times \mathbb {R}^{3}))^3$, the momentum equations (1.10)$_{2}$ hold in the sense:

  $$\begin{aligned}&\int _{\mathbb {R}_{+}^4}\Big (\mathcal {M}^{\varepsilon } \cdot \Psi _{t} +\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} \cdot \big (\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} \cdot \nabla \big ) \Psi +P(\rho ^{\varepsilon }) \nabla \cdot \Psi \Big )\,\textrm{d} \textbf{x} \textrm{d} t +\int _{\mathbb {R}^{3}} \mathcal {M}_{0}^{\varepsilon }(\textbf{x}) \cdot \Psi (0, \textbf{x})\,\textrm{d} \textbf{x} \\&\quad =-\varepsilon \int _{\mathbb {R}_{+}^{4}}\Big (\frac{1}{2} \mathcal {M}^{\varepsilon } \cdot \big (\Delta \Psi +\nabla (\nabla \cdot \Psi )\big ) +\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}}\cdot \big (\nabla \sqrt{\rho ^{\varepsilon }} \cdot \nabla \big )\Psi \Big )\,\textrm{d} \textbf{x} \textrm{d} t\\&\qquad -\varepsilon \int _{\mathbb {R}_{+}^{4}} \nabla \sqrt{\rho ^{\varepsilon }} \cdot \big (\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} \cdot \nabla \big ) \Psi \,\textrm{d}\textbf{x}\textrm{d}t +\int _{\mathbb {R}_{+}^{4}}\big ( \rho ^{\varepsilon } \nabla \Phi ^{\varepsilon } \cdot \Psi \big )(t, \textbf{x})\,\textrm{d}\textbf{x}. \end{aligned}$$

• (iv) For any $t\ge 0$ and $\xi (\textbf{x})\in C_{0}^1(\mathbb {R}^3)$,

  $$\begin{aligned} \int _{\mathbb {R}^3}\nabla \Phi ^{\varepsilon }(t,\textbf{x})\cdot \nabla \xi (\textbf{x})\,\textrm{d}\textbf{x}=-\int _{\mathbb {R}^3}\rho ^{\varepsilon }(t,\textbf{x})\xi (\textbf{x})\,\textrm{d}\textbf{x}. \end{aligned}$$

Then we have

Theorem 2.1

(Global existence of spherically symmetric solutions for CNSPEs). Assume that the initial data function $(\rho _{0}^{\varepsilon },\mathcal {M}_{0}^{\varepsilon },\Phi _{0}^{\varepsilon })(\textbf{x})$ is given in (2.17)–(2.18) with $(\rho _{0}^{\varepsilon },m_{0}^{\varepsilon },\Phi _{0}^{\varepsilon })(r)$ satisfying (2.9)–(2.10). Then, for each fixed $\varepsilon \in (0, 1]$, there exists a global weak solution $(\rho ^{\varepsilon },\mathcal {M}^{\varepsilon }, \Phi ^{\varepsilon })(t,\textbf{x})$ of the Cauchy problem (1.10) and (2.17)–(2.18) in the sense of Definition 2.1 with following spherically symmetric form:

$$\begin{aligned} (\rho ^{\varepsilon },\mathcal {M}^{\varepsilon },\Phi ^{\varepsilon })(t,\textbf{x}) =(\rho ^{\varepsilon }(t,r),m^{\varepsilon }(t,r)\frac{\textbf{x}}{r},\Phi ^{\varepsilon }(t,r))\qquad \text {for }r=|\textbf{x}| \end{aligned}$$

(2.19)

such that, for $t\ge 0$,

$$\begin{aligned}&\int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}}\Big |^{2}+\rho ^{\varepsilon } e(\rho ^{\varepsilon })-\frac{1}{2}|\nabla \Phi ^{\varepsilon }|^2\Big )\,\textrm{d}\textbf{x} \le \int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}_{0}^{\varepsilon }}{\sqrt{\rho _{0}^{\varepsilon }}}\Big |^{2} +\rho _{0}^{\varepsilon } e(\rho _{0}^{\varepsilon })-\frac{1}{2}|\nabla \Phi _{0}^{\varepsilon }|^2\Big )\, \textrm{d}\textbf{x}. \end{aligned}$$

(2.20)

Furthermore, for $(\rho ^{\varepsilon }, m^{\varepsilon }, \Phi ^{\varepsilon })(t, r)$, there exists a measurable function $u^{\varepsilon }(t,r)$ with

$$\begin{aligned} u^{\varepsilon }(t,r):=\frac{m^{\varepsilon }(t,r)}{\rho ^{\varepsilon }(t,r)}\qquad { a.e.}\text { on }\big \{(t,r):\,\rho ^{\varepsilon }(t,r)\ne 0\big \}, \end{aligned}$$

and $u^{\varepsilon }(t,r):=0~{ a.e.}\text { on }\big \{(t,r)\,:\,\rho ^{\varepsilon }(t,r)=0\text { or }r=0\big \}$ such that $m^{\varepsilon }(t,r)=(\rho ^{\varepsilon }u^{\varepsilon })(t,r)$ a.e. on $\mathbb {R}_{+}^2:=\mathbb {R}_+\times \mathbb {R}_+$. Moreover, the following properties hold:

$$\begin{aligned}&(\textrm{i})~ \int _{0}^{\infty } \rho ^{\varepsilon }(t, r)\, r^{2}\textrm{d}r=\int _{0}^{\infty } \rho _{0}^{\varepsilon }(r)\, r^{2}\textrm{d} r=\frac{M}{\omega _{3}} \qquad \text {for } t \ge 0, \end{aligned}$$

(2.21)

$$\begin{aligned}&(\textrm{ii})~ \int _{0}^{\infty } \eta ^{*}(\rho ^{\varepsilon }, m^{\varepsilon })(t, r)\, r^{2}\textrm{d} r+\varepsilon \int _{\mathbb {R}_{+}^{2}}(\rho ^{\varepsilon }|u^{\varepsilon }|^{2})(t, r)\,r^2\textrm{d}r \textrm{d} t + \Vert \nabla \Phi ^{\varepsilon }\Vert _{L^2(\mathbb {R}^3)}\nonumber \\&\qquad +\Vert \Phi ^{\varepsilon }\Vert _{L^{6}(\mathbb {R}^3)}+\int _{0}^{\infty }\Big (\int _{0}^{r} \rho ^{\varepsilon }(t, z)\, z^{2}\textrm{d}z \Big ) \rho ^{\varepsilon }(t, r)\, r\textrm{d} r\le C\left( M, E_{0}\right) \qquad \text{ for } t \ge 0, \end{aligned}$$

(2.22)

$$\begin{aligned}&(\textrm{iii})~ \sup _{t\in [0,T]}\varepsilon ^{2} \int _{0}^{\infty }\left| \left( \sqrt{\rho ^{\varepsilon }}\right) _{r}\right| ^{2}\, r^{2}\textrm{d} r+\varepsilon \int _{0}^{T} \int _{0}^{\infty }\frac{P'(\rho ^{\varepsilon })}{\rho ^{\varepsilon }}\left| \rho _{r}^{\varepsilon }\right| ^{2}\, r^{2}\textrm{d}r \textrm{d} t\nonumber \\&\qquad \, \le C(M, E_{0}, T), \end{aligned}$$

(2.23)

$$\begin{aligned}&(\textrm{iv})~~ \int _{0}^{T} \int _{d}^{D} \rho ^{\varepsilon }\left| u^{\varepsilon }\right| ^{3}\,r^{2}\textrm{d} r \textrm{d} t \le C(d, D, M, E_{0}, T), \end{aligned}$$

(2.24)

$$\begin{aligned}&(\textrm{v})~~ \int _{0}^{T} \int _{d}^{\infty }(\rho ^{\varepsilon })^{\gamma _2+1}\, r^{2}\textrm{d} r \textrm{d} t \le C(d, M, E_{0}, T), \end{aligned}$$

(2.25)

for any $T \in \mathbb {R}_{+}$ and interval $[d, D]\Subset (0, \infty )$, where $C(M, E_{0})$, $C(M, E_{0}, T)$, and $C(d, D, M, E_{0}, T)$ are positive constants independent of $\varepsilon$. In addition, for $\varepsilon \in (0,1]$,

$$\begin{aligned} \partial _{t} \eta ^{\psi }\left( \rho ^{\varepsilon }, m^{\varepsilon }\right) +\partial _{r} q^{\psi }\left( \rho ^{\varepsilon }, m^{\varepsilon }\right) \qquad \text{ is } \text{ compact } \text{ in } W_{{\textrm{loc}}}^{-1,p}( \mathbb {R}_{+}^{2}) \end{aligned}$$

(2.26)

for any $p\in [1,2)$, where $\psi (s)$ is any smooth function with compact support on $\mathbb {R}$.

Remark 2.1

In this paper, we require the density-dependent viscosity coefficients $\mu (\rho )$ and $\lambda (\rho )$ to satisfy the BD entropy relation (cf. [2,3,4,5]):

$$\begin{aligned} \rho \mu '(\rho )=\mu (\rho )+\lambda (\rho ), \end{aligned}$$

(2.27)

which is important for us to derive the estimate for the derivative of the density. Under the physical restriction (1.11) and the BD entropy relation (2.27), $\lambda (\rho )$ cannot be a non-zero constant. Since we focus mainly on the global existence of weak solutions for CEPEs by the vanishing viscosity limit of weak solutions of CNSPEs which means the viscous terms will vanish eventually, we consider only the special case $(\mu (\rho ),\lambda (\rho ))=(\rho ,0)$ in the present paper, which corresponds to the well-known Saint-Venant model of shallow water.

Recently, in [6, 27], the global existence of weak solutions was established for the compressible Navier–Stokes equations and CNSPEs for a class of general density-dependent viscous coefficients satisfying the BD entropy relation, respectively. Motivated by [6, 27], it should be able to extend our results to a class of more general viscous coefficients. However, for such general viscous coefficients $\mu (\rho )$ and $\lambda (\rho )$ satisfying the BD relation, we have to check the uniform estimates of the solutions and the validity of vanishing viscosity limit $\varepsilon \rightarrow 0$ so that major modifications to our present paper are required, which is out of scope of this paper.

Now we introduce the notion of finite-energy solutions of CEPEs (1.1).

Definition 2.2

A measurable vector function $(\rho ,\mathcal {M},\Phi )$ is said to be a finite-energy solution of the Cauchy problem (1.1) and (1.8)–(1.9) provided that

• (i) $\rho (t,\textbf{x})\ge 0$ a.e., and $(\mathcal {M},\frac{\mathcal {M}}{\sqrt{\rho }})(t, \textbf{x})=\textbf{0}$ a.e. on $\{(t,\textbf{x})\in \mathbb {R}_{+}^{4}:\,\rho (t,\textbf{x})=0\}$ (vacuum).

• (ii) For a.e. $t>0$, the total energy is finite:

  $$\begin{aligned} \left\{ \begin{aligned}&\int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}}{\sqrt{\rho }}\Big |^{2} +\rho e(\rho )+\frac{1}{2}|\nabla \Phi |^{2}\Big )(t, {\textbf {x}}) \,\text {d} {\textbf {x}} \le C(E_{0}, M), \\&\int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}}{\sqrt{\rho }}\Big |^{2} +\rho e(\rho )-\frac{1}{2}|\nabla \Phi |^{2}\Big )(t, {\textbf {x}})\,\text {d} {\textbf {x}}\\ {}&\quad \,\, \le \int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}_{0}}{\sqrt{\rho _{0}}}\Big |^{2} +\rho _{0} e(\rho _{0})-\frac{1}{2}|\nabla \Phi _{0}|^{2}\Big )({\textbf {x}})\,\text {d} {\textbf {x}}. \end{aligned}\right. \end{aligned}$$

  (2.28)

• (iii) For any $\zeta (t, \textbf{x})\in C_{0}^{1}([0,\infty )\times \mathbb {R}^{3})$,

  $$\begin{aligned} \int _{\mathbb {R}_{+}^{4}}(\rho \zeta _{t}+\mathcal {M}\cdot \nabla \zeta )\,\textrm{d}\textbf{x}\textrm{d}t+\int _{\mathbb {R}^3}(\rho _{0}\zeta )(0,\textbf{x})\,\textrm{d}\textbf{x}=0. \end{aligned}$$

  (2.29)

• (iv) For any $\Psi (t, \textbf{x})=(\Psi _{1},\Psi _{2},\Psi _{3})(t,\textbf{x})\in (C_{0}^{1}([0,\infty )\times \mathbb {R}^3))^{3}$,

  $$\begin{aligned}&\int _{\mathbb {R}_{+}^{4}}\Big (\mathcal {M}\cdot \partial _{t}\Psi +\frac{\mathcal {M}}{\sqrt{\rho }}\cdot (\frac{\mathcal {M}}{\sqrt{\rho }}\cdot \nabla )\Psi +P(\rho )\, \nabla \cdot \Psi \Big )\,\textrm{d}\textbf{x}\textrm{d}t+\int _{\mathbb {R}^3}\mathcal {M}_{0}(\textbf{x})\cdot \Psi (0,\textbf{x})\,\textrm{d}\textbf{x}\nonumber \\&\quad =\int _{\mathbb {R}_{+}^{4}}(\rho \nabla \Phi \cdot \Psi )(t, \textbf{x})\,\textrm{d} \textbf{x}. \end{aligned}$$

  (2.30)

• (v) For any $\xi (\textbf{x})\in C_{0}^{1}(\mathbb {R}^3)$,

  $$\begin{aligned} \int _{\mathbb {R}^{3}} \nabla \Phi (t, {\textbf {x}}) \cdot \nabla \xi ({\textbf {x}}) \,\text {d}{} {\textbf {x}} =- \int _{\mathbb {R}^{3}} \rho (t, {\textbf {x}}) \xi ({\textbf {x}})\,\text {d}{} {\textbf {x}}\qquad \,\, \text{ for }\,\,{ a.e.}\,\, t \ge 0. \end{aligned}$$

  (2.31)

Remark 2.2

In the spherically symmetric form, Definition 2.2 becomes the following: A measurable vector function $(\rho ,\mathcal {M},\Phi )(t,\mathbf{{x}})=(\rho (t,r),m(t,r)\frac{\mathbf{{x}}}{r},\Phi (t,r))$ is said to be a spherically symmetric finite-energy solution of the Cauchy problem (1.1) and (1.13)–(1.14) provided that

• (i) $\rho (t,r)\ge 0$ a.e., and $(m,\frac{m}{\sqrt{\rho }})(t,r)=\textbf{0}$ a.e. on $\{(t,r)\in \mathbb {R}_{+}^2:\,\rho (t,r)=0\}$ (vacuum).

• (ii) For a.e. $t>0$, the total energy is finite:

  $$\begin{aligned} \left\{ \begin{aligned}&\int _{0}^{\infty }\Big (\frac{1}{2}\Big |\frac{m}{\sqrt{\rho }}\Big |^{2} +\rho e(\rho )+\frac{1}{2}|\Phi _{r}|^{2}\Big )(t, r) \,r^2\text {d} r \le C(E_{0}, M), \\ {}&\int _{0}^{\infty }\Big (\frac{1}{2}\Big |\frac{m}{\sqrt{\rho }}\Big |^{2} +\rho e(\rho )-\frac{1}{2}|\Phi _{r}|^{2}\Big )(t, r)\,r^2\text {d}r\\ {}&\quad \,\, \le \int _{0}^{\infty }\Big (\frac{1}{2}\Big |\frac{m_{0}}{\sqrt{\rho _{0}}}\Big |^{2}+\rho _{0} e(\rho _{0})-\frac{1}{2}| \Phi _{0r}|^{2}\Big )(r)\,r^2\text {d}r. \end{aligned}\right. \end{aligned}$$

  (2.32)

• (iii) For any $\zeta (t, r)\in C_{0}^{1}([0,\infty )\times \mathbb {R})$,

  $$\begin{aligned}&\int _{\mathbb {R}_{+}^2}(\rho \zeta _{t}+m\zeta _{r})(t,r)\,r^2\textrm{d}r\textrm{d}t +\int _{0}^{\infty }\rho _{0}(r)\zeta (0,r)\,r^2\textrm{d}r=0. \end{aligned}$$

  (2.33)

• (iv) For any $\psi (t,r)\in C_{0}^{1}([0,\infty )\times \mathbb {R})$ with $\psi (t,0)=0$ for all $t\ge 0$,

  $$\begin{aligned}&\int _{\mathbb {R}_{+}^2}m(t,r)\psi _{t}(t,r)\,r^2\textrm{d}r\textrm{d}t +\int _{\mathbb {R}_{+}^2}\big (\frac{m^2}{\rho }\big )(t,r)\,\psi _{r}(t,r)\,r^2\textrm{d}r\textrm{d}t\nonumber \\&\qquad +\int _{\mathbb {R}_{+}^2}P(\rho (t,r))\,(\psi _{r}+\frac{2}{r}\psi )(t,r)\,r^2\textrm{d}r\textrm{d}t+\int _{0}^{\infty }m_{0}(r)\,\psi (0,r)\,r^2\textrm{d}r\nonumber \\&\quad =\int _{\mathbb {R}_{+}^2}(\rho \Phi _{r})(t,r)\,\psi (t,r)\,r^2\textrm{d} r\textrm{d}t. \end{aligned}$$

  (2.34)

• (v) For any $\xi (r)\in C_{0}^{1}(\mathbb {R})$ and a.e. $t\ge 0$,

  $$\begin{aligned}&\int _{0}^{\infty }\Phi _{r}(t,r)\,\xi _{r}(r)\,r^2\textrm{d}r=-\int _{0}^{\infty } \rho (t,r)\,\xi ( r)\,r^2\textrm{d}r. \end{aligned}$$

  (2.35)

To establish the strong convergence of the inviscid limit of solutions $(\rho ^{\varepsilon }, \mathcal {M}^{\varepsilon }, \Phi ^{\varepsilon })(t,\textbf{x})$ of CNSPEs (1.10) obtained in Theorem 2.1 as $\varepsilon \rightarrow 0$, we establish the following $L^p$ compensated compactness framework for the 1-D Euler equations (2.11) with general pressure law (1.4)–(1.6), in which restriction $\gamma _{2}\in (\frac{6}{5}, \gamma _{1}]$ in (1.6) can be relaxed to $\gamma _{2}\in (1, \gamma _1]$.

Theorem 2.2

($L^p$ compensated compactness framework). Let

$$\begin{aligned} (\rho ^{\varepsilon },m^{\varepsilon })(t,r)=(\rho ^\varepsilon , \rho ^\varepsilon u^\varepsilon )(t,r) \end{aligned}$$

be a sequence of measurable functions with $\rho ^{\varepsilon }\ge 0$ a.e. on $\mathbb {R}_{+}^2$ satisfying the following two conditions:

• (i) For any $T>0$ and $K\Subset \mathbb {R}_+$, there exists $C(K,T)>0$ independent of $\varepsilon$ such that

  $$\begin{aligned} \int _0^T\int _K\big ((\rho ^\varepsilon )^{\gamma _2+1}+\rho ^{\varepsilon }|u^{\varepsilon }|^3\big )\,\textrm{d}r\textrm{d}t\le C(K,T). \end{aligned}$$

• (ii) For any entropy pair $(\eta ^\psi ,q^\psi )$ defined in (2.15)–(2.16) with any smooth function $\psi (s)$ of compact support on $\mathbb {R}$,

  $$\begin{aligned} \partial _{t} \eta ^{\psi }(\rho ^{\varepsilon }, m^{\varepsilon }) +\partial _{r} q^{\psi }(\rho ^{\varepsilon }, m^{\varepsilon }) \qquad \text{ is } \text{ compact } \text{ in }\, W_{\textrm{loc}}^{-1,1}(\mathbb {R}_{+}^{2}). \end{aligned}$$

Then there exists a subsequence (still denoted) $(\rho ^{\varepsilon },m^{\varepsilon })(t,r)$ and a vector function $(\rho ,m)(t,r)$ such that, as $\varepsilon \rightarrow 0$,

$$\begin{aligned} \begin{aligned} \rho ^{\varepsilon }(t,r)&\rightarrow \rho (t,r)~~ \textrm{in}~L^{q_1}_\textrm{loc}(\mathbb {R}_{+}^2)\qquad \,\,\,\,\textrm{for}\ q_1\in [1,\gamma _2+1),\\ m^{\varepsilon }(t,r)&\rightarrow m(t,r)~~ \textrm{in}~L^{q_2}_\textrm{loc}(\mathbb {R}_{+}^2)\qquad \,\,\, \textrm{for}\ q_2\in [1,\frac{3(\gamma _2+1)}{\gamma _2+3}), \end{aligned} \end{aligned}$$

(2.36)

where $L_{\textrm{loc}}^{p}(\mathbb {R}_{+}^{2})$ represents $L^{p}([0, T] \times K)$ for any $T>0$ and compact set $K \Subset \mathbb {R}_+$.

Now, we are ready to state our main theorem.

Theorem 2.3

(Global existence of finite-energy solutions). Let the pressure function $P(\rho )$ satisfy (1.4)–(1.6), and let the spherically symmetric initial data $(\rho _0,\mathcal {M}_0, \Phi _{0})(\textbf{x})$ be given in (1.13)–(1.14) with $(\rho _{0},m_{0},\Phi _{0})(r)$ satisfying (2.1)–(2.2) and (2.4). Assume that $\gamma _2>\frac{4}{3}$, or $M<M_\textrm{c}$ as $\gamma _2\in (\frac{6}{5},\frac{4}{3}]$. Then there exists a global finite-energy solution $(\rho ,\mathcal {M},\Phi )(t,\textbf{x})$ of (1.1) and (1.13)–(1.14) with spherical symmetry form (1.12) in the sense of Definition 2.2.

Remark 2.3

For the steady gaseous star problem, there is no white dwarf star if the total mass is larger than the so-called Chandrasekhar limit when $\gamma \in (\frac{6}{5},\frac{4}{3}]$; see [7]. Theorem 2.3 requires similar restriction on the total mass when $\gamma _2\in (\frac{6}{5},\frac{4}{3}]$ for non-steady gaseous stars. Moreover, in view of (2.5), for the non-steady white dwarf star, the critical mass is exactly the Chandrasekhar limit in the case that $P(\rho )=\kappa _2\rho ^{\frac{4}{3}}$. It would be interesting to analyze whether the critical mass defined in (2.6)–(2.8) for $\gamma _2\in (\frac{6}{5},\frac{4}{3})$ is optimal.

Remark 2.4

Theorem 2.3 can be extended to the 3-D compressible Euler equations, i.e., (1.1) with $\Phi =0$. Moreover, the inviscid limit from the compressible Navier–Stokes equations to Euler equations with far-field vacuum can also be justified.

Remark 2.5

Theorem 2.3 also holds for the plasmas case, i.e., $k_{g}=-1$ in (1.1), by a similar proof. In this case, the restriction: $M<M_\textrm{c}$ can be removed, and condition $\gamma _{2}>\frac{6}{5}$ can be relaxed to $\gamma _2>1$ if the additional assumption: $\rho _{0}\in L^{\frac{6}{5}}(\mathbb {R}^3)$ is imposed. We omit the proof in this paper for brevity and, instead, refer the reader to [10] for details.

3 Properties of the General Pressure Law and Related Internal Energy

In this section, we present some useful estimates involving the general pressure $P(\rho )$ with (1.4)–(1.6) and the corresponding internal energy $e(\rho )$, which are used in the subsequent development.

Denote $c(\rho ):=\sqrt{P'(\rho )}$ as the speed of sound, and

$$\begin{aligned} k(\rho ):=\int _{0}^{\rho }\frac{\sqrt{P'(y)}}{y}\,\textrm{d} y. \end{aligned}$$

(3.1)

By direct calculation, we can obtain the following asymptotic behaviors of $P(\rho )$, $e(\rho )$, and $k(\rho )$.

Lemma 3.1

Assume that $\rho _{*}$ given in (1.5) is small enough and $\rho ^{*}$ given in (1.6) is large enough such that the following estimates hold:

• (i) When $\rho \in (0,\rho _{*}]$,

  $$\begin{aligned} \left\{ \begin{aligned}&\underline{\kappa }_{1}\rho ^{\gamma _1}\le P(\rho )\le \bar{\kappa }_{1}\rho ^{\gamma _1},\\&\underline{\kappa }_{1}\gamma _1\rho ^{\gamma _1-1}\le P'(\rho )\le \bar{\kappa }_{1}\gamma _1\rho ^{\gamma _1-1},\\&\underline{\kappa }_{1}\gamma _1(\gamma _1-1)\rho ^{\gamma _1-2}\le P''(\rho )\le \bar{\kappa }_{1}\gamma _1(\gamma _1-1)\rho ^{\gamma _1-2}, \end{aligned}\right. \end{aligned}$$

  (3.2)

  and when $\rho \in [\rho ^{*},\infty )$,

  $$\begin{aligned} \left\{ \begin{aligned}&\underline{\kappa }_{2}\rho ^{\gamma _2}\le P(\rho )\le \bar{\kappa }_{2}\rho ^{\gamma _2},\\&\underline{\kappa }_{2}\gamma _2\rho ^{\gamma _2-1}\le P'(\rho )\le \bar{\kappa }_{2}\gamma _2\rho ^{\gamma _2-1},\\&\underline{\kappa }_{2}\gamma _2(\gamma _2-1)\rho ^{\gamma _2-2}\le P''(\rho )\le \bar{\kappa }_{2}\gamma _2(\gamma _2-1)\rho ^{\gamma _2-2}, \end{aligned}\right. \end{aligned}$$

  (3.3)

  where we have denoted $\,\underline{\kappa }_{i}:=(1-\mathfrak {a}_0)\kappa _{i}$ and $\bar{\kappa }_{i}:=(1+\mathfrak {a}_0) \kappa _{i}$ with $\mathfrak {a}_0=\frac{3-\gamma _1}{2(\gamma _1+1)}$ and $i=1,2$.

• (ii) For $e(\rho )$ and $k(\rho )$, there exists $C>0$ depending on $(\gamma _1, \gamma _2, \kappa _1,\kappa _2, \rho _{*}, \rho ^{*})$ such that

  $$\begin{aligned}&C^{-1}\rho ^{\gamma _1-1}\le e(\rho )\le C\rho ^{\gamma _1-1},\,\, C^{-1}\rho ^{\gamma _1-2}\le e'(\rho )\le C\rho ^{\gamma _1-2} \,\,\,\,\, \text { for }\rho \in (0,\rho _{*}], \end{aligned}$$

  (3.4)
  $$\begin{aligned}&C^{-1}\rho ^{\gamma _2-1}\le e(\rho )\le C\rho ^{\gamma _2-1},\,\, C^{-1}\rho ^{\gamma _2-2}\le e'(\rho )\le C\rho ^{\gamma _2-2} \,\,\,\,\, \text { for }\rho \in [\rho ^{*},\infty ), \end{aligned}$$

  (3.5)

  and, for $i=0,1$,

  $$\begin{aligned}&\frac{\rho ^{\theta _{1}-i}}{C}\le k^{(i)}(\rho )\le C\rho ^{\theta _{1}-i},\,\, \frac{\rho ^{\theta _{1}-2}}{C}\le |k''(\rho )|\le C\rho ^{\theta _1-2}\,\,\,\,\, \text {for } \rho \in (0,\rho _{*}],\nonumber \\ \end{aligned}$$

  (3.6)
  $$\begin{aligned}&\frac{\rho ^{\theta _{2}-i}}{C}\le k^{(i)}(\rho )\le C\rho ^{\theta _{2}-i}, \,\, \frac{\rho ^{\theta _{2}-2}}{C}\le |k''(\rho )|\le C\rho ^{\theta _2-2} \,\,\,\,\,\text {for } \rho \in [\rho ^{*},\infty ), \nonumber \\ \end{aligned}$$

  (3.7)

  where $\theta _{1}=\frac{\gamma _1-1}{2}$ and $\theta _2=\frac{\gamma _2-1}{2}$.

It follows from (3.2)–(3.3) that

$$\begin{aligned} \frac{(3\gamma _1-1)(\gamma _1-1)}{\gamma _1+5}P'(\rho )\le \rho P''(\rho )\le \frac{(5+\gamma _1)(\gamma _1-1)}{3\gamma _1-1}P'(\rho )<2P'(\rho ), \end{aligned}$$

(3.8)

when $\rho \in [0,\rho _{*}]\cup [\rho ^{*},\infty )$. For later use, we denote

$$\begin{aligned} \nu :=1-\frac{(3\gamma _{1}-1)(\gamma _{1}-1)}{2(5+\gamma _{1})} <1, \qquad d(\rho ):=2+\frac{\rho k''(\rho )}{k'(\rho )}. \end{aligned}$$

(3.9)

Then it follows from (3.8) that

$$\begin{aligned} 0<\Big \vert \frac{\rho k''(\rho )}{k'(\rho )}\Big \vert =1-\frac{\rho P''(\rho )}{2P'(\rho )}\le \nu <1\qquad \text {for }\rho \in (0,\rho _{*}]\cup [\rho ^{*},\infty ). \end{aligned}$$

(3.10)

Motivated by [64], we have

Lemma 3.2

$0<d(\rho )\le C$ for all $\rho >0$, and

$$\begin{aligned} \big |d(\rho )-(1+\theta _2)\big |\le C\rho ^{-\epsilon }\qquad \text{ for }\,\, \rho \gg 1. \end{aligned}$$

(3.11)

Proof

It follows from (1.4) that $d(\rho )=1+\frac{\rho P''(\rho )}{2P'(\rho )}>0.$ Moreover, by (3.10), it is direct to see that $d(\rho )$ is bounded. Using (1.6), we see that, for $\rho \ge \rho ^{*}$,

$$\begin{aligned} \left\{ \begin{aligned}&P'(\rho )=\gamma _2\kappa _2\rho ^{\gamma _2-1}\big (1+\mathcal {P}_2(\rho )+\rho \mathcal {P}_2'(\rho )\big ),\\&P''(\rho )=\gamma _2(\gamma _2-1)\kappa _2\rho ^{\gamma _2-2}\big (1+\mathcal {P}_2(\rho )+3\rho \mathcal {P}_2'(\rho )+\rho ^2\mathcal {P}_2''(\rho )\big ). \end{aligned}\right. \end{aligned}$$

Then, for $\rho \ge \max \{\rho ^{*},(8C^{*})^{1/\epsilon }\}$,

$$\begin{aligned} \Big |d(\rho )-(1+\theta _2)\Big |=\Big \vert \frac{\rho P''(\rho )}{2P'(\rho )}-\theta _2\Big \vert =\Big \vert \frac{\theta _2\big (2\rho \mathcal {P}_2'(\rho )+\rho ^2\mathcal {P}_2''(\rho )\big )}{1+P_2(\rho ) +3\rho \mathcal {P}_2(\rho )+\rho ^2\mathcal {P}_2''(\rho )}\Big \vert \le C(\theta _2, C^{*})\rho ^{-\epsilon }, \end{aligned}$$

where we have used that $|\mathcal {P}_2^{(j)}(\rho )|\le C^{*}\rho ^{-\epsilon -j}$ for $j=0,1,2,$ in the last inequality. $\square$

Hereafter, for simplicity of notation, we assume that (3.11) holds for $\rho \ge \rho ^{*}$. Furthermore, using (1.6) and $e'(\rho )=\frac{P(\rho )}{\rho ^2}$, we obtain that, for $\rho \ge \rho ^{*}$,

$$\begin{aligned} e(\rho ) =\frac{\kappa _{2}}{\gamma _2-1}\big (\rho ^{\gamma _2-1}-(\rho ^{*})^{\gamma _2-1}\big ) +\kappa _2\int _{\rho ^{*}}^{\rho }s^{\gamma _2-2}\mathcal {P}_{2}(s)\,\textrm{d}s+\int _{0}^{\rho ^{*}}\frac{P(s)}{s^2}\,\textrm{d}s, \end{aligned}$$

(3.12)

which, with $e(0)=0$ and $|\mathcal {P}_{2}(\rho )|\le C^{*}\rho ^{-\epsilon }$, yields that, for any parameter $\beta >0$,

$$\begin{aligned} \lim \limits _{\rho \rightarrow 0}\rho ^{\frac{\gamma _2-1}{5\gamma _2-6}}(\beta +e(\rho ))^{-\frac{1}{5\gamma _2-6}}=0, \quad \,\,\lim \limits _{\rho \rightarrow \infty }\rho ^{\frac{\gamma _2-1}{5\gamma _2-6}}(\beta +e(\rho ))^{-\frac{1}{5\gamma _2-6}} =(\frac{\kappa _2}{\gamma _2-1})^{-\frac{1}{5\gamma _2-6}}. \end{aligned}$$

Then we see that

$$\begin{aligned}&C_{\max }(\beta ):=\sup _{\rho \ge 0}\rho ^{\frac{\gamma _2-1}{5\gamma _2-6}}(\beta +e(\rho ))^{-\frac{1}{5\gamma _2-6}}\in [(\frac{\kappa _2}{\gamma _2-1})^{-\frac{1}{5\gamma _2-6}},\,\infty ), \end{aligned}$$

(3.13)

$$\begin{aligned}&\rho ^{\frac{6(\gamma _2-1)}{5\gamma _2-6}}(\beta \rho +\rho e(\rho ))^{-\frac{1}{5\gamma _2-6}}\le C_{\max }(\beta )\rho \qquad \, \text {for }\rho >0. \end{aligned}$$

(3.14)

With a careful analysis of $C_{\max }(\beta )$, we obtain some estimates of $M_\textrm{c}$ defined in (2.6).

Proposition 3.3

Let $h(\rho )=P(\rho )\rho ^{-1}-(\gamma _2-1)e(\rho )$, and let $\tilde{M}_\textrm{c}$ be the critical mass obtained in [10, (2.8)] for the polytropic gases in (1.2) with $\gamma \in (\frac{6}{5},\frac{4}{3})$. Then $M_\textrm{c}$ defined in (2.6)–(2.8) satisfies that $M_\textrm{c}\le \tilde{M}_\textrm{c}$; in particular, $M_\textrm{c}<\tilde{M}_\textrm{c}$ when $h'(\rho )> 0\ \mathrm{for~all}\ \rho > 0$. For example,

$$\begin{aligned} P_{\delta }(\rho ):=\int _{0}^{\rho ^{\frac{1}{3}}}\frac{s^4}{\sqrt{\delta +s^{2+\epsilon _{0}}}}\,\textrm{d}s\qquad \,\, \text {for}\, \delta >0\,\text { and}\, \epsilon _{0}\in (0,\frac{4}{5}) \end{aligned}$$

(3.15)

satisfies conditions (1.4)–(1.6). If $M_\textrm{c}(\delta )$ is the critical mass defined in (2.6)–(2.8) for pressure $P_{\delta }(\rho )$, then $M_\textrm{c}(\delta )<\tilde{M}_\textrm{c}$ for any $\delta >0$.

Proof

For $\gamma _2\in (\frac{6}{5},\frac{4}{3})$, it follows from (2.6)–(2.8) and (3.13) that, for any fixed $\beta >0$,

$$\begin{aligned} M_\textrm{c}(\beta )&\!=\! \Big (\frac{2}{9(\gamma _2\!-\!1)}(C_{\max }(\beta ))^{\frac{5\gamma _2-6}{3(\gamma _2-1)}}\omega _{3}^{-\frac{4-3\gamma _2}{3(\gamma _2-1)}} \omega _{4}^{-\frac{2}{3}}\Big )^{-\frac{3(\gamma _2-1)}{5\gamma _2-6}} \Big (\frac{E_{0}\!+\!\omega _{3}^{-1}\beta M_\textrm{c}(\beta )}{4-3\gamma _2}\Big )^{-\frac{4-3\gamma _2}{5\gamma _2-6}}\nonumber \\&\le \Big (\frac{2}{9(\gamma _2-1)}\Big (\frac{\kappa _2}{\gamma _2-1}\Big )^{-\frac{1}{3(\gamma _2-1)}}\omega _{3}^{-\frac{4-3\gamma _2}{3(\gamma _2-1)}}\omega _{4}^{-\frac{2}{3}}\Big )^{-\frac{3(\gamma _2-1)}{5\gamma _2-6}} \Big (\frac{E_{0}}{4-3\gamma _2}\Big )^{-\frac{4-3\gamma _2}{5\gamma _2-6}}\nonumber \\&=\tilde{M}_\textrm{c}, \end{aligned}$$

(3.16)

which yields that $M_\textrm{c}\le \tilde{M}_\textrm{c}$.

Let $g(\rho ):=\rho ^{\frac{\gamma _2-1}{5\gamma _2-6}}\big (\beta +e(\rho )\big )^{-\frac{1}{5\gamma _2-6}}$. Then $C_{\max }(\beta )=\max _{\rho \ge 0}g(\rho )$. Since $e'(\rho )=\frac{P(\rho )}{\rho ^2}$, a direct calculation shows that

$$\begin{aligned} g'(\rho )=\frac{1}{5\gamma _2-6}\rho ^{5-4\gamma _2}\big (\beta +e(\rho )\big )^{-\frac{5(\gamma _2-1)}{5\gamma _2-6}} \big ((\gamma _2-1)\beta -h(\rho )\big ). \end{aligned}$$

(3.17)

If $h'(\rho )> 0$ for all $\rho > 0$, then $h(\rho )\ge h(0)=0$. Let $K_{0}:=\max _{\rho> 0}h(\rho )>0$. For $\beta$ small enough such that $0<\beta <\frac{K_{0}}{\gamma _2-1}$, there exists a unique point $\rho _{\beta }>0$ such that $g'(\rho _{\beta })=0$, i.e.,

$$\begin{aligned} h(\rho _{\beta })=(\gamma _2-1)\beta , \end{aligned}$$

(3.18)

and $C_{\max }(\beta )=g(\rho _{\beta }) =(\gamma _2-1)^{\frac{1}{5\gamma _2-6}} \big (P(\rho _{\beta })\rho _{\beta }^{-\gamma _2}\big )^{-\frac{1}{5\gamma _2-6}}.$ Moreover, it follows from (3.18) that $\lim _{\beta \rightarrow 0+}\rho _{\beta }=0$. Thus, we see from (1.5) that

$$\begin{aligned} \lim \limits _{\beta \rightarrow 0+}C_{\max }(\beta ) =(\gamma _2-1)^{\frac{1}{5\gamma _2-6}} \lim \limits _{\rho _{\beta }\rightarrow 0}\big (P(\rho _{\beta })\rho _{\beta }^{-\gamma _2}\big )^{-\frac{1}{5\gamma _2-6}}=\infty , \end{aligned}$$

which, with (3.16$_{1}$), implies that $\lim _{\beta \rightarrow 0+}M_\textrm{c}(\beta )=0$.

On the other hand, it follows directly from (3.13) and (3.16)$_{1}$ that $\lim _{\beta \rightarrow \infty }M_\textrm{c}(\beta )=0$. Therefore, the maximum value $M_\textrm{c}$ of $M_\textrm{c}(\beta )$ must be attained at some point $\beta _{0}\in (0,\infty )$ with $M_\textrm{c}=M_\textrm{c}(\beta _0)< \tilde{M}_\textrm{c}$ due to (3.16)$_{1}$.

For the pressure function $P_{\delta }(\rho )$ in (3.15), it is direct to check that conditions (1.4)–(1.6) are satisfied and $\gamma _2=\frac{4}{3}-\frac{\epsilon _{0}}{6}\in (\frac{6}{5},\frac{4}{3})$. Let $e_{\delta }(\rho )$ be the corresponding internal energy with $e_{\delta }'(\rho )=\frac{P_{\delta }(\rho )}{\rho ^2}$ and $e_{\delta }(0)=0$, and $h_{\delta }(\rho ):=P_{\delta }(\rho )\rho ^{-1}-(\gamma _2-1)e_{\delta }(\rho )$. It follows from a direct calculation that

$$\begin{aligned} h_{\delta }'(\rho )=\rho ^{-2}\big (-\gamma _2P_{\delta }(\rho )+\rho P_{\delta }'(\rho )\big ) =:\rho ^{-2}T_{\delta }(\rho ), \end{aligned}$$

$T_{\delta }(0)=0$, and $T_{\delta }'(\rho )=-(\gamma _2-1)P_{\delta }'(\rho )+\rho P_{\delta }''(\rho )=\frac{2+\epsilon _{0}}{18}\delta \rho ^{\frac{2}{3}}(\delta +\rho ^{\frac{2+\epsilon _{0}}{3}})^{-\frac{3}{2}}>0$ for $\rho >0$. Thus, $T_{\delta }(\rho )>0$ for $\rho >0$, which implies that $h_{\delta }'(\rho )> 0$ for $\rho >0$, so that $M_\textrm{c}(\delta )<\tilde{M}_\textrm{c}$ for any $\delta >0$. $\square$

4 Entropy Analysis: Weak Entropy Pairs

Compared with the polytropic gas case in [10], there is no explicit formula of the entropy kernel for the general pressure law (1.4)–(1.6) so that we have to analyze the entropy equation (2.14) carefully to obtain several desired estimates.

4.1 A special entropy pair

In order to obtain the higher integrability of the velocity, we are going to construct a special entropy pair such that $\rho |u|^3$ can be controlled by the entropy flux. Indeed, such a special entropy $\hat{\eta }(\rho ,u)$ is constructed as

$$\begin{aligned} \hat{\eta }(\rho ,u)={\left\{ \begin{array}{ll} \frac{1}{2}\rho u^2+\rho e(\rho )\quad &{}\text {for }u\ge k(\rho ),\\ -\frac{1}{2}\rho u^2-\rho e(\rho )\,\,\, &{}\text {for }u\le -k(\rho ), \end{array}\right. } \end{aligned}$$

(4.1)

for $k(\rho )=\int _{0}^{\rho }\frac{\sqrt{P'(y)}}{y}\,\textrm{d}y$ and, in the intermediate region $-k(\rho )\le u\le k(\rho )$, $\hat{\eta }(\rho ,u)$ is the unique solution of the Goursat problem of the entropy equation (2.14):

$$\begin{aligned} \left\{ \begin{aligned} \displaystyle&\eta _{\rho \rho }-k'(\rho )^2\eta _{uu}=0,\\ \displaystyle&\eta (\rho ,u)\vert _{u=\pm k(\rho )}=\pm \big (\frac{1}{2}\rho u^2+\rho e(\rho )\big )\textrm{;} \end{aligned} \right. \end{aligned}$$

(4.2)

see Fig. 1. Set

$$\begin{aligned} V_1=\frac{1}{2k'(\rho )}\big (\eta _{\rho }+k'(\rho )\eta _{u}\big ),\qquad V_2=\frac{1}{2k'(\rho )}\big (\eta _{\rho }-k'(\rho )\eta _u\big ). \end{aligned}$$

(4.3)

Then (4.2) can be rewritten as

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \frac{\partial V_1}{\partial \rho }-k'(\rho )\frac{\partial V_1}{\partial u} =-\frac{k''(\rho )}{2k'(\rho )}(V_1+V_2),\\ \displaystyle \frac{\partial V_2}{\partial \rho }+k'(\rho )\frac{\partial V_2}{\partial u}=-\frac{k''(\rho )}{2k'(\rho )}(V_1+V_2). \end{array}\right. } \end{aligned}$$

(4.4)

The corresponding characteristic boundary conditions become

$$\begin{aligned} \left\{ \begin{aligned}&V_1\vert _{u=\pm k(\rho )}=\pm \frac{1}{2k'(\rho )}\big (\frac{1}{2}u^2+e(\rho )+\rho e'(\rho )\big )\pm \frac{1}{2}\rho u,\\&V_2\vert _{u=\pm k(\rho )}=\pm \frac{1}{2k'(\rho )}\big (\frac{1}{2}u^2+e(\rho )+\rho e'(\rho )\big )\mp \frac{1}{2}\rho u. \end{aligned} \right. \end{aligned}$$

(4.5)

Since $\frac{k''(\rho )}{k'(\rho )}$ has the singularity at vacuum $\rho =0$, the Goursat problem (4.4)–(4.5) is singular, which requires a careful analysis.

It follows from (4.4) that there exist two characteristic curves originating from origin O(0, 0) in the $(\rho ,u)$–plane:

$$\begin{aligned} \ell _{+}:=\{(\rho ,u)\, :\, u=k(\rho )\}, \qquad \ell _{-}:=\{(\rho ,u)\,:\ u=-k(\rho )\}. \end{aligned}$$

(4.6)

For any given point $O_1(\rho _0,u_0)$ with $u_0=0$, we can draw two backward characteristic curves $\ell _{0}^{\pm }$ through $O_1(\rho _0,u_0)$; see Fig. 1. Let $O_2(\rho _{0}^{+},u_{0}^{+})$ be the intersection point of $\ell _{0}^{+}$ and $\ell _{+}$, and let $O_3(\rho _{0}^{-},u_{0}^{-})$ be the intersection point of $\ell _{0}^{-}$ and $\ell _{-}$. Let $\Sigma$ be the region surrounded by arc $\widehat{OO_2O_{1}O_{3}}$, and let $\overline{\Sigma }$ be the closure of $\Sigma$.

Fig. 1

The schematic diagram of the characteristic curves of (4.4)

Lemma 4.1

The Goursat problem (4.2) admits a unique solution $\hat{\eta }\in C^2(\mathbb {R}_{+} \times \mathbb {R})$ such that

• (i) $|\hat{\eta }(\rho ,u)|\le C\big (\rho |u|^2+\rho ^{\gamma (\rho )}\big )$ for $(\rho ,u)\in \mathbb {R}_{+}\times \mathbb {R}$, where $\gamma (\rho )=\gamma _1$ if $\rho \in [0,\rho _{*}]$ and $\gamma (\rho )=\gamma _2$ if $\rho \in (\rho _{*},\infty )$.

• (ii) If $\hat{\eta }$ is regarded as a function of $(\rho , u)$,

  $$\begin{aligned} |\hat{\eta }_{\rho }(\rho ,u)|\le C\big (|u|^2+\rho ^{2\theta (\rho )}\big ),\,\,\, |\hat{\eta }_{u}(\rho ,u)|\le C\big (\rho |u|+\rho ^{\theta (\rho )+1}\big ) \quad \,\,\,\,\text {for }(\rho ,u)\in \mathbb {R}_{+}\times \mathbb {R}, \end{aligned}$$

  and, if $\hat{\eta }$ is regarded as a function of $(\rho , m)$,

  $$\begin{aligned} |\hat{\eta }_{\rho }(\rho ,m)|\le C\big (|u|^2+\rho ^{2\theta (\rho )}\big ),\,\,\,|\hat{\eta }_{m}(\rho ,m)|\le C(|u|+\rho ^{\theta (\rho )}) \quad \,\,\,\,\text {for}\, (\rho ,m)\in \mathbb {R}_{+}\times \mathbb {R}, \end{aligned}$$

  where $\theta (\rho ):=\frac{\gamma (\rho )-1}{2}$.

• (iii) If $\hat{\eta }_{m}$ is regarded as a function of $(\rho ,u)$,

  $$\begin{aligned} \qquad |\hat{\eta }_{m\rho }(\rho ,u)|\le C\rho ^{\theta (\rho )-1},\quad |\hat{\eta }_{mu}(\rho ,u)|\le C, \end{aligned}$$

  and, if $\hat{\eta }_{m}$ is regarded as a function of $(\rho ,m)$,

  $$\begin{aligned} |\hat{\eta }_{m\rho }(\rho ,m)|\le C\rho ^{\theta (\rho )-1},\quad |\hat{\eta }_{mm}(\rho ,m)|\le C\rho ^{-1}. \end{aligned}$$

• (iv) If $\hat{q}$ is the corresponding entropy flux determined by (2.12), then $\hat{q}\in C^2(\mathbb {R}_{+}\times \mathbb {R})$ and

  $$\begin{aligned}&\hat{q}(\rho ,u)=\frac{1}{2}\rho |u|^3\pm \rho u\big (e(\rho )+\rho e'(\rho )\big )\qquad{} & {} \text{ for } \pm u\ge k(\rho ),\\ {}&|\hat{q}(\rho ,u)|\le C\rho ^{\gamma (\rho )+\theta (\rho )}\qquad \qquad \qquad \,\,\,\,\,\,{} & {} \text{ for } |u|<k(\rho ),\\ {}&\hat{q}(\rho ,u)\ge \frac{1}{2}\rho |u|^3\qquad \qquad \qquad \qquad \quad \,\,\,{} & {} \text{ for } |u|\ge k(\rho ),\\ {}&|\hat{q}-u\hat{\eta }|\le C\big (\rho ^{\gamma (\rho )}|u|+\rho ^{\gamma (\rho )+\theta (\rho )}\big )\qquad{} & {} \text{ for } (\rho ,u)\in \mathbb {R}_{+}\times \mathbb {R}. \end{aligned}$$

Proof

To prove that (4.2) has a unique $C^2$–solution $\hat{\eta }$ in $\mathbb {R}_{+}\times \mathbb {R}$, it suffices to prove that (4.4)–(4.5) admits a unique $C^1$–solution $(V_1, V_2)$ in $\Sigma$ for any given point $O(\rho _0, u_0)$. We use the Picard iteration and divide the proof into six steps.

1. For any point $A(\rho ,u)\in \Sigma$, there are two backward characteristic curves through $A(\rho , u)$:

$$\begin{aligned} \begin{aligned} \ell _1&:=\big \{(s,\,u^{(1)}(s))\,:\,u^{(1)}(s)=-k(s)+u+k(\rho ),\, 0< s\le \rho \big \},\\ \ell _2&:=\big \{(s,\,u^{(2)}(s))\,:\,u^{(2)}(s)=-k(\rho )+u+k(s),\, 0< s\le \rho \big \}. \end{aligned} \end{aligned}$$

(4.7)

Let $B(\rho _1,u_1)$ be the intersection point of $\ell _{+}$ and $\ell _1$, and let $C(\rho _2,u_2)$ be the intersection point of $\ell _{-}$ and $\ell _2$. It follows from (4.6)–(4.7) that

$$\begin{aligned} u_{1}=k(\rho _1)=\frac{k(\rho )+u}{2},\qquad u_2=-k(\rho _2)=\frac{u-k(\rho )}{2}. \end{aligned}$$

(4.8)

Using (4.4) and integrating $V_1$ and $V_2$ along the characteristic curves $\ell _{1}$ and $\ell _2$ respectively, we have

$$\begin{aligned} V_i(\rho ,u) =V_i(\rho _i,u_i)-\int _{\rho _i}^{\rho }\frac{k''(s)}{2k'(s)}\, \sum \limits _{j=1}^2V_j(s,{u}^{(i)}(s))\,\textrm{d}s\qquad \text{ for } i=1,2. \end{aligned}$$

(4.9)

Denote $V_{i}^{(0)}(\rho ,u):=V_{i}(\rho _{i},u_{i})$. It follows from (4.5) and (4.8) that

$$\begin{aligned} V_i^{(0)}(\rho ,u)=(-1)^{i+1}\Big (\frac{1}{2}\rho _ik(\rho _i)+\frac{1}{4}\frac{k^2(\rho _i)}{k'(\rho _i)}+\frac{e(\rho _i)+\rho _ie'(\rho _i)}{2k'(\rho _i)}\Big )\qquad \text{ for }\, i=1,2. \end{aligned}$$

(4.10)

We define the iterated scheme:

$$\begin{aligned} V_i^{(n+1)}(\rho ,u):=V_i(\rho _i,u_i)-\int _{\rho _i}^{\rho }\frac{k''(s)}{2k'(s)}\, \sum \limits _{j=1}^{2}V_j^{(n)}(s,u^{(i)}(s))\, \textrm{d}s\qquad \text{ for }\, i=1,2. \end{aligned}$$

(4.11)

Then we obtain two sequences $\{V_{i}^{(n)}\}_{n=0}^{\infty }$ for $i=1,2$. We now prove that $\{V_{i}^{(n)}\}_{n=0}^{\infty }$ are uniformly convergent in $\overline{\Sigma }$, which is equivalent to proving that

$$\begin{aligned} V_{i}^{(0)}(\rho ,u)+\sum \limits _{n=1}^{\infty }\big (V_{i}^{(n)}-V_{i}^{(n-1)}\big )(\rho ,u), \quad i=1,2, \end{aligned}$$

(4.12)

are uniformly convergent in $\overline{\Sigma }$.

From Lemma 3.1 and (4.10), we know that $V_{i}^{(0)}, i=1,2$, are continuous in $\overline{\Sigma }$ and there exists a constant $C_1>0$ depending only on $\rho _{*}$ and $\rho ^{*}$ such that

$$\begin{aligned} |V_i^{(0)}(\rho ,u)| \le {\left\{ \begin{array}{ll} C_{1}\rho _{i}^{1+\theta _{1}}\,\,\, &{}\text {for }\rho _{i}\le \rho _{*},\\ C_{1}\rho _{i}^{1+\theta _{2}}\,\,\, &{}\text {for }\rho _{i}\ge \rho _{*}, \end{array}\right. } \qquad i=1,2, \end{aligned}$$

which, with the fact that $\rho _{i}\le \rho$, yields that

$$\begin{aligned} |V_i^{(0)}(\rho ,u)|&\le {\left\{ \begin{array}{ll} C_{1}\rho ^{1+\theta _{1}}\quad &{}\text {for }\rho \le \rho _{*},\\ \tilde{C}_{1}\rho ^{1+M_{1}}\,\,\, &{}\text {for }\rho _{*}\le \rho \le \rho ^{*},\\ \hat{C}_{1}\rho ^{1+\theta _2}\quad &{}\text {for }\rho \ge \rho ^{*}, \end{array}\right. } \qquad i=1,2, \end{aligned}$$

(4.13)

where $\tilde{C}_{1}\ge C_{1}(\rho _{*})^{\theta _{2}-M_{1}}$, $\hat{C}_{1}\ge C_{1}$, and $M_{1}$ are positive constants to be chosen later.

It follows from (3.9) and (3.10) that there exist a constant $\nu <1$ and a constant $C_{0}\gg 1$ depending on $\rho _{*}$ and $\rho ^{*}$ such that

$$\begin{aligned} \Big \vert \frac{k''(\rho )}{k'(\rho )}\Big \vert \le {\left\{ \begin{array}{ll} \nu \rho ^{-1}\quad &{}\text {for}\, 0<\rho \le \rho _{*}\,\text { and}\, \rho \ge \rho ^{*},\\ C_{0}\rho ^{-1}\,\,\,&{}\text {for}\, \rho _{*}< \rho < \rho ^{*}. \end{array}\right. } \end{aligned}$$

(4.14)

For the estimate of $|V_{i}^{(1)}-V_{i}^{(0)}|$, we divide it into six cases:

Case 1. $\rho _{i}\le \rho \le \rho _{*}$: It follows from (4.11) and (4.13)–(4.14) that

$$\begin{aligned} \begin{aligned} \big |(V_i^{(1)}-V_{i}^{(0)})(\rho ,u)\big |&\le \int _{\rho _i}^{\rho } C_{1}\nu \,s^{\theta _1}\,\textrm{d}s \le C_1 \rho ^{1+\theta _1}\,\varpi _1, \end{aligned} \end{aligned}$$

(4.15)

where $\varpi _1:=\frac{\nu }{1+\theta _1}\in (0,1)$.

Case 2. $\rho _{i}\le \rho _{*}\le \rho \le \rho ^{*}$: Then

$$\begin{aligned}&\big |(V_i^{(1)}-V_{i}^{(0)})(\rho ,u)\big | \le \Big (\int _{\rho _{i}}^{\rho _{*}}+\int _{\rho _{*}}^{\rho }\Big ) \Big \vert \frac{k''(s)}{2k'(s)}\Big \vert \, \sum \limits _{j=1}^2\big |V_{j}^{(0)}(s,u^{(i)}(s))\big |\,\textrm{d}s \nonumber \\&\quad \le \int _{\rho _i}^{\rho _{*}}\frac{\nu }{2s}\, (2C_{1}s^{1+\theta _1})\,\textrm{d}s+\int _{\rho _{*}}^{\rho }\frac{C_{0}}{2s}\, (2\tilde{C}_{1}s^{1+M_{1}})\,\textrm{d}s\nonumber \\&\quad \le \tilde{C}_{1}\big (\rho ^{1+M_{1}}-(\rho _{*})^{1+M_{1}}\big )\varpi _{M_1} +C_{1}(\rho _{*})^{1+\theta _1}\varpi _1 \le \tilde{C}_{1}\rho ^{1+M_{1}}\,\varpi _{M_1}, \end{aligned}$$

(4.16)

where $\varpi _{M_1}:=\frac{C_0}{1+M_{1}}$ and, in the last inequality of (4.16), we have chosen

$$\begin{aligned} \tilde{C}_{1}\ge C_{1}(\rho _{*})^{\theta _1-M_{1}}\varpi _1 \varpi _{M_1}^{-1}. \end{aligned}$$

(4.17)

Case 3. $\rho _{*}\le \rho _{i}\le \rho \le \rho ^{*}$: It is direct to see that

$$\begin{aligned} \big |(V_i^{(1)}-V_{i}^{(0)})(\rho ,u)\big | \le \int _{\rho _{i}}^{\rho }\tilde{C}_{1}C_{0}\,s^{M_1}\,\textrm{d}s \le \tilde{C}_{1} \rho ^{1+M_{1}}\,\varpi _{M_1}. \end{aligned}$$

(4.18)

Case 4. $\rho _{i}\le \rho _{*}<\rho ^{*}\le \rho$: Then

$$\begin{aligned}&\big |(V_i^{(1)}-V_{i}^{(0)})(\rho ,u)\big | \le \Big (\int _{\rho _{i}}^{\rho _{*}}+\int _{\rho _{*}}^{\rho ^{*}}+\int _{\rho ^{*}}^{\rho }\Big ) \Big \vert \frac{k''(s)}{2k'(s)}\Big \vert \, \sum \limits _{j=1}^2\big |V_{j}^{(0)}(s,u^{(i)}(s))\big |\,\textrm{d}s \nonumber \\&\quad \le \int _{\rho _{i}}^{\rho _{*}}\frac{\nu }{2s}\, (2C_{1}s^{1+\theta _{1}})\,\textrm{d}s +\int _{\rho _{*}}^{\rho ^{*}}\frac{C_{0}}{2s}\,(2\tilde{C}_{1}s^{1+M_{1}})\,\textrm{d}s +\int _{\rho ^{*}}^{\rho }\frac{\nu }{2s}\, (2\hat{C}_{1}s^{1+\theta _{2}})\,\textrm{d}s\nonumber \\&\quad \le \hat{C}_{1}\big (\rho ^{1+\theta _2}-(\rho ^{*})^{1+\theta _2}\big )\varpi _2 +\tilde{C}_{1}\big ((\rho ^{*})^{1+M_{1}}- (\rho _{*})^{1+M_{1}}\big )\varpi _{M_1} +C_{1}(\rho _{*})^{1+\theta _{1}}\varpi _1\nonumber \\&\quad \le \hat{C}_{1}\rho ^{1+\theta _2}\,\varpi _2, \end{aligned}$$

(4.19)

where $\varpi _2:=\frac{\nu }{1+\theta _2}\in (0,1)$ and, in the last inequality of (4.19), we have used (4.17) and chosen

$$\begin{aligned} \hat{C}_{1}\ge \tilde{C}_{1}(\rho ^{*})^{M_{1}-\theta _{2}}\varpi _{M_1}\varpi _2^{-1}. \end{aligned}$$

(4.20)

Case 5. $\rho _{*}\le \rho _{i}\le \rho ^{*}\le \rho$: It follows similarly that

$$\begin{aligned}&\big |(V_i^{(1)}-V_{i}^{(0)})(\rho ,u)\big | \le \Big (\int _{\rho _{i}}^{\rho ^{*}}+\int _{\rho ^{*}}^{\rho }\Big ) \Big \vert \frac{k''(s)}{2k'(s)}\Big \vert \,\sum \limits _{j=1}^2\big |V_{j}^{(0)}(s,u^{(i)}(s))\big |\,\textrm{d}s \nonumber \\&\quad \le \hat{C}_{1}\big (\rho ^{1+\theta _2}- (\rho ^{*})^{1+\theta _2}\big )\varpi _2+\tilde{C}_{1}(\rho ^{*})^{1+M_{1}}\varpi _{M_1} \le \hat{C}_{1}\rho ^{1+\theta _2}\,\varpi _2, \end{aligned}$$

(4.21)

where we have used (4.20) in the last inequality of (4.21).

Case 6. $\rho ^{*}\le \rho _{i}\le \rho$: We see that

$$\begin{aligned} \big |(V_i^{(1)}-V_{i}^{(0)})(\rho ,u)\big |&\le \int _{\rho _{i}}^{\rho } \hat{C}_{1}\nu \, s^{\theta _{2}}\,\textrm{d}s \le \hat{C}_{1}\rho ^{1+\theta _2}\,\varpi _2. \end{aligned}$$

(4.22)

Combining (4.15)–(4.22), we obtain

$$\begin{aligned} \big |(V_i^{(1)}-V_{i}^{(0)})(\rho ,u)\big |\le {\left\{ \begin{array}{ll} C_{1}\rho ^{1+\theta _{1}}\,\varpi _1\quad &{}\text {for }\rho \le \rho _{*},\\ \tilde{C}_{1}\rho ^{1+M_{1}}\,\varpi _{M_1}\,\,\, &{}\text {for }\rho _{*}\le \rho \le \rho ^{*},\\ \hat{C}_{1}\rho ^{1+\theta _2}\,\varpi _2 \quad &{}\text {for }\rho \ge \rho ^{*}, \end{array}\right. } \quad i=1,2, \end{aligned}$$

(4.23)

if (4.17) and (4.20) hold.

To utilize the induction arguments, we make the induction assumption for $n=k$:

$$\begin{aligned} \big |(V_i^{(k)}-V_{i}^{(k-1)})(\rho ,u)\big | \le {\left\{ \begin{array}{ll} C_{1}\rho ^{1+\theta _{1}}\,\varpi _1^k\quad &{}\text {for }\rho \le \rho _{*},\\ \tilde{C}_{1}\rho ^{1+M_{1}}\,\varpi _{M_1}^k\,\,\,&{}\text {for }\rho _{*}\le \rho \le \rho ^{*},\\ \hat{C}_{1}\rho ^{1+\theta _2}\,\varpi _2^k \quad &{}\text {for }\rho \ge \rho ^{*}, \end{array}\right. } \quad i=1,2. \end{aligned}$$

(4.24)

We now make the estimate for $n=k+1$. To estimate $\big |V_{i}^{(k+1)}(\rho ,u)-V_{i}^{(k)}(\rho ,u)\big |$, it suffices to consider the case: $\rho _{i}\le \rho _{*}<\rho ^{*}\le \rho$, since the other cases can be done by similar arguments as in (4.15)–(4.22). Noting (4.24) and $\rho _{i}\le \rho _{*}<\rho ^{*}\le \rho$, and using similar arguments in (4.19), we have

$$\begin{aligned}&\big |(V_i^{(k+1)}-V_{k}^{(k)})(\rho ,u)\big |\\&\quad \le \Big (\int _{\rho _{i}}^{\rho _{*}}+\int _{\rho _{*}}^{\rho ^{*}}+\int _{\rho ^{*}}^{\rho }\Big ) \Big \vert \frac{k''(s)}{2k'(s)}\Big \vert \, \sum \limits _{j=1}^2\big |V_{j}^{(k)}(s,u^{(i)}(s))-V_{j}^{(k-1)}(s,u^{(i)}(s))\big |\,\textrm{d}s\\&\quad \le \int _{\rho _{i}}^{\rho _{*}} \nu C_{1}\varpi _1^ks^{\theta _{1}}\,\textrm{d}s +\int _{\rho _{*}}^{\rho ^{*}} C_{0}\tilde{C}_{1}\varpi _{M_1}^ks^{M_{1}}\,\textrm{d}s +\int _{\rho ^{*}}^{\rho } \nu \hat{C}_{1}\varpi _2^ks^{\theta _{2}}\,\textrm{d}s\\&\quad \le \hat{C}_{1}\big (\rho ^{1+\theta _2}-(\rho ^{*})^{1+\theta _2}\big )\varpi _2^{k+1} +\tilde{C}_{1}\big ((\rho ^{*})^{1+M_{1}}-(\rho _{*})^{1+M_{1}}\big )\varpi _{M_1}^{k+1} +C_{1}(\rho ^{*})^{1+\theta _{1}}\,\varpi _1^{k+1}\\&\quad \le \hat{C}_{1}\rho ^{1+\theta _2}\varpi _2^{k+1}, \end{aligned}$$

where we have chosen $\tilde{C}_1$ and $\hat{C}_1$ such that

$$\begin{aligned} \tilde{C}_{1}\ge C_{1}(\rho _{*})^{\theta _{1}-M_{1}}\big (\varpi _1\varpi _{M_1}^{-1}\big )^{k+1},\qquad \hat{C}_{1}\ge \tilde{C}_{1}(\rho ^{*})^{M_{1}-\theta _{2}} \big (\varpi _{M_1}\varpi _2^{-1}\big )^{k+1}. \end{aligned}$$

(4.25)

Therefore, under assumption (4.25), we obtain

$$\begin{aligned} \big |(V_i^{(k+1)}-V_{i}^{(k)})(\rho ,u)\big |\le {\left\{ \begin{array}{ll} C_{1}\rho ^{1+\theta _{1}}\,\varpi _1^{k+1}\quad &{}\text {for }\rho \le \rho _{*},\\ \tilde{C}_{1}\rho ^{1+M_{1}}\,\varpi _{M_1}^{k+1}\,\,\, &{}\text {for }\rho _{*}\le \rho \le \rho ^{*},\\ \hat{C}_{1}\rho ^{1+\theta _2}\,\varpi _2^{k+1}\,\,\, &{}\text {for }\rho \ge \rho ^{*}. \end{array}\right. } \quad i=1,2. \end{aligned}$$

Recalling that $\theta _{1}\ge \theta _2$, we can take $C_{0}$ and $M_{1}$ large enough such that

$$\begin{aligned} 0<\varpi _1\le \varpi _{M_1}\le \varpi _2<1. \end{aligned}$$

(4.26)

Combining (4.13) with (4.23)–(4.26), and taking

$$\begin{aligned} \varpi _{M_1}=\varpi _2,\qquad \tilde{C}_{1}=C_{1}(\rho _{*})^{\theta _{2}-M_{1}}, \qquad \hat{C}_{1}=\tilde{C}_{1}(\rho ^{*})^{M_{1}-\theta _2}, \end{aligned}$$

(4.27)

by induction, we conclude that, for any $n\ge 1$,

$$\begin{aligned} \big |(V_{i}^{(n)}-V_{i}^{(n-1)})(\rho ,u)\big |\le {\left\{ \begin{array}{ll} C_{1}\rho ^{1+\theta _{1}}\,\varpi _1^n\quad &{}\text {for }\rho \le \rho _{*},\\ \tilde{C}_{1}\rho ^{1+M_{1}}\,\varpi _{M_1}^n\,\,\, &{}\text {for }\rho _{*}\le \rho \le \rho ^{*},\\ \hat{C}_{1}\rho ^{1+\theta _2}\,\varpi _2^n\,\,\, &{}\text {for }\rho \ge \rho ^{*}, \end{array}\right. } \quad \,i=1,2. \end{aligned}$$

(4.28)

Noting that (4.26) and $\rho \le \rho _0$ for $(\rho ,u)\in \overline{\Sigma }$, we have proved that the two sequences in (4.12), $i=1,2$, are uniformly convergent in $\overline{\Sigma }$ so that sequence $\{(V_1^{(n)}, V_2^{(n)})\}$ is uniformly convergent in $\overline{\Sigma }$. Let $(V_1, V_2)$ be the limit function of sequence $(V_{1}^{(n)},V_{2}^{(n)})$. Noting the continuity and the uniform convergence of $(V_{1}^{(n)},V_{2}^{(n)})$, $(V_{1},V_2)$ is continuous in $\overline{\Sigma }$. Taking the limit: $n\rightarrow \infty$ in (4.11), we conclude that $(V_{1},V_{2})$ is the continuous solution of (4.9).

2. It follows from (4.13) and (4.28) that, for $(\rho , u)\in \{\rho \ge 0,\,|u|\le k(\rho )\}$ and $i=1,2$,

$$\begin{aligned} \big |V_{i}(\rho ,u)\big |&\le \big \vert V_{i}^{(0)} (\rho ,u)\big \vert +\sum \limits _{n=1}^{\infty }\big \vert (V_{i}^{(n)}-V_{i}^{(n-1)})(\rho ,u)\big \vert \nonumber \\&\le {\left\{ \begin{array}{ll} C\rho ^{1+\theta _{1}}\quad &{}\text {for }\rho \le \rho _{*},\\ C\rho ^{1+M_{1}}\,\,\,&{}\text {for }\rho _{*}\le \rho \le \rho ^{*},\\ C\rho ^{1+\theta _2}\quad &{}\text {for }\rho \ge \rho ^{*}. \end{array}\right. } \end{aligned}$$

(4.29)

On the other hand, we see from (4.3) that, for $|u|\le k(\rho )$,

$$\begin{aligned} \begin{aligned} |\hat{\eta }_{\rho }(\rho ,u)|&=\left| k'(\rho )(V_1(\rho ,u)+V_2(\rho ,u))\right| \le C\rho ^{\gamma (\rho )-1},\\ |\hat{\eta }_{u}(\rho ,u)|&= |V_1(\rho ,u)-V_2(\rho ,u)|\le C\rho ^{1+\theta (\rho )}. \end{aligned} \end{aligned}$$

(4.30)

Hence, for $|u|\le k(\rho )$, it holds that

$$\begin{aligned} \begin{aligned}&|\hat{\eta }(\rho ,u)|\le \int _{\bar{\rho }}^{\rho } |\hat{\eta }_{\rho }(s,u)|\,\textrm{d}s+|\hat{\eta }(\bar{\rho },u)|\le C\rho ^{\gamma (\rho )},\\&|\hat{\eta }_{m}(\rho ,m)|=|\rho ^{-1}\hat{\eta }_{u}(\rho ,u)|\le C\rho ^{\theta (\rho )}, \end{aligned} \end{aligned}$$

(4.31)

where $(\bar{\rho },u)$ is the point satisfying $k(\bar{\rho })=|u|$, and we have used the boundary data in (4.2).

3. We now show that $V_1$ and $V_2$ have continuous first-order derivatives with respect to $(\rho , u)$. Using (4.7)–(4.8) and Lemma 3.1, we have

$$\begin{aligned} \frac{\partial u^{(i)}(s)}{\partial u}=1,\qquad \Big \vert \frac{\partial \rho _{i}}{\partial u}\Big \vert = \frac{1}{2 |k'(\rho _{i})|} \le C_{2}\rho _{i}^{1-\theta (\rho _i)}. \end{aligned}$$

(4.32)

Applying $\partial _u$ to (4.11) and using (4.32) yield that, for $i=1,2$,

$$\begin{aligned} \frac{\partial V_{i}^{(n+1)}}{\partial u}(\rho ,u)&=\frac{\partial V_{i}(\rho _{i},u_{i})}{\partial u} + \frac{k''(\rho _{i})}{2k'(\rho _{i})}\, \sum \limits _{j=1}^{2}V_{j}^{(n)}(\rho _{i},u_{i})\,\frac{\partial \rho _{i}}{\partial u}\nonumber \\&\quad -\int _{\rho _{i}}^{\rho }\frac{k''(s)}{2k'(s)}\,\sum \limits _{j=1}^{2}\frac{\partial V_{j}^{(n)}}{\partial u}(s, u^{(i)}(s))\,\textrm{d}s. \end{aligned}$$

(4.33)

It follows from (4.10), (4.32), Lemma 3.1, and a direct calculation that, for $i=1,2$,

$$\begin{aligned} \Big |\frac{\partial V_{i}^{(0)}}{\partial u}(\rho ,u)\Big |&=\Big \vert \frac{\mathrm{{d}} V_{i}(\rho _i,u_i)}{\mathrm{{d}} \rho _i}\frac{\partial \rho _{i}}{\partial u}\Big \vert \le C_{2}\rho _i \le {\left\{ \begin{array}{ll} \bar{C}_2\rho \qquad \,&{}\text {for }\rho \le \rho _{*},\\ \tilde{C}_2\rho ^{1+M_2}\,\,\, &{}\text {for }\rho _{*}\le \rho \le \rho ^{*},\\ \hat{C}_2\rho \qquad \,&{}\text {for }\rho \ge \rho ^{*}, \end{array}\right. } \end{aligned}$$

(4.34)

where $C_{2}$ is chosen to be a common, fixed, and large enough constant in (4.32) and (4.34) depending only on $\rho _{*}$ and $\rho ^{*}$, and $\bar{C}_2 \ge C_2,\tilde{C}_2\ge C_2(\rho _{*})^{-M_2}, \hat{C}_2\ge C_2$, and $M_2$ are some large positive constants to be chosen later.

To estimate $\big |(\frac{\partial V_{i}^{(1)}}{\partial u})(\rho ,u)-(\frac{\partial V_{i}^{(0)}}{\partial u})(\rho ,u)\big |$, we divide it into six cases:

Case 1. $\rho _{i}\le \rho \le \rho _{*}$: It follows from (4.13)–(4.14) and (4.33)–(4.34) that

$$\begin{aligned} \Big |\frac{\partial (V_{i}^{(1)}- V_{i}^{(0)})}{\partial u}(\rho ,u)\Big |&\le \int _{\rho _{i}}^{\rho }\Big \vert \frac{k''(s)}{2k'(s)} \Big \vert \, \sum \limits _{j=1}^2\Big \vert \Big (\frac{\partial V_{j}^{(0)}}{\partial u}\Big )(s, u^{(i)}(s))\Big \vert \,\textrm{d}s\nonumber \\&\qquad + \Big \vert \frac{k''(\rho _{i})}{2k'(\rho _{i})}\Big \vert \, \sum \limits _{j=1}^2\big \vert V_{j}^{(0)}(\rho _{i},u_{i})\big \vert \, \Big \vert \frac{\partial \rho _{i}}{\partial u}\Big \vert \nonumber \\&\le \int _{\rho _{i}}^{\rho }\frac{\nu }{2s}\,(2\bar{C}_2s)\,\textrm{d}s+\frac{\nu }{2\rho _{i}}\,(2C_{1}\rho _{i}^{1+\theta _{1}})\,(C_{2}\rho _{i}^{1-\theta _1}) \nonumber \\&=\bar{C}_{2}\nu \rho -\bar{C}_{2}\nu \rho _{1}+C_{1}C_{2}\nu \rho _{i}\le \bar{C}_{2}\nu \rho , \end{aligned}$$

(4.35)

where, in the last inequality of (4.35), we have chosen

$$\begin{aligned} \bar{C}_{2}\ge C_{1}C_{2}. \end{aligned}$$

(4.36)

Case 2. $\rho _{i}\le \rho _{*}\le \rho \le \rho ^{*}$: Then, similarly, we have

$$\begin{aligned}&\Big |\frac{\partial (V_{i}^{(1)}- V_{i}^{(0)})}{\partial u}(\rho ,u)\Big |\nonumber \\&\qquad \le \Big (\int _{\rho _{i}}^{\rho _{*}}+\int _{\rho _{*}}^{\rho }\Big ) \Big \vert \frac{k''(s)}{2k'(s)}\Big \vert \, \sum \limits _{j=1}^2\Big \vert \Big (\frac{\partial V_{j}^{(0)}}{\partial u}\Big )(s, u^{(i)}(s))\Big \vert \,\textrm{d}s \nonumber \\&\qquad \quad + \Big \vert \frac{k''(\rho _{i})}{2k'(\rho _{i})}\Big \vert \,\sum \limits _{j=1}^2\big \vert V_{j}^{(0)}(\rho _{i},u_{i})\big \vert \, \Big \vert \frac{\partial \rho _{i}}{\partial u}\Big \vert \nonumber \\&\quad \le \tilde{C}_{2}\big (\rho ^{1+M_{2}}- (\rho _{*})^{1+M_2}\big )\varpi _{M_2} +\bar{C}_{2}\nu (\rho _{*}- \rho _{i})+C_{1}C_{2}\nu \rho _{i} \le \tilde{C}_{2}\rho \, \varpi _{M_2}, \end{aligned}$$

(4.37)

where $\varpi _{M_2}:=\frac{C_{0}}{1+M_{2}}$ and, in the last inequality of (4.37), we have used (4.36) and chosen

$$\begin{aligned} \tilde{C}_{2}\ge \bar{C}_2(\rho _{*})^{-M_2}\,\nu \varpi _{M_2}^{-1}. \end{aligned}$$

(4.38)

Case 3. $\rho _{*}\le \rho _{i}\le \rho \le \rho ^{*}$: It follows that

$$\begin{aligned}&\Big |\frac{\partial (V_{i}^{(1)}- V_{i}^{(0)})}{\partial u}(\rho ,u)\Big |\nonumber \\&\quad \le \int _{\rho _{i}}^{\rho }\Big \vert \frac{k''(s)}{2k'(s)}\Big \vert \, \sum \limits _{j=1}^2\Big \vert \Big (\frac{\partial V_{j}^{(0)}}{\partial u}\Big )(s, u^{(i)}(s))\Big \vert \,\textrm{d}s + \Big \vert \frac{k''(\rho _{i})}{2k'(\rho _{i})}\Big \vert \, \sum \limits _{j=1}^2\big \vert V_{j}^{(0)}(\rho _{i},u_{i})\big \vert \, \Big \vert \frac{\partial \rho _{i}}{\partial u}\Big \vert \nonumber \\&\quad \le \tilde{C}_{2}\big (\rho ^{1+M_{2}}- \rho _{i}^{1+M_2}\big )\varpi _{M_2}+\tilde{C_1}C_2C_{0}\rho _{i}^{1+M_{1}-\theta _2} \le \tilde{C}_{2}\rho \,\varpi _{M_2}, \end{aligned}$$

(4.39)

where, in the last inequality of (4.39), we have chosen

$$\begin{aligned} M_{1}\ge M_2+\theta _2,\qquad \tilde{C}_2\ge \tilde{C}_1C_2(1+M_2)(\rho ^{*})^{M_{1}-M_2-\theta _2}. \end{aligned}$$

(4.40)

Case 4. $\rho _{i}\le \rho _{*}<\rho ^{*}\le \rho$: For this case, similarly, we have

$$\begin{aligned}&\Big |\frac{\partial (V_{i}^{(1)}- V_{i}^{(0)})}{\partial u}(\rho ,u)\Big |\nonumber \\ {}&\quad \le \Big (\int _{\rho _{i}}^{\rho _{*}}+\int _{\rho _{*}}^{\rho ^{*}}+\int _{\rho ^{*}}^{\rho }\Big ) \Big \vert \frac{k''(s)}{2k'(s)}\Big \vert \, \sum \limits _{j=1}^2\Big \vert \Big (\frac{\partial V_{j}^{(0)}}{\partial u}\Big )(s, u^{(i)}(s))\Big \vert \,\text {d}s \nonumber \\ {}&\qquad + \Big \vert \frac{k''(\rho _{i})}{2k'(\rho _{i})}\Big \vert \,\sum \limits _{j=1}^2\big \vert V_{j}^{(0)}(\rho _{i},u_{i})\big \vert \, \Big \vert \frac{\partial \rho _{i}}{\partial u}\Big \vert \nonumber \\ {}&\quad \le \hat{C}_2(\rho - \rho ^{*})\nu + \tilde{C}_{2} \big ((\rho ^{*})^{1+M_2}- (\rho _{*})^{1+M_2}\big ) \varpi _{M_2} +\bar{C}_{2}(\rho _{*}- \rho _{i})\nu +C_{1}C_2\rho _{i}\nu \nonumber \\ {}&\quad \le \hat{C}_2\rho \, \nu , \end{aligned}$$

(4.41)

where, in the last inequality of (4.41), we have used (4.36) and (4.38) and chosen

$$\begin{aligned} \hat{C}_2\ge \tilde{C}_2(\rho ^{*})^{M_2}\,\nu ^{-1}\varpi _{M_2}. \end{aligned}$$

(4.42)

Case 5. $\rho _{*}\le \rho _{i}\le \rho ^{*}\le \rho$: Then

$$\begin{aligned}&\Big |\frac{\partial (V_{i}^{(1)}- V_{i}^{(0)})}{\partial u}(\rho ,u)\Big |\nonumber \\ {}&\quad \le \Big (\int _{\rho _{i}}^{\rho ^{*}}+\int _{\rho ^{*}}^{\rho }\Big ) \Big \vert \frac{k''(s)}{2k'(s)}\Big \vert \, \sum \limits _{j=1}^2\Big \vert \Big (\frac{\partial V_{j}^{(0)}}{\partial u}\Big )(s, u^{(i)}(s))\Big \vert \,\text {d}s\nonumber \\ {}&\qquad + \Big \vert \frac{k''(\rho _{i})}{2k'(\rho _{i})}\Big \vert \,\sum \limits _{j=1}^2\big \vert V_{j}^{(0)}(\rho _{i},u_{i})\big \vert \, \Big \vert \frac{\partial \rho _{i}}{\partial u}\Big \vert \nonumber \\ {}&\quad \le \hat{C}_2(\rho - \rho ^{*})\nu +\tilde{C}_{2}\big ((\rho ^{*})^{1+M_2} - \rho _{i}^{1+M_2}\big )\varpi _{M_2}+\tilde{C}_{1}C_{2}C_{0}\rho _{i}^{1+M_1-\theta _2} \le \hat{C}_2 \rho \,\nu , \end{aligned}$$

(4.43)

where we have used (4.40) and (4.42) in the last inequality of (4.43).

Case 6. $\rho ^{*}\le \rho _{i}\le \rho$: It follows similarly that

$$\begin{aligned}&\Big |\frac{\partial (V_{i}^{(1)}- V_{i}^{(0)})}{\partial u}(\rho ,u)\Big |\nonumber \\ {}&\quad \le \int _{\rho _{i}}^{\rho }\Big \vert \frac{k''(s)}{2k'(s)}\Big \vert \, \sum \limits _{j=1}^2\Big \vert \Big (\frac{\partial V_{j}^{(0)}}{\partial u}\Big )(s, u^{(i)}(s))\Big \vert \,\text {d}s + \Big \vert \frac{k''(\rho _{i})}{2k'(\rho _{i})}\Big \vert \, \sum \limits _{j=1}^2\big \vert V_{j}^{(0)}(\rho _{i},u_{i})\big \vert \, \Big \vert \frac{\partial \rho _{i}}{\partial u}\Big \vert \nonumber \\ {}&\quad \le \hat{C}_2\rho \nu -\hat{C}_2 \rho _{i}\nu +\hat{C}_{1}C_{2}\rho _i\nu \le \hat{C}_2\rho \,\nu , \end{aligned}$$

(4.44)

where, in the last inequality of (4.44), we have chosen

$$\begin{aligned} \hat{C}_2\ge \hat{C}_1C_2. \end{aligned}$$

(4.45)

Combining (4.35)–(4.45), we conclude that, for $i=1,2$,

$$\begin{aligned} \Big |\frac{\partial (V_{i}^{(1)}- V_{i}^{(0)})}{\partial u}(\rho ,u)\Big |\le {\left\{ \begin{array}{ll} \bar{C}_{2}\rho \,\nu \quad &{}\text {for }\rho \le \rho _{*},\\ \tilde{C}_{2}\rho ^{1+M_{2}}\,\varpi _{M_2}\,\,\, &{}\text {for }\rho _{*}\le \rho \le \rho _{*},\\ \hat{C}_{2}\rho \,\nu \quad &{}\text {for }\rho \ge \rho _{*}, \end{array}\right. } \end{aligned}$$

(4.46)

provided (4.36), (4.38), (4.40), (4.42), and (4.45) hold.

To use the induction arguments, we make the induction assumption for $n=k$: For $i=1,2$,

$$\begin{aligned} \Big |\frac{\partial (V_{i}^{(k)}- V_{i}^{(k-1)})}{\partial u}(\rho ,u)\Big |\le {\left\{ \begin{array}{ll} \bar{C}_{2}\rho \,\nu ^{k}\quad &{}\text {for }\rho \le \rho _{*},\\ \tilde{C}_{2}\rho ^{1+M_{2}}\,\varpi _{M_2}\,\,\, &{}\text {for }\rho _{*}\le \rho \le \rho _{*},\\ \hat{C}_{2}\rho \,\nu ^{k}\quad &{}\text {for }\rho \ge \rho _{*}. \end{array}\right. } \end{aligned}$$

(4.47)

To estimate $\vert \frac{\partial (V_{i}^{(k+1)}- V_{i}^{(k)})}{\partial u}(\rho ,u)\vert$, it suffices to consider the case: $\rho _{i}\le \rho _{*}<\rho ^{*}\le \rho$ for simplicity of presentation, since the other cases can be estimated by similar arguments in (4.35)–(4.45). In fact, for the case: $\rho _{i}\le \rho _{*}<\rho ^{*}\le \rho$, it follows from (4.28) and (4.47) that

$$\begin{aligned}&\Big \vert \frac{\partial (V_{i}^{(k+1)}- V_{i}^{(k)})}{\partial u}(\rho ,u)\Big \vert \\&\quad \le \Big (\int _{\rho _{i}}^{\rho _{*}}+\int _{\rho _{*}}^{\rho ^{*}}+\int _{\rho ^{*}}^{\rho }\Big ) \Big \vert \frac{k''(s)}{2k'(s)}\Big \vert \,\sum \limits _{j=1}^{2} \Big \vert \frac{\partial (V_{j}^{(k)}- V_{j}^{(k-1)})}{\partial u}(s,u^{(i)}(s))\Big \vert \textrm{d}s\\&\qquad +\Big \vert \frac{k''(\rho _{i})}{2k'(\rho _{i})}\Big \vert \,\Big \vert \frac{\partial \rho _{i}}{\partial u}\Big \vert \, \sum \limits _{j=1}^2\big \vert (V_{j}^{(k)}-V_{j}^{(k-1)})(\rho _{i},u_{i})\big \vert \\&\quad \le \int _{\rho _{i}}^{\rho _{*}} \bar{C}_2\nu ^{k+1}\,\textrm{d}s +\int _{\rho _{*}}^{\rho ^{*}} C_{0}\tilde{C}_2\varpi _{M_2}^ks^{+M_2}\,\textrm{d}s +\int _{\rho ^{*}}^{\rho } \hat{C}_2\nu ^{k+1}\,\textrm{d}s + C_{2}\nu \rho _{i}^{-\theta _1}\, C_{1}\varpi _{M_2}^k\rho _{i}^{1+\theta _1}\\&\quad \le \hat{C}_{2}(\rho - \rho ^{*})\,\nu ^{k+1} +\tilde{C}_{2} \big ((\rho ^{*})^{1+M_2}- (\rho _{*})^{1+M_2}\big )\varpi _{M_2}^{k+1} +\bar{C}_{2}(\rho _{*}- \rho _{i})\nu ^{k+1} +C_{1}C_{2}\rho _i \nu ^{k+1}\\&\quad \le \hat{C}_{2}\rho \, \nu ^{k+1}, \end{aligned}$$

where we have chosen

$$\begin{aligned} M_{1}&\ge M_{2}+\theta _2,\qquad \bar{C}_{2}\ge C_{1}C_{2}, \nonumber \\ \tilde{C}_2&\ge \max \Big \{\tilde{C}_{1}C_{2}(1+M_2)(\rho ^{*})^{M_{1}-M_2-\theta _2}\Big (\frac{1+M_{2}}{1+M_1}\Big )^{k}, \,\bar{C}_2\frac{\big (\nu \varpi _{M_2}^{-1}\big )^{k+1}}{(\rho _{*})^{M_2}}\Big \}, \nonumber \\ \hat{C}_2&\ge \max \Big \{\hat{C}_{1}C_2,\,\tilde{C}_2(\rho ^{*})^{M_2} \big (\nu ^{-1}\varpi _{M_2}\big )^{k+1}\Big \}. \end{aligned}$$

(4.48)

Thus, under assumption (4.48), we conclude that, for $i=1,2$,

$$\begin{aligned} \Big |\frac{\partial (V_{i}^{(k+1)}- V_{i}^{(k)})}{\partial u}(\rho ,u)\Big |\le {\left\{ \begin{array}{ll} \bar{C}_{2}\rho \,\nu ^{k+1}\quad &{}\text {for }\rho \le \rho _{*},\\ \tilde{C}_{2}\rho ^{1+M_{2}}\,\varpi _{M_2}^{k+1}\,\,\, &{}\text {for }\rho _{*}\le \rho \le \rho _{*},\\ \hat{C}_{2}\rho \,\nu ^{k+1}\quad &{}\text {for }\rho \ge \rho _{*}. \end{array}\right. } \end{aligned}$$

Combining (4.27) with (4.46)–(4.48) and taking

$$\begin{aligned}&\varpi _{M_2}=\nu ,\qquad \bar{C}_2=\max \{C_{2},\,C_{1}C_2\},\\&\tilde{C}_{2}\!=\!\max \big \{\bar{C}_2(\rho _{*})^{-M_2},\tilde{C}_1C_2(1\!+\!M_2)(\rho ^{*})^{M_{1}-M_2-\theta _2}\big \}, \quad \! \hat{C}_{2}\!=\!\max \big \{\tilde{C}_2(\rho ^{*})^{M_2},\hat{C}_1C_2\big \}, \end{aligned}$$

we have proved that, for any $n\ge 1$ and $i=1,2$,

$$\begin{aligned} \Big |\frac{\partial (V_{i}^{(n)}- V_{i}^{(n-1)})}{\partial u}(\rho ,u)\Big |\le {\left\{ \begin{array}{ll} \bar{C}_{2}\rho \,\nu ^{n}\quad &{}\text {for }\rho \le \rho _{*},\\ \tilde{C}_{2}\rho ^{1+M_{2}}\,\nu ^{n}\,\,\, &{}\text {for }\rho _{*}\le \rho \le \rho _{*},\\ \hat{C}_{2}\rho \, \nu ^{n}\quad \,&{}\text {for }\rho \ge \rho _{*}. \end{array}\right. } \end{aligned}$$

(4.49)

Noting that $\nu <1$ and $\rho \le \rho _0$ for $(\rho ,u)\in \overline{\Sigma }$, we know that $\big \{\frac{\partial V_{i}^{(n)}}{\partial u}\big \}$ is uniformly convergent in $\overline{\Sigma }$. It is direct to check that the limit function is $\frac{\partial V_{i}}{\partial u}$. Due to the continuity and uniform convergence of $\{\frac{\partial V_{i}^{(n)}}{\partial u}\}$, it is clear that $\frac{\partial V_{i}}{\partial u}$ is continuous in $\overline{\Sigma }$.

On the other hand, it follows from (4.9) that

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\partial V_{1}^{(n)}}{\partial \rho }=k'(\rho )\frac{\partial V_{1}^{(n)}}{\partial u} -\frac{k''(\rho )}{2k'(\rho )}\big (V_{1}^{(n-1)}+V_{2}^{(n-1)}\big ),\\&\frac{\partial V_{2}^{(n)}}{\partial \rho }=-k'(\rho )\frac{\partial V_{2}^{(n)}}{\partial u}-\frac{k''(\rho )}{2k'(\rho )}\big (V_{1}^{(n-1)}+V_{2}^{(n-1)}\big ), \end{aligned}\right. \end{aligned}$$

which, with (4.14), (4.28), and (4.49), yields that, for $k\ge 0$ and $i=1,2$,

$$\begin{aligned}&\Big \vert \frac{\partial (V_{i}^{(n)}- V_{i}^{(n-1)})}{\partial \rho }(\rho ,u)\Big \vert \nonumber \\&\quad \le k'(\rho )\Big \vert \frac{\partial (V_{i}^{(n)}- V_{i}^{(n-1)})}{\partial u}(\rho ,u)\Big \vert +\Big \vert \frac{k''(\rho )}{2k'(\rho )}\Big \vert \, \sum \limits _{j=1}^2\big \vert (V_{j}^{(n)}-V_{j}^{(n-1)})(\rho ,u)\big \vert \nonumber \\&\quad \le {\left\{ \begin{array}{ll} C\rho ^{\theta _1}\,\nu ^{n}\qquad &{}\text {for }\rho \le \rho _{*},\\ C\rho ^{M_1}\,\nu ^{n}\,\,\, &{}\text {for }\rho _{*}\le \rho \le \rho _{*},\\ C\rho ^{\theta _2}\,\nu ^{n}\qquad &{}\text {for }\rho \ge \rho _{*}, \end{array}\right. } \end{aligned}$$

(4.50)

for some large constant $C>0$. Thus, $\frac{\partial V_{i}^{(n)}}{\partial \rho }$ converges uniformly to $\frac{\partial V_{i}}{\partial \rho }$ in $\overline{\Sigma }$. It is clear that $\frac{\partial V_{i}}{\partial \rho }$ is continuous. Therefore, $(V_{1}(\rho ,u), V_{2}(\rho ,u))$ is a $C^1$–solution of the Goursat problem (4.4)–(4.5), which implies that $\hat{\eta }$ is a $C^2$–solution of (4.2).

4. From (4.34) and (4.49), we obtain that, for $i=1,2$,

$$\begin{aligned} \Big \vert \frac{\partial V_{i}}{\partial u}(\rho ,u)\Big \vert \le \Big \vert \frac{\partial V_{i}^{(0)}}{\partial u}(\rho ,u)\Big \vert +\sum \limits _{n=1}^{\infty }\Big \vert \frac{\partial (V_{i}^{(n)}-V_{i}^{(n-1)})}{\partial u}(\rho ,u)\Big \vert \le C\rho \end{aligned}$$

(4.51)

for $\rho \ge 0$ and $|u|\le k(\rho )$. Similarly, using (4.50), we see that, for $i=1,2$,

$$\begin{aligned} \Big \vert \frac{\partial V_{i}}{\partial \rho }\Big \vert \le C\rho ^{\theta (\rho )}\qquad \text {for}\, \rho \ge 0\,\text { and}\, |u|\le k(\rho ). \end{aligned}$$

Therefore, for $|u|\le k(\rho )$, it follows from (4.3) and (4.51) that

$$\begin{aligned} \begin{aligned} |\hat{\eta }_{uu}(\rho ,u)|&=|\partial _{u}V_1(\rho ,u)-\partial _{u}V_2(\rho ,u)|\le C\rho ,\\ |\hat{\eta }_{\rho u}(\rho ,u)|&=|k'(\rho )(\partial _{u}V_1(\rho ,u)+\partial _{u}V_2(\rho ,u))|\le C\rho ^{\theta (\rho )}. \end{aligned} \end{aligned}$$

If $\hat{\eta }_{m}$ is regarded as a function of $(\rho ,u)$, we have

$$\begin{aligned} |\hat{\eta }_{m\rho }(\rho ,u)|\le C\rho ^{\theta (\rho )-1},\quad |\hat{\eta }_{mu}(\rho ,u)|\le C \qquad \,\, \text { for }|u|\le k(\rho ). \end{aligned}$$

If $\hat{\eta }_{m}$ is regarded as a function of $(\rho ,m)$, we see that, for $|u|\le k(\rho )$,

$$\begin{aligned} \begin{aligned} |\hat{\eta }_{m\rho }(\rho ,m)|&=|\hat{\eta }_{m\rho }(\rho ,u)+u\hat{\eta }_{mu}(\rho ,u)|\le C\rho ^{\theta (\rho )-1},\\ |\hat{\eta }_{mm}(\rho ,m)|&=|\rho ^{-1}\hat{\eta }_{mu}(\rho ,u)|\le C\rho ^{-1}. \end{aligned} \end{aligned}$$

5. We now prove the uniqueness of $\hat{\eta }$, which is equivalent to the uniqueness of solutions of (4.4)–(4.5) in the class of $C^1$–solutions satisfying (4.29). Suppose that there exist two $C^{1}$ solutions $(V_{1},V_{2})$ and $(\tilde{V}_{1},\tilde{V}_{2})$ of (4.4)–(4.5) satisfying the uniform estimate (4.29). Then it follows from (4.9) that, for $i=1,2$,

$$\begin{aligned} V_{i}(\rho ,u)-\tilde{V}_{i}(\rho ,u)=-\int _{\rho _{i}}^{\rho }\frac{k''(s)}{2k'(s)}\,\sum \limits _{j=1}^2\big (V_{j}(s,u^{(i)}(s))-\tilde{V}_{j}(s,u^{(i)}(s))\big )\,\textrm{d}s. \end{aligned}$$

(4.52)

Applying the uniform estimates (4.29) and similar arguments as in (4.28)–(4.52) yields

$$\begin{aligned} \max _{(\rho ,u)\in \mathbb {R}_{+}\times \mathbb {R}\atop |u|\le k(\rho )}\big \vert V_{i}(\rho ,u)-\tilde{V}_{i}(\rho ,u)\big \vert \le \left\{ \begin{aligned}&C\rho ^{1+\theta _{1}}\,\varpi _1^n\quad \,\,\,\,\,\,\,\text {for }\rho \le \rho _{*},\\&C\rho ^{1+M_{1}}\,\varpi _{M_1}^n \quad \,\text {for }\rho _{*}\le \rho \le \rho ^{*},\\&C\rho ^{1+\theta _2}\,\varpi _2^n\qquad \,\text {for }\rho \ge \rho ^{*}, \end{aligned} \right. \end{aligned}$$

for any $n\ge 0$, where $C \gg 1$ is independent of n. Taking $n\rightarrow \infty$, we obtain that $V_{i}(\rho ,u)\equiv \tilde{V}_{i}(\rho ,u)$ for $|u|\le k(\rho )$ which, with (4.3) and $\hat{\eta }(0,u)\equiv 0$, yields the uniqueness of $\hat{\eta }$.

6. We now estimate the entropy flux $\hat{q}$. It follows from (2.12) that, for all entropy pairs,

$$\begin{aligned} q_{\rho }=u\eta _{\rho }+\rho k'(\rho )^2\eta _u,\qquad q_{u}=\rho \eta _{\rho }+u\eta _{u}. \end{aligned}$$

(4.53)

Then there exists an entropy flux $\hat{q}(\rho ,u)\in C^2(\mathbb {R}_{+}\times \mathbb {R})$ corresponding to the special entropy $\hat{\eta }$:

$$\begin{aligned} \hat{q}(\rho ,u)=\frac{1}{2}\rho |u|^3\pm \rho u\big (e(\rho )+\rho e'(\rho )\big )\qquad \, \text {for } \pm u\ge k(\rho ). \end{aligned}$$

It follows from (4.30) and (4.53) that $|\hat{q}_{\rho }(\rho ,u)|=|u\hat{\eta }_{\rho }+\rho k'(\rho )^2\hat{\eta }_{u}|\le C\rho ^{\gamma (\rho )+\theta (\rho )-1}$ for $|u|\le k(\rho )$, which implies

$$\begin{aligned} |\hat{q}(\rho ,u)| = \Big |\int _{\bar{\rho }}^{\rho }\hat{q}_{\rho }(s,u) \,\textrm{d}s+\hat{q}(\bar{\rho },u) \Big |\le C\rho ^{\gamma (\rho )+\theta (\rho )}\qquad \, \text {for } |u|\le k(\rho ), \end{aligned}$$

(4.54)

where $(\bar{\rho },u)$ is the point satisfying $k(\bar{\rho })=|u|$.

For $|u|\le k(\rho )$, it follows from (4.31) and (4.54) that $|\hat{q}-u\hat{\eta }|\le |\hat{q}|+|u||\hat{\eta }|\le C\rho ^{\gamma (\rho )+\theta (\rho )}$. In region $\{(\rho ,u):\,|u|\ge k(\rho )\}$, it is direct to check that all the estimates in Lemma 4.1 hold by using (4.1). Therefore, the proof of Lemma 4.1 is now complete. $\square$

4.2 Estimates of the weak entropy pairs

In order to show the compactness of the weak entropy dissipation measures below, we now derive some estimates of the weak entropy pairs. To achieve this, from (2.15)–(2.16), it requires to analyze the entropy kernel and entropy flux kernel, respectively.

The entropy kernel $\chi =\chi (\rho ,u,s)$ is a fundamental solution of the entropy equation (2.14):

$$\begin{aligned} \left\{ \begin{aligned} \displaystyle&\chi _{\rho \rho }-\frac{P'(\rho )}{\rho ^2}\chi _{uu}=0,\\ \displaystyle&\chi \vert _{\rho =0}=0,\quad \chi _{\rho }\vert _{\rho =0}=\delta _{u=s}. \end{aligned} \right. \end{aligned}$$

(4.55)

As pointed out in [11] that equation (4.55) is invariant under the Galilean transformation, which implies that $\chi (\rho ,u,s)=\chi (\rho ,u-s,0)=\chi (\rho ,0,s-u)$. For simplicity, we write it as $\chi (\rho ,u,s)=\chi (\rho ,u-s)$ below when no confusion arises.

The corresponding entropy flux kernel $\sigma (\rho ,u,s)$ satisfies the Cauchy problem for $\sigma -u\chi$:

$$\begin{aligned} \left\{ \begin{aligned} \displaystyle&(\sigma -u\chi )_{\rho \rho }-\frac{P'(\rho )}{\rho ^2}(\sigma -u\chi )_{uu}=\frac{P''(\rho )}{\rho }\chi _{u},\\ \displaystyle&(\sigma -u\chi )\vert _{\rho =0}=0,\quad (\sigma -u\chi )_{\rho }\vert _{\rho =0}=0. \end{aligned} \right. \end{aligned}$$

(4.56)

We recall from [11] that $\sigma -u\chi$ is also Galilean invariant. From (1.4)–(1.6), $P(\rho )$ satisfies all the conditions in [11, 12].

For later use, we introduce the definition of fractional derivatives (cf. [8, 11, 48]). For any real $\alpha >0$, the fractional derivative $\partial _{s}^{\alpha }f$ of a function $f=f(s)$ is

$$\begin{aligned} \partial _{s}^{\alpha }f(s)=\Gamma (-\alpha )f*[s]_{+}^{-\alpha -1}, \end{aligned}$$

where $\Gamma (x)$ is the Gamma function and the convolution should be understood in the sense of distributions. The following formula:

$$\begin{aligned} \partial _{s}^{\alpha }(sg(s))=s\partial _{s}^{\alpha +1}g+(\alpha +1)\partial _{s}^{\alpha }g \end{aligned}$$

holds for fractional derivatives. We now present two useful lemmas for the entropy kernel $\chi (\rho ,u)$ and the entropy flux kernel $\sigma (\rho , u)$ when $\rho$ is bounded.

Lemma 4.2

([11, Theorems 2.1–2.2]). The entropy kernel $\chi (\rho ,u)$ admits the expansion:

$$\begin{aligned} \chi (\rho ,u)=a_{1}(\rho )G_{\lambda _1}(\rho ,u)+a_2(\rho )G_{\lambda _1+1}(\rho ,u)+g_1(\rho ,u)\qquad \text {for}\, \rho \in [0,\infty ), \end{aligned}$$

(4.57)

where $k(\rho )=\int _{0}^{\rho }\frac{\sqrt{P'(y)}}{y}\,\textrm{d}y$ and

$$\begin{aligned} \begin{aligned}&G_{\lambda _1}(\rho ,u)=[k(\rho )^2-u^2]_{+}^{\lambda _1},\qquad \lambda _1=\frac{3-\gamma _1}{2(\gamma _1-1)}>0,\\ {}&a_1(\rho )=M_{\lambda _1}k(\rho )^{-\lambda _1}k'(\rho )^{-\frac{1}{2}},\quad M_{\lambda _1} =\left( \frac{2\lambda _1}{\sqrt{2\lambda _1+1}}\int _{-1}^{1}(1-z^2)^{\lambda _1}\,\text {d}z\right) ^{-1},\\ {}&a_2(\rho )=-\frac{1}{4(\lambda _1+1)}k(\rho )^{-\lambda _1-1}k'(\rho )^{-\frac{1}{2}} \int _{0}^{\rho }k(s)^{\lambda _1}k'(s)^{-\frac{1}{2}}a_{1}''(s)\,\text {d} s. \end{aligned} \end{aligned}$$

(4.58)

Moreover, ${\text {supp}}\chi (\rho ,u)\subset \{(\rho ,u)\,:\, |u|\le k(\rho )\}$, and $\chi (\rho ,u)>0$ in $\{(\rho ,u)\,: \, |u|<k(\rho )\}$. The remainder term $g_1(\rho ,\cdot )$ and its fractional derivative $\partial _{u}^{\lambda _1+1}g_1(\rho ,\cdot )$ are Hölder continuous. Furthermore, for any fixed $\rho _{\max }>0$, there exists $C(\rho _{\max })>0$ depending only on $\rho _\textrm{max}$ such that

$$\begin{aligned} |g_1(\rho ,u-s)|\le C(\rho _{\max })[k(\rho )^2-(u-s)^2]_{+}^{\lambda _1+\alpha _0+1}, \end{aligned}$$

(4.59)

for any $0\le \rho \le \rho _{\max }$ and some $\alpha _0\in (0,1)$. In addition, for any $0\le \rho \le \rho _{\max }$,

$$\begin{aligned} |a_1(\rho )|+\rho ^{1-2\theta _1}|a_1'(\rho )|+\rho ^{2-2\theta _1}|a_1''(\rho )| + |a_2(\rho )|+\rho |a_2'(\rho )|+\rho ^2|a_2''(\rho )|\le C(\rho _{\max }). \end{aligned}$$

(4.60)

Proof

Since (4.57)–(4.59) have been derived in [11, Theorem 2.2], it suffices to prove (4.60). From (3.6), we find that $|a_{1}(\rho )|\le C(\rho _{\max })$ for $0\le \rho \le \rho _{\max }$. For $|a_1'(\rho )|$, a direct calculation shows that

$$\begin{aligned} a_1'(\rho )=-\lambda _1M_{\lambda _1}k(\rho )^{-\lambda _1-1}k'(\rho )^{\frac{1}{2}} -\frac{1}{2}M_{\lambda _1}k(\rho )^{-\lambda _1}k'(\rho )^{-\frac{3}{2}}k''(\rho ). \end{aligned}$$

It follows from (1.5) that $k(\rho )=C_{1}\rho ^{\theta _1}\big (1+O(\rho ^{2\theta _1})\big )$ as $\rho \in [0,\rho _{\max }]$ for some constant $C_1>0$ that may depend on $\kappa _1$ and $\gamma _1$. Then, by direct calculation, we observe that the term involving $\rho ^{-1}$ in $a_1'(\rho )$ vanishes so that $|a_1'(\rho )|\le C(\rho _{\max })\rho ^{2\theta _1-1}$. Similarly, we obtain that $|a_1''(\rho )|\le C(\rho _{\max })\rho ^{2\theta _1-2}$. Finally, using (4.58)$_3$, we can obtain the estimates of $a_2(\rho )$ in (4.60) by a direct calculation. This completes the proof. $\square$

Lemma 4.3

([11, Theorem 2.3]). The entropy flux kernel $\sigma (\rho ,u)$ admits the expansion

$$\begin{aligned} (\sigma -u\chi )(\rho ,u) =-u\big (b_{1}(\rho )G_{\lambda _1}(\rho ,u)+b_2(\rho )G_{\lambda _1+1}(\rho ,u)\big ) +g_2(\rho ,u)\qquad \text {for}\, \rho \in [0,\infty ), \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&b_1(\rho )=M_{\lambda _1}\rho k(\rho )^{-\lambda _1-1}k'(\rho )^{\frac{1}{2}}>0,\\&\begin{aligned} b_2(\rho )&=-\frac{1}{4(\lambda _1+1)}\rho k'(\rho )^{\frac{1}{2}}k(\rho )^{-(\lambda _1+2)} \int _{0}^{\rho }k(s)^{\lambda _1}k'(s)^{-\frac{1}{2}}a_1''(s)\,\textrm{d} s\\&\quad \, -\frac{1}{4(\lambda _1+1)}k(\rho )^{-(\lambda _1+2)}k'(\rho )^{-\frac{1}{2}} \int _{0}^{\rho }k(s)^{\lambda _1+1}k'(s)^{-\frac{1}{2}}b_1''(s)\,\textrm{d}s\\&\quad \, +\frac{1}{4(\lambda _1+1)}k(\rho )^{-(\lambda _1+2)}k'(\rho )^{-\frac{1}{2}} \int _{0}^{\rho } s k(s)^{\lambda _1}k'(s)^{\frac{1}{2}}a_1''(s)\,\textrm{d}s. \end{aligned} \end{aligned} \end{aligned}$$

(4.61)

The remainder term $g_2(\rho ,\cdot )$ and its fractional derivative $\partial _{u}^{\lambda _1+1}g_2(\rho ,\cdot )$ are Hölder continuous. Moreover, for any fixed $\rho _{\max }>0$, there exists $C(\rho _{\max })>0$ depending only on $\rho _{max}$ such that

$$\begin{aligned} |g_2(\rho ,u-s)|\le C(\rho _{\max })[k(\rho )^2-(u-s)^2]_{+}^{\lambda _1+\alpha _0+1}, \end{aligned}$$

for any $0\le \rho \le \rho _{\max }$ and some $\alpha _0\in (0,1)$. Furthermore, similar to the proof of (4.60), for any $0\le \rho \le \rho _{\max }$,

$$\begin{aligned} |b_1(\rho )|+\rho ^{1-2\theta _1}|b_1'(\rho )|+\rho ^{2-2\theta _1}|b_2''(\rho )| +|b_2(\rho )|+\rho |b_2'(\rho )|+\rho ^2|b_2''(\rho )|\le C(\rho _{\max }). \end{aligned}$$

(4.62)

Remark 4.1

In [11, Theorem 2.2], it is proved that $a_2(\rho )$ and $b_{2}(\rho )$ satisfy $|a_2(\rho )|+|b_2(\rho )|\le C\rho k(\rho )^{-2}$ for the pressure law given in [11, (2.1)]. In this paper, we have improved them to be (4.60) and (4.62) under conditions (1.4)–(1.6).

For later use, we recall a useful representation formula for $\chi (\rho ,u)$.

Lemma 4.4

(First representation formula, [64, Lemma 3.4]). Given any $(\rho ,u)$ with $|u|\le k(\rho )$ and $0\le \rho _0<\rho$,

$$\begin{aligned} \begin{aligned} \chi (\rho , u)&=\frac{1}{2(\rho -\rho _0) k^{\prime }(\rho )} \int _{\rho _0}^{\rho } k^{\prime }(s)\, \tilde{d}(s)\chi (s, u+k(\rho )-k(s)) \,\textrm{d}s\\&\quad + \frac{1}{2(\rho -\rho _0) k^{\prime }(\rho )} \int _{\rho _0}^{\rho } k^{\prime }(s)\, \tilde{d}(s)\chi (s, u-k(\rho )+k(s)) \,\textrm{d}s \\&\quad -\frac{1}{2(\rho -\rho _0) k^{\prime }(\rho )} \int _{-(k(\rho )-k(\rho _0))}^{k(\rho )-k(\rho _0)} \chi (\rho _0, u-s) \,\textrm{d}s, \end{aligned} \end{aligned}$$

where $\tilde{d}(\rho ):=2+(\rho -\rho _0)\frac{k''(\rho )}{k'(\rho )}.$

Remark 4.2

In the statement of [64, Lemma 3.4], $\rho _{0}$ is positive. However, the proof of [64, Lemma 3.4] is also valid for $\rho _{0}=0$ without modification; see also [11, (3.38)].

Given any $\psi \in C_{0}^2(\mathbb {R})$, a regular weak entropy pair $(\eta ^{\psi },\,q^{\psi })$ can be given by

$$\begin{aligned} \eta ^{\psi }(\rho ,u)=\int _{\mathbb {R}} \psi (s)\,\chi (\rho ,u,s)\, \textrm{d}s,\qquad q^{\psi }(\rho ,u)=\int _{\mathbb {R}}\psi (s)\,\sigma (\rho ,u,s) \,\textrm{d}s. \end{aligned}$$

(4.63)

It follows from (4.55) that

$$\begin{aligned} \left\{ \begin{aligned} \displaystyle&\eta _{\rho \rho }^{\psi }-k'(\rho )^2\eta _{uu}^{\psi }=0,\\ \displaystyle&\eta ^{\psi }\vert _{\rho =0}=0,\quad \eta _{\rho }^{\psi }\vert _{\rho =0}=\psi (u). \end{aligned} \right. \end{aligned}$$

(4.64)

Using Lemmas 4.2–4.4, we obtain the following lemma for the weak entropy pair $(\eta ^{\psi },q^{\psi })$.

Lemma 4.5

For any weak entropy $(\eta ^{\psi },q^{\psi })$ defined in (4.63), there exists a constant $C_{\psi }>0$ depending only on $\rho ^{*}$ and $\psi$ such that, for all $\rho \in [0,2\rho ^{*}]$,

$$\begin{aligned} |\eta ^{\psi }(\rho ,u)|+ |q^{\psi }(\rho ,u)|\le C_{\psi }\rho . \end{aligned}$$

If $\eta ^{\psi }$ is regarded as a function of $(\rho ,m)$, then

$$\begin{aligned} |\eta _{m}^{\psi }(\rho ,m)|+|\rho \eta _{mm}^{\psi }(\rho ,m)|\le C_{\psi },\qquad |\eta _{\rho }^{\psi }(\rho ,m)|\le C_{\psi }\big (1+\rho ^{\theta _1}\big ). \end{aligned}$$

Moreover, if $\eta _m^{\psi }$ is regarded as a function of $(\rho ,u)$, then

$$\begin{aligned} |\eta _{mu}^{\psi }(\rho ,u)|+|\rho ^{1-\theta _1}\eta _{m\rho }^{\psi }(\rho ,u)|\le C_{\psi }. \end{aligned}$$

Proof

All the estimates can be found in [63, Lemma 3.8] or [64, Lemma 4.13] except the estimate of $\eta _{\rho }^{\psi }(\rho ,m)$. In fact, applying Lemma 4.4 to (4.64) and using $\displaystyle d(\rho ):=2+\frac{\rho k''(\rho )}{k'(\rho )}$, we have

$$\begin{aligned} \eta ^{\psi }(\rho , u)=&\frac{1}{2\rho k^{\prime }(\rho )} \int _{0}^{\rho } k^{\prime }(s)\, d(s)\, \eta ^{\psi }(s, u+k(\rho )-k(s))\, \textrm{d}s \nonumber \\&+\frac{1}{2\rho k^{\prime }(\rho )} \int _{0}^{\rho } k^{\prime }(s)\, d(s)\, \eta ^{\psi }(s, u-k(\rho )+k(s)) \,\textrm{d}s := I_1+I_2. \end{aligned}$$

(4.65)

We regard $\eta ^{\psi }$ as a function of $(\rho ,m)$. Then we have

$$\begin{aligned} \partial _{\rho }\eta ^{\psi }(\rho ,m)=\partial _{\rho }\eta ^{\psi }(\rho ,u)-\frac{u}{\rho }\partial _{u}\eta ^{\psi }(\rho ,u). \end{aligned}$$

(4.66)

Without loss of generality, we assume that ${\text {supp}}\psi \subset [-L,L]$ for some $L>0$. Then a direct calculation shows that $\eta ^{\psi }(\rho ,u)=0$ if $|u|\ge k(\rho )+L$. Noticing $\eta _{u}^{\psi }(\rho ,u)=\rho \eta _{m}^{\psi }(\rho ,m)$, we have

$$\begin{aligned} \Big \vert \frac{u}{\rho }\partial _{u}\eta ^{\psi }(\rho ,u)\Big \vert \le |u|\,|\eta _{m}^{\psi }(\rho ,m)| \le C_{\psi }(1+\rho ^{\theta _1})\qquad \text{ for } 0\le \rho \le 2\rho ^{*}. \end{aligned}$$

(4.67)

Thus, it suffices to calculate $\partial _{\rho }\eta ^{\psi }(\rho ,u)$. It follows from (4.65) that $\partial _{\rho }\eta ^{\psi }(\rho ,u)=\partial _{\rho }I_1+\partial _{\rho }I_2$. A direct calculation shows that

$$\begin{aligned} \partial _{\rho }I_1&=\frac{1}{2}\big (-\rho ^{-2}(k'(\rho ))^{-1}-\rho ^{-1}(k'(\rho ))^{-2}k''(\rho )\big )\nonumber \\ {}&\qquad \times \int _{0}^{\rho } k^{\prime }(s)\, d(s)\, \eta ^{\psi }(s, u+k(\rho )-k(s)) \, \text {d}s\nonumber \\ {}&\quad +\frac{1}{2\rho }\int _{0}^{\rho }k'(s)\,d(s)\,\eta _{u}^{\psi }(s,u+k(\rho )-k(s)) \,\text {d}s + \frac{1}{2\rho }d(\rho )\eta ^{\psi }(\rho ,u). \end{aligned}$$

(4.68)

Using (3.6) and Lemma 3.2, we obtain that

$$\begin{aligned} |\partial _{\rho }I_1| \le C_\psi (1+\rho ^{\theta _1}) \qquad \,\, \text { for }\,\, 0\le \rho \le 2\rho ^{*}. \end{aligned}$$

Similarly, we obtain that $|\partial _{\rho }I_2|\le C_{\psi }(1+\rho ^{\theta _1})$. Thus, we conclude that $|\partial _{\rho }\eta ^{\psi }(\rho ,u)|\le |\partial _{\rho }I_1|+|\partial _{\rho }I_2|\le C_{\psi }(1+\rho ^{\theta _1}),$ which, with (4.66)–(4.67), implies that $|\partial _{\rho }\eta ^{\psi }(\rho ,m)|\le C_\psi (1+\rho ^{\theta _1}).$ $\square$

We notice that all the above estimates for the weak entropy pairs in Lemmas 4.2–4.5 hold when the density is bounded. To establish the $L^p$-compensated compactness framework, we need the entropy pair estimates when the density is large, namely $\rho \ge \rho ^{*}$. From now on in this subsection, we use the representation formula of Lemma 4.4 to estimate $(\eta ^{\psi },q^{\psi })$ in the large density region $\rho \ge \rho ^{*}$.

Lemma 4.6

There exists a positive constant $C>0$ depending only on $\rho ^{*}$ such that

$$\begin{aligned} \Vert \chi (\rho ,\cdot )\Vert _{L_{u}^{\infty }}\le C\rho \qquad \text {for}\, \rho \ge \rho ^{*}. \end{aligned}$$

Proof

For $\rho \ge \rho ^{*}$, $\chi (\rho ,u)$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\displaystyle \chi _{\rho \rho }-k'(\rho )^2\chi _{uu}=0,\\&\displaystyle \chi (\rho ,u)\vert _{\rho =\rho ^{*}}=\chi (\rho ^{*},u),\quad \chi _{\rho }(\rho ,u)\vert _{\rho =\rho ^{*}}=\chi _{\rho }(\rho ^{*},u). \end{aligned}\right. \end{aligned}$$

where $\chi (\rho ^{*},u)$ and $\chi _{\rho }(\rho ^{*},u)$ are given in Lemma 4.2. Then, applying Lemma 4.4, we obtain that, for $\rho > \rho ^{*}$,

$$\begin{aligned} \Vert k'(\rho )\chi (\rho ,\cdot )\Vert _{L_{u}^{\infty }}&\le \frac{1}{\rho -\rho ^{*}}\int _{\rho ^{*}}^{\rho }d_{*}(s)\Vert k'(s)\chi (s,\cdot )\Vert _{L_{u}^{\infty }}\,\textrm{d}s\\&\quad +\frac{1}{2(\rho -\rho ^{*})}\int _{-(k(\rho )-k(\rho ^{*}))}^{k(\rho )-k(\rho ^{*})}\Vert \chi (\rho ^{*},\cdot )\Vert _{L_{u}^{\infty }}\,\textrm{d}s\\&\le \frac{1}{\rho -\rho ^{*}}\int _{\rho ^{*}}^{\rho }d_{*}(s)\Vert k'(s)\chi (s,\cdot )\Vert _{L_{u}^{\infty }}\,\textrm{d}s+C\rho ^{\theta _2-1}, \end{aligned}$$

where $d_{*}(\rho ):=2+(\rho -\rho ^{*})\frac{k''(\rho )}{k'(\rho )}$. By a similar proof to that for Lemma A.3, we have

$$\begin{aligned} \Vert k'(\rho )\chi (\rho ,\cdot )\Vert _{L_{u}^{\infty }}\le C\rho ^{\theta _{2}}\qquad \text {for}\, \rho \ge 2\rho ^{*}, \end{aligned}$$

which, with (3.7), yields that $\Vert \chi (\rho ,\cdot )\Vert _{L_{u}^{\infty }}\le C\rho$ for $\rho \ge 2\rho ^{*}$. For $\rho ^{*}\le \rho \le 2\rho ^{*}$, it follows from Lemma 4.2 that $\Vert \chi (\rho ,\cdot )\Vert _{L_{u}^{\infty }}\le C\le C\rho$. $\square$

Lemma 4.7

Let $\rho \ge \rho ^{*}$ and $\psi \in C_{0}^2(\mathbb {R})$. Then, in the $(\rho ,u)$–coordinates,

$$\begin{aligned} |\eta ^{\psi }(\rho ,u)|+|\eta _{u}^{\psi }(\rho ,u)|+|\eta _{uu}^{\psi }(\rho ,u)|\le C_{\psi }\rho , \quad |\eta _{\rho }^{\psi }(\rho ,u)|+\rho ^{1-\theta _2}|\eta _{\rho \rho }^{\psi }(\rho ,u)|\le C_{\psi }\rho ^{\theta _2}. \end{aligned}$$

In the $(\rho ,m)$–coordinates, $\,|\eta ^{\psi }_{\rho }(\rho ,m)|+ \rho ^{\theta _2}|\eta _{m}^{\psi }(\rho ,m)|+\rho ^{1+\theta _2}|\eta _{mm}^{\psi }(\rho ,m)|\le C_{\psi }\rho ^{\theta _2}$.

If $\eta ^{\psi }_{m}(\rho ,m)$ is regarded as a function of $(\rho , u)$, then $\,|\eta _{mu}^{\psi }|+\rho ^{1-\theta _2}|\eta _{m\rho }^{\psi }|\le C_{\psi }$.

All the above constants $C_{\psi }>0$ depend only on $\Vert \psi \Vert _{C^2}$ and ${\text {supp}}\psi$.

Proof

We divide the proof into five steps.

1. Using (4.63) and Lemma 4.6, we obtain that, for $\rho \ge \rho ^{*}$,

$$\begin{aligned} |{\eta }^{\psi }(\rho ,u)|+|{\eta }_{u}^{\psi }(\rho ,u)|+|{\eta }_{uu}^{\psi }(\rho ,u)| \le \Vert {\chi }(\rho ,\cdot )\Vert _{L^{\infty }(\mathbb {R})}\Vert (\psi ,\psi ',\psi '')\Vert _{L^1(\mathbb {R})} \le C_{\psi }\rho . \end{aligned}$$

(4.69)

2. For the estimate of $\eta _{\rho }^{\psi }(\rho ,u)$, the proof is similar to Lemma 4.5. Indeed, $\eta ^{\psi }$ satisfies

$$\begin{aligned} \left\{ \begin{aligned} \displaystyle&\eta _{\rho \rho }^{\psi }-k'(\rho )^2\eta _{uu}^{\psi }=0,\\ \displaystyle&\eta ^{\psi }(\rho ,u)\vert _{\rho =\rho ^{*}}=\eta ^{\psi }(\rho ^{*},u),\quad \eta _{\rho }^{\psi }(\rho ,u)\vert _{\rho =\rho ^{*}}=\eta _{\rho }^{\psi }(\rho ^{*},u). \end{aligned}\right. \end{aligned}$$

(4.70)

It follows from (4.70) and Lemma 4.4 that

$$\begin{aligned} \eta ^{\psi }(\rho , u)&=\frac{1}{2(\rho -\rho ^{*}) k^{\prime }(\rho ) }\int _{\rho ^{*}}^{\rho } d_{*}(s)k^{\prime }(s)\eta ^{\psi }(s, u+k(\rho )-k(s))\,\textrm{d}s\nonumber \\&\quad -\frac{1}{2(\rho -\rho ^{*}) k^{\prime }(\rho ) }\int _{\rho ^{*}}^{\rho } d_{*}(s)k^{\prime }(s)\eta ^{\psi }(s, u-k(\rho )+k(s))\,\textrm{d}s\nonumber \\&\quad -\frac{1}{2(\rho -\rho ^{*}) k^{\prime }\left( \rho \right) }\int _{-(k(\rho )-k(\rho ^{*}))}^{k(\rho )-k(\rho ^{*})}\eta ^{\psi }(\rho ^{*},u-s)\,\textrm{d}s, \end{aligned}$$

(4.71)

where $d_{*}(\rho )=2+(\rho -\rho ^{*})\frac{k''(\rho )}{k'(\rho )}$ and $0<d_{*}(\rho )\le 3$ for $\rho \ge \rho ^{*}$ from (3.10). Then, following the similar arguments as in the proof of Lemma 4.5, we can obtain that $|\eta _{\rho }^{\psi }(\rho ,u)|\le C_{\psi }\rho ^{\theta _2}$ for $\rho \ge 2\rho ^{*}$. Moreover, from Lemma 4.5, $|\eta _{\rho }^{\psi }(\rho ,u)|\le C_{\psi }\le C_{\psi }\rho ^{\theta _2}$ for $\rho \in [\rho ^{*},2\rho ^{*}]$ so that

$$\begin{aligned} |\eta _{\rho }^{\psi }(\rho ,u)|\le C_{\psi }\rho ^{\theta _2}\qquad \text {for }\rho \ge \rho ^{*}. \end{aligned}$$

(4.72)

3. For $\eta _{\rho \rho }^{\psi }(\rho ,u)$, it follows from (4.69)–(4.70) that

$$\begin{aligned} |\eta _{\rho \rho }^{\psi }(\rho ,u)|\le |k'(\rho )^2|\,|\eta _{uu}^{\psi }(\rho ,u)| \le C_{\psi }\rho ^{2\theta _2-1}=C_{\psi }\rho ^{\gamma _2-2}\qquad \text { for }\rho \ge \rho ^{*}. \end{aligned}$$

4. In the $(\rho ,m)$–coordinates, it is clear that

$$\begin{aligned}&\eta _{m}^{\psi }(\rho ,m)=\rho ^{-1}\eta _{u}^{\psi }(\rho ,u),\quad \eta _{mm}^{\psi }(\rho ,m)=\rho ^{-2}\eta _{uu}^{\psi }(\rho ,u),\quad \\&\eta _{\rho }^{\psi }(\rho ,m)=\eta _{\rho }^{\psi }(\rho ,u)-\frac{m}{\rho ^2}\eta _{u}^{\psi }(\rho ,u). \end{aligned}$$

On the other hand, if $\eta _{m}^{\psi }$ is regarded as a function $(\rho ,u)$, it is direct to obtain

$$\begin{aligned} \eta _{mu}^{\psi }(\rho , u)=\partial _{u}\big (\rho ^{-1}\eta _{u}^{\psi }(\rho ,u)\big )=\rho ^{-1}\eta _{uu}^{\psi }(\rho ,u). \end{aligned}$$

Thus, using (4.69) and (4.72),

$$\begin{aligned} |\eta _{m}^{\psi }(\rho ,m)|+ |\eta _{mu}^{\psi }(\rho ,u)|+\rho |\eta _{mm}^{\psi }(\rho ,m)|&\le C_{\psi }\rho ^{-1},\\ |\eta _{\rho }^{\psi }(\rho ,m)|\le |\eta _{\rho }^{\psi }(\rho ,u)|+\frac{|m|}{\rho ^2}\big |\eta _{u}^{\psi }(\rho ,u)\big |&\le C_{\psi }\rho ^{\theta _2}+C_{\psi }(L+k(\rho ))\le C_{\psi }\rho ^{\theta _2}, \end{aligned}$$

where we have used that ${\text {supp}}\psi \subset [-L,L]$ and $\eta _{u}^{\psi }(\rho ,u)=\rho \eta _{m}^{\psi }(\rho ,m)$.

5. For the estimates of $\eta _{m\rho }^{\psi }(\rho ,u)=\partial _{\rho }\eta _{m}^{\psi }(\rho ,u)$, it follows from (4.71) that

$$\begin{aligned} \eta _m^{\psi }(\rho ,m)=\frac{1}{\rho }\eta _u^{\psi }(\rho ,u)&=\frac{1}{2\rho (\rho -\rho ^{*})k'(\rho )}\int _{\rho ^{*}}^{\rho }d_{*}(s)\,k'(s)\,\eta _{u}^{\psi }(s,u+k(\rho )-k(s))\,\textrm{d}s\nonumber \\&\quad +\frac{1}{2\rho (\rho -\rho ^{*})k'(\rho )}\int _{\rho ^{*}}^{\rho }d_{*}(s)\,k'(s)\,\eta _{u}^{\psi }(s,u-k(\rho )+k(s))\,\textrm{d}s\nonumber \\&\quad -\frac{1}{2\rho (\rho -\rho ^{*})k'(\rho )}\int _{-(k(\rho )-k(\rho ^{*}))}^{k(\rho )-k(\rho ^{*})}\eta _{u}^{\psi }(\rho ^{*},u-s)\,\textrm{d}s\nonumber \\&:=J_1+J_2+J_3, \end{aligned}$$

(4.73)

and $\partial _{\rho }\eta _{m}^{\psi }(\rho ,u)=\partial _{\rho }J_1+\partial _{\rho }J_2+\partial _{\rho }J_3$.

A direct calculation shows that

$$\begin{aligned} \partial _{\rho }J_1&=\partial _{\rho }\Big (\frac{1}{2\rho (\rho -\rho ^{*})k'(\rho )}\Big ) \int _{\rho ^{*}}^{\rho }d_{*}(s)\,k'(s)\,\eta _{u}^{\psi }(s,u+k(\rho )-k(s))\,\textrm{d}s\nonumber \\&\quad +\frac{1}{2\rho (\rho -\rho ^{*})}\int _{\rho ^{*}}^{\rho }d_{*}(s)\,k'(s)\,\eta _{uu}^{\psi }(s,u+k(\rho )-k(s))\,\textrm{d}s\nonumber \\&\quad +\frac{1}{2\rho (\rho -\rho ^{*})}d_{*}(\rho )\,\eta _{u}^{\psi }(\rho ,u)\nonumber \\&:=J_{1,1}+J_{1,2}+J_{1,3}, \end{aligned}$$

(4.74)

$$\begin{aligned} \partial _{\rho }J_2&=\partial _{\rho }\Big (\frac{1}{2\rho (\rho -\rho ^{*})k'(\rho )}\Big )\int _{\rho ^{*}}^{\rho }d_{*}(s)\,k'(s)\,\eta _{u}^{\psi }(s,u-k(\rho )+k(s))\,\textrm{d}s \nonumber \\&\quad -\frac{1}{2\rho (\rho -\rho ^{*})}\int _{\rho ^{*}}^{\rho }d_{*}(s)\,k'(s)\,\eta _{uu}^{\psi }(s,u-k(\rho )+k(s))\,\textrm{d}s\nonumber \\&\quad +\frac{1}{2\rho (\rho -\rho ^{*})}d_{*}(\rho )\eta _{u}^{\psi }(\rho ,u)\nonumber \\&:=J_{2,1}+J_{2,2}+J_{2,3}. \end{aligned}$$

(4.75)

Clearly, we have

$$\begin{aligned} \Big \vert \partial _{\rho }\Big (\frac{1}{2\rho (\rho -\rho ^{*})k'(\rho )}\Big )\Big \vert= & {} \frac{1}{2}\Big \vert \frac{\rho -\rho ^*}{2\rho ^{2}\,(\rho -\rho ^{*})^2\,k'(\rho )} +\frac{k''(\rho )}{\rho \,(\rho -\rho ^{*})\,k'(\rho )^2}\Big \vert \\\le & {} \frac{C}{(\rho -\rho ^{*})^2\rho ^{\theta _2}}, \end{aligned}$$

which, with (4.69) and $0<d_{*}(\rho )\le 3$ for $\rho \ge \rho ^{*}$, yields

$$\begin{aligned} |J_{1,1}+J_{2,1}|&=\Big \vert \partial _{\rho }\Big (\frac{1}{2\rho \,(\rho -\rho ^{*})\,k'(\rho )}\Big ) \int _{\rho ^{*}}^{\rho }k'(s)\,d_{*}(s)\,\eta _{u}^{\psi }(s,u+k(\rho )-k(s))\,\textrm{d}s\Big \vert \nonumber \\&\quad +\Big \vert \partial _{\rho }\Big (\frac{1}{2\rho \,(\rho -\rho ^{*})\,k'(\rho )}\Big ) \int _{\rho ^{*}}^{\rho }k'(s)\,d_{*}(s)\,\eta _{u}^{\psi }(s,u-k(\rho )+k(s))\,\textrm{d}s\Big \vert \nonumber \\&\le \frac{C_{\psi }}{(\rho -\rho ^{*})^2\,\rho ^{\theta _2}}\int _{\rho ^{*}}^{\rho }s^{\theta _2}\,\textrm{d}s\le \frac{C_{\psi }}{\rho -\rho ^{*}}. \end{aligned}$$

(4.76)

It follows from (4.69) and $0<d_{*}(\rho )\le 3$ for $\rho \ge \rho ^{*}$ that

$$\begin{aligned} \begin{aligned} |J_{1,2}|+|J_{2,2}|&=\Big \vert \frac{1}{2\rho (\rho -\rho ^{*})}\int _{\rho ^{*}}^{\rho }d_{*}(s)\,k'(s)\,\eta _{uu}^{\psi }(s,u+k(\rho )-k(s))\,\textrm{d}s\Big \vert \\&\quad +\Big \vert \frac{1}{2\rho (\rho -\rho ^{*})}\int _{\rho ^{*}}^{\rho }d_{*}(s)\,k'(s)\,\eta _{uu}^{\psi }(s,u-k(\rho )+k(s))\,\textrm{d}s\Big \vert \\&\le \frac{C_{\psi }}{\rho (\rho -\rho ^{*})}\int _{\rho ^{*}}^{\rho }s^{\theta _2}\,\textrm{d}s \le \frac{C_{\psi }}{\rho (\rho -\rho ^{*})}\rho ^{\theta _2}(\rho -\rho ^{*})\le C_{\psi }\rho ^{\theta _2-1}. \end{aligned} \end{aligned}$$

(4.77)

For $J_{1,3}+J_{2,3}$, it is direct to see that

$$\begin{aligned} |J_{1,3}+J_{2,3}|\le \Big \vert \frac{1}{\rho (\rho -\rho ^{*})}\,d_{*}(\rho )\,\eta _{u}^{\psi }(\rho ,u)\Big \vert \le \frac{C_{\psi }}{\rho -\rho ^{*}}. \end{aligned}$$

(4.78)

For $\partial _{\rho }J_3$, we notice that

$$\begin{aligned} \begin{aligned} \partial _{\rho }J_3&=-\partial _{\rho }\Big (\frac{1}{2\rho \,(\rho -\rho ^{*})\,k'(\rho )}\Big ) \int _{-(k(\rho )-k(\rho ^{*}))}^{k(\rho )-k(\rho ^{*})}\eta _{u}^{\psi }(\rho ^{*},u-s)\,\textrm{d}s\\&\quad -\frac{1}{2\rho \,(\rho -\rho ^{*})}\,\big (\eta _{u}^{\psi }(\rho ^{*},u-k(\rho )+k(\rho ^{*})) +\eta _{u}^{\psi }(\rho ^{*},u+k(\rho )-k(\rho ^{*}))\big ), \end{aligned} \end{aligned}$$

which, with $0<\theta _2\le 1$, yields

$$\begin{aligned} |\partial _{\rho }J_3| \le \frac{C_{\psi }\rho ^{*}}{(\rho -\rho ^{*})^2}\,|k(\rho )-k(\rho ^{*})|+\frac{C_{\psi }\rho ^{*}}{\rho \,(\rho -\rho ^{*})} \le \frac{C_{\psi }}{\rho -\rho ^{*}}(1+\rho ^{\theta _2-1})\le \frac{C_{\psi }}{\rho -\rho ^{*}}. \end{aligned}$$

(4.79)

Combining (4.74)–(4.79) with (4.73) yields that $|\eta _{m\rho }^{\psi }(\rho ,u)|\le C_{\psi }\rho ^{\theta _2-1}$ for $\rho \ge 2\rho ^{*}$. For $\rho ^{*}\le \rho \le 2\rho ^{*}$, it follows from Lemma 4.5 that $|\eta _{m\rho }^{\psi }(\rho ,u)|\le C_{\psi }\rho ^{\theta _1-1}\le C_{\psi }$ for $\rho ^{*}\le \rho \le 2\rho ^{*}$. Thus, we conclude that $|\eta _{m\rho }^{\psi }(\rho ,u)|\le C_{\psi }\rho ^{\theta _2-1}$ for $\rho \ge \rho ^{*}$. $\square$

We now estimate $q^{\psi }$ for $\rho \ge \rho ^{*}$. It follows from (4.56) that $h:=\sigma -u\chi$ satisfies

$$\begin{aligned} \left\{ \begin{aligned} \displaystyle&h_{\rho \rho }-k'(\rho )^2h_{uu}=\frac{P''(\rho )}{\rho }\chi _{u},\\ \displaystyle&h(\rho ^{*},u)=(\sigma -u\chi )(\rho ^{*},u),\quad h_{\rho }(\rho ^{*},u)=(\sigma -u\chi )_{\rho }(\rho ^{*},u), \end{aligned}\right. \end{aligned}$$

where $(\sigma -u\chi )(\rho ^{*},u)$ and $(\sigma -u\chi )_{\rho }(\rho ^{*},u)$ are given by Lemma 4.3. Similar to Lemma 4.4, we have the following representation formula for h.

Lemma 4.8

(Second representation formula [64, Lemmas 3.4 and 3.9]). For any $(\rho ,u)$ with $|u|\le k(\rho )$ and $\rho >\rho ^{*}$,

$$\begin{aligned} h(\rho , u)&=\frac{1}{2(\rho -\rho ^{*}) k^{\prime }(\rho )} \int _{\rho ^{*}}^{\rho } k^{\prime }(s) d_{*}(s) \big (h(s, u+k(\rho )-k(s))+ h(s, u-k(\rho )+k(s))\big ) \,\textrm{d}s\nonumber \\&\quad +\frac{1}{2(\rho -\rho ^{*})k'(\rho )}\int _{\rho ^{*}}^{\rho }(s-\rho ^{*})\frac{P''(s)}{s} \chi (s,u+k(\rho )-k(s))\,\textrm{d}s\nonumber \\&\quad -\frac{1}{2(\rho -\rho ^{*})k'(\rho )}\int _{\rho ^{*}}^{\rho }(s-\rho ^{*})\frac{P''(s)}{s}\chi (s,u-k(\rho )+k(s))\,\textrm{d}s\nonumber \\&\quad -\frac{1}{2(\rho -\rho ^{*}) k^{\prime }(\rho )} \int _{-(k(\rho )-k(\rho ^{*}))}^{k(\rho )-k(\rho ^{*})} h(\rho ^{*}, u-s) \,\textrm{d}s, \end{aligned}$$

(4.80)

where $d_{*}(\rho )=2+(\rho -\rho ^{*})\frac{k''(\rho )}{k'(\rho )}$.

Lemma 4.9

There exists a constant $C>0$ depending only on $\rho ^{*}$ such that

$$\begin{aligned} \Vert (\sigma -u\chi )(\rho ,u)\Vert _{L_{u}^{\infty }}\le C\rho ^{1+\theta _2} \qquad \,\, \text {for }\rho \ge \rho ^{*}. \end{aligned}$$

Proof

It follows from (3.3), (4.80), and Lemma 4.6 that

$$\begin{aligned} \Vert k'(\rho )h(\rho ,\cdot )\Vert _{L_{u}^{\infty }}&\le \frac{1}{\rho -\rho ^{*}}\int _{\rho ^{*}}^{\rho }d_{*}(s)\Vert k'(s)h(s,\cdot )\Vert _{L_{u}^{\infty }}\,\textrm{d}s\\&\quad +\frac{C}{\rho -\rho ^{*}}\int _{\rho ^{*}}^{\rho }s^{\gamma _2-1}\,\textrm{d}s+C\rho ^{\theta _2-1}\\&\le \frac{1}{\rho -\rho ^{*}}\int _{\rho ^{*}}^{\rho }d_{*}(s)\Vert k'(s)h(s,\cdot )\Vert _{L_{u}^{\infty }}\,\textrm{d}s+C\rho ^{2\theta _2}, \end{aligned}$$

which, with (3.7) and a similar proof to that for Lemma A.3, yields

$$\begin{aligned} \Vert k'(\rho )h(\rho ,\cdot )\Vert _{L_{u}^{\infty }}\le C\rho ^{2\theta _2} \,\Longrightarrow \,\Vert h(\rho ,\cdot )\Vert _{L_{u}^{\infty }}\le C\rho ^{1+\theta _2}\qquad \text {for }\rho \ge 2\rho ^{*}. \end{aligned}$$

For $\rho ^{*}\le \rho \le 2\rho ^{*}$, it follow from Lemma 4.3 that $\Vert h(\rho ,\cdot )\Vert _{L_{u}^{\infty }}\le C\le C\rho ^{1+\theta _2}$. $\square$

Lemma 4.10

For $\rho \ge \rho ^{*}$ and $\psi \in C_{0}^2(\mathbb {R})$,

$$\begin{aligned} |q^{\psi }(\rho ,u)|\le C_{\psi }\rho ^{1+\theta _2}. \end{aligned}$$

(4.81)

Proof

Recall that

$$\begin{aligned} \begin{aligned} q^{\psi }(\rho ,u)&=\int _{\mathbb {R}}\big (\sigma (\rho ,u,s)-u\chi (\rho ,u-s)\big )\psi (s)\,\textrm{d}s+u\int _{\mathbb {R}}\chi (\rho ,u-s)\psi (s)\,\textrm{d}s\\&:=h^{\psi }(\rho ,u)+u\,\eta ^{\psi }(\rho ,u). \end{aligned} \end{aligned}$$

(4.82)

It follows from Lemma 4.9 that

$$\begin{aligned} |h^{\psi }(\rho ,u)|\le C\Vert (\sigma -u\chi )(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\Vert \psi \Vert _{L^1(\mathbb {R})}\le C_{\psi }\rho ^{1+\theta _2}. \end{aligned}$$

(4.83)

Since there exists $L>0$ such that ${\text {supp}}\psi \subset [-L,L]$, then it follows from Lemma 4.7 that $|u\eta ^{\psi }(\rho ,u)|\le (k(\rho )+L)|\eta ^{\psi }(\rho ,u)|\le C_{\psi }\rho ^{1+\theta _2}$ for $\rho \ge \rho ^{*}$, which, with (4.82)–(4.83), yields (4.81). $\square$

4.3 Singularities of the entropy kernel and the entropy flux kernel

As indicated in [11, 63, 64], understanding the singularities of the entropy kernel and the entropy flux kernel is essential for the reduction of the Young measure. Thus, it requires some detailed estimates of the singularities of the entropy kernel and the entropy flux kernel. The arguments in this subsection are similar to [64, Sect. 6], the main difference is that a more subtle Grönwall inequality (see Lemma A.3) is needed to obtain the desired estimates of the singularities.

Lemma 4.11

For $\rho \ge \rho ^{*}$, the coefficient functions $a_1(\rho )$ and $a_2(\rho )$ and the remainder term $g_1(\rho ,u)$ in Lemma 4.2 satisfy

$$\begin{aligned}&|a_1(\rho )|+\rho ^{\theta _2}|a_2(\rho )|\le C\rho ^{\frac{1}{2}-\frac{\theta _2}{2\theta _1}}, \qquad \,\, \Vert g_1(\rho ,u)\Vert _{L_{u}^{\infty }(\mathbb {R})} \le {\left\{ \begin{array}{ll} C\rho \quad &{}\text {if }\,\,\theta _2<\theta _1,\\ C\rho \ln \rho \,\,\, &{}\text {if }\,\, \theta _2=\theta _1, \end{array}\right. }\\&\Vert \partial _ug_1(\rho ,u)\Vert _{L_{u}^{\infty }(\mathbb {R})} + \Vert \partial _u^{\lambda _1+1}g_1(\rho ,u)\Vert _{L_{u}^{\infty }(\mathbb {R})} +\Vert \partial _u^{\lambda _1+1+\alpha _0}g_1(\rho ,u)\Vert _{L_{u}^{\infty }(\mathbb {R})}\le C\rho , \end{aligned}$$

where $\alpha _0\in (0,1)$ is the Hölder exponent.

Proof

We divide the proof into five steps.

1. It follows from (4.60) that, for $\rho \in [0,\rho _{*}]$,

$$\begin{aligned} |a_1(\rho )|+\rho ^{1-2\theta _1}|a_1'(\rho )|+\rho ^{2-2\theta _1}|a_1''(\rho )| +|a_2(\rho )|+\rho |a_2'(\rho )|+\rho ^{2}|a_2''(\rho )|\le C. \end{aligned}$$

(4.84)

For $\rho \ge \rho ^{*}$, it follows from (3.7), (4.58), and a direct calculation that

$$\begin{aligned}&|a_1(\rho )|+\rho | a_1'(\rho )|+\rho ^2| a_1''(\rho )|\le C\rho ^{\frac{1}{2}-\frac{\theta _2}{2\theta _1}}, \end{aligned}$$

(4.85)

$$\begin{aligned}&|a_{2}(\rho )|\,=\Big \vert \frac{1}{4\lambda _1+1}k(\rho )^{-\lambda _1-1}k'(\rho )^{-\frac{1}{2}} \int _{0}^{\rho }k(s)^{\lambda _1}k'(s)^{-\frac{1}{2}}a_1''(s)\,\text {d}s\Big \vert \nonumber \\ {}&\quad \qquad \quad \le C\rho ^{\frac{1}{2}-\frac{\theta _2}{2\theta _1}-\theta _2}\Big (\rho _{*}^{\theta _1}+1 +\int _{\rho ^{*}}^{\rho }s^{-1-\theta _2}\,\text {d}s\Big ) \le C\rho ^{\frac{1}{2}-\theta _2-\frac{\theta _2}{2\theta _1}}. \end{aligned}$$

(4.86)

Moreover, calculating the derivatives explicitly, we conclude

$$\begin{aligned} |a_{2}(\rho )|+\rho |a_2'(\rho )|+\rho ^2|a_2''(\rho )|\le C\rho ^{\frac{1}{2}-\theta _2-\frac{\theta _2}{2\theta _1}} \qquad \,\, \text {for }\rho \ge \rho ^{*}. \end{aligned}$$

2. For the remainder term $g_1(\rho ,u)$, it follows from [11, Proof of Theorem 2.1] that

$$\begin{aligned} \left\{ \begin{aligned} \displaystyle&\partial _{\rho \rho }g_1(\rho ,u)-k'(\rho )^2\partial _{uu}g_1(\rho ,u)=A(\rho )k(\rho )^{-1}f_{\lambda _1+1}(\frac{u}{k(\rho )}),\\ \displaystyle&g_1(0,\cdot )=0,\quad \partial _{\rho }g_1(0,\cdot )=0, \end{aligned}\right. \end{aligned}$$

where $f_{\lambda _1}(y)=[1-y^2]_{+}^{\lambda _1}$ and $A(\rho )=-a_2''(\rho )k(\rho )^{2\lambda _1+3}$. By (4.84), $A(\rho )\sim O(\rho ^{-1+2\theta _1})$ as $\rho \rightarrow 0$ and $|A(\rho )|\le \rho ^{-\frac{3}{2}+\theta _2(1+\frac{1}{2\theta _1})}$ for $\rho \ge \rho ^{*}$. Similar to Lemma 4.4, we have the following representation formula for $g_1(\rho ,u)$:

$$\begin{aligned} k'(\rho )g_1(\rho ,u)&=\frac{1}{2\rho }\int _{0}^{\rho }d(s)\,k'(s)\, \big (g_1(s,u+k(\rho )-k(s))+g_1(s,u-k(\rho )+k(s))\big )\,\textrm{d}s\nonumber \\&\quad +\frac{1}{2\rho }\int _{0}^{\rho }s\,A(s)\,k(s)^{-1}\, \Big (\int _{u-k(\rho )+k(s)}^{u+k(\rho )-k(s)}f_{\lambda _1+1}(\frac{y}{k(s)})\,\textrm{d}y\Big )\,\textrm{d}s, \end{aligned}$$

(4.87)

where $d(\rho )=2+\rho \frac{k''(\rho )}{k'(\rho )}$ satisfying (3.11). Since $\rho A(\rho )k(\rho )^{-1}\sim O(\rho ^{\theta _1})$ as $\rho \rightarrow 0$, the second integral is well-defined. Then it follows directly from (3.6)–(3.7) and (4.87) that

$$\begin{aligned}&\Vert k'(\rho )g_1(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\nonumber \\&\quad \le \frac{1}{\rho }\int _{0}^{\rho }d(s)\Vert k'(s)g_1(s,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\,\textrm{d}s+\frac{2k(\rho )}{\rho } \Big (\int _{0}^{\rho _{*}}+\int _{\rho _{*}}^{\rho ^{*}}+\int _{\rho ^{*}}^{\rho }\Big )|sA(s)k(s)^{-1}|\,\textrm{d}s\nonumber \\&\quad \le \frac{1}{\rho }\int _{0}^{\rho }d(s)\Vert k'(s)g_1(s,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\,\textrm{d}s +C\rho ^{\theta _2-1}\big (\rho _{*}^{1+\theta _1}+1+\rho ^{\frac{1}{2}+\frac{\theta _2}{2\theta _1}}\big )\nonumber \\&\quad \le \frac{1}{\rho }\int _{0}^{\rho }d(s)\Vert k'(s)g_1(s,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\,\textrm{d}s +C\rho ^{-\frac{1}{2}+\frac{\theta _2}{2\theta _1}+\theta _2}\qquad \,\, \text {for }\rho \ge \rho ^{*}. \end{aligned}$$

(4.88)

Since $g_1(\rho ,u)$ is Hölder continuous and ${\text {supp}}g_1(\rho ,\cdot )\subset [-k(\rho ),k(\rho )]$, it follows from (3.6)–(3.7) that $\Vert k'(\rho )g_1(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}$ is locally integrable with respect to $\rho \in [0,\infty )$. Applying Lemma A.3 to (4.88), we obtain that, for $\rho \ge \rho ^{*}$,

$$\begin{aligned} \Vert k'(\rho )g_1(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\le \left\{ \begin{array}{ll} C\rho ^{\theta _2}&{}\quad \text {if }\, \theta _2<\theta _1,\\ C\rho ^{\theta _2}\ln \rho &{}\quad \text {if }\,\theta _2=\theta _1, \end{array}\right. \end{aligned}$$

which, with (3.7), yields that, for $\rho \ge \rho ^{*}$,

$$\begin{aligned} \Vert g_1(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\le \left\{ \begin{array}{ll} C\rho &{}\quad \text {if }\theta _2<\theta _1,\\ C\rho \ln \rho &{}\quad \text {if }\theta _2=\theta _1. \end{array}\right. \end{aligned}$$

3. Applying $\partial _{u}$ to (4.87), we have

$$\begin{aligned}&k'(\rho )\partial _ug_1(\rho ,u)\nonumber \\&\quad =\frac{1}{2\rho }\int _{0}^{\rho }d(s)k'(s)\Big (\partial _ug_1(s,u+k(\rho )-k(s)) +\partial _ug_1(s,u-k(\rho )+k(s))\Big )\,\textrm{d}s\nonumber \\&\qquad +\frac{1}{2\rho }\int _{0}^{\rho }sA(s)k(s)^{-1}f_{\lambda _1+1}(\frac{u+k(\rho )-k(s)}{k(s)}) \,\textrm{d}s\nonumber \\&\qquad -\frac{1}{2\rho }\int _{0}^{\rho }sA(s)k(s)^{-1}f_{\lambda _1+1}(\frac{u-k(\rho )+k(s)}{k(s)})\,\textrm{d}s. \end{aligned}$$

(4.89)

Since $|f_{\lambda _1+1}(s)|\le 1$, by similar arguments as in Step 2, we can obtain

$$\begin{aligned} \Vert \partial _{u}g_1(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\le C\rho \qquad \,\, \text {for }\rho \ge \rho ^{*}. \end{aligned}$$

4. Applying the fractional derivative $\partial _{u}^{\lambda _1}$ to (4.89), we have

$$\begin{aligned}&k'(\rho )\partial _u^{\lambda _1+1}g_1(\rho ,u)\\&\quad =\frac{1}{2\rho }\int _{0}^{\rho }d(s)k'(s) \Big ((\partial _u^{\lambda _1+1}g_1)(s,u+k(\rho )-k(s)) +(\partial _u^{\lambda _1+1}g_1)(s,u-k(\rho )+k(s))\Big )\,\textrm{d}s\\&\qquad +\frac{1}{2\rho }\int _{0}^{\rho }sA(s)k(s)^{-1-\lambda _1} (\partial _{u}^{\lambda _1}f_{\lambda _1+1})(\frac{u+k(\rho )-k(s)}{k(s)})\,\textrm{d}s\\&\qquad -\frac{1}{2\rho }\int _{0}^{\rho }sA(s)k(s)^{-1-\lambda _1}(\partial _{u}^{\lambda _1}f_{\lambda _1+1})(\frac{u-k(\rho )+k(s)}{k(s)})\,\textrm{d}s, \end{aligned}$$

where we have taken into account the homogeneity of the factional derivative in the last term. Using the Fourier transform relation as in [48, (I.26)–(I.27)], we can obtain

$$\begin{aligned} \big |\mathscr {F}\big ((\partial _{u}^{\lambda _1}f_{\lambda _1+1})(u)\big )(\xi )\big | = C_{\lambda _1+1}|\xi |^{-\frac{3}{2}}\,\vert J_{\lambda _1+\frac{3}{2}}(|\xi |)\vert \le \frac{\tilde{C}_{\lambda _1+1}}{1+\xi ^2} \end{aligned}$$

for some positive constants $C_{\lambda _1+1}$ and $\tilde{C}_{\lambda _1+1}$ depending only on $\lambda _1+1$, where we have used the asymptotic relations for the first kind of Bessel functions $J_{\lambda _1+\frac{3}{2}}(|\xi |)$ to obtain the final inequality. Since $(1+|\xi |^2)^{-1}$ is integrable, applying the Fourier inversion theorem, we see that $(\partial _{u}^{\lambda _1}f_{\lambda _1+1})(u)$ is uniformly bounded. Hence, by similar arguments as in Step 2, we have

$$\begin{aligned} \Vert \big (\partial _u^{\lambda _1+1}g_1\big )(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\le C\rho \qquad \,\, \text {for}\, \rho \ge \rho ^{*}. \end{aligned}$$

5. By Lemma 4.2, we assume that $\alpha _0\in (0,1)$ is the Hölder exponent of $(\partial _{u}^{\lambda _1+1})g_1(\rho ,u)$. Then, applying the fractional derivative $\partial _{u}^{\lambda _1}$ to (4.89), we have

$$\begin{aligned}&k'(\rho )\partial _u^{\lambda _1+1+\alpha _0}g_1(\rho ,u)\\&\quad =\frac{1}{2\rho }\int _{0}^{\rho }d(s)k'(s)(\partial _u^{\lambda _1+1+\alpha _0}g_1)(s,u+k(\rho )-k(s))\,\textrm{d}s\\&\qquad +\frac{1}{2\rho }\int _{0}^{\rho }d(s)k'(s)(\partial _u^{\lambda _1+1+\alpha _0}g_1)(s,u-k(\rho )+k(s))\,\textrm{d}s\\&\qquad +\frac{1}{2\rho }\int _{0}^{\rho }sA(s)k(s)^{-1-\lambda _1-\alpha _0} (\partial _{u}^{\lambda _1+\alpha _0}f_{\lambda _1+1})(\frac{u+k(\rho )-k(s)}{k(s)})\,\textrm{d}s\\&\qquad -\frac{1}{2\rho }\int _{0}^{\rho }sA(s)k(s)^{-1-\lambda _1-\alpha _0}(\partial _{u}^{\lambda _1+\alpha _0}f_{\lambda _1+1})(\frac{u-k(\rho )+k(s)}{k(s)})\,\textrm{d}s. \end{aligned}$$

Noting

$$\begin{aligned} \big |\mathscr {F}\big ((\partial _{u}^{\lambda _1+\alpha _0}f_{\lambda _1+1})(u)\big )(\xi )\big | =C_{\lambda _1+1}|\xi |^{-\frac{3}{2}+\alpha _0}\big |J_{\lambda _1+\frac{3}{2}}(|\xi |)\big | \le \frac{\tilde{C}_{\lambda _1+1}}{1+\xi ^{2-\alpha _0}}, \end{aligned}$$

and using the Fourier inversion theorem, we find that $(\partial _{u}^{\lambda _1+\alpha _0}f_{\lambda _1+1})(u)$ is uniformly bounded. By similar arguments as in Step 2, we obtain that $\Vert \big (\partial _u^{\lambda _1+1+\alpha _0}g_1\big )(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\le C\rho$ for $\rho \ge \rho ^{*}$. This completes the proof. $\square$

From Lemmas 4.2 and  4.11, we conclude

Corollary 4.12

$\chi (\rho , \cdot )$ is Hölder continuous and

$$\begin{aligned} \Vert \chi (\rho ,\cdot )\Vert _{C_{u}^{\tilde{\alpha }}}\le C(1+\rho |\ln \rho |)\qquad \,\, \text {for}\,\,\, \tilde{\alpha }\in (0,\min \{\lambda _1,1\}]\text { and }\rho \ge 0. \end{aligned}$$

Lemma 4.13

For $\rho \ge \rho ^{*}$, the coefficient functions $b_{1}(\rho )$ and $b_2(\rho )$ and the remainder term $g_2(\rho ,u)$ in Lemma 4.3 satisfy

$$\begin{aligned} \begin{aligned}&|b_1(\rho )| +\rho ^{\theta _2}|b_2(\rho )|\le C\rho ^{\frac{1}{2}-\frac{\theta _2}{2\theta _1}},\qquad \,\, \Vert g_2(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\le {\left\{ \begin{array}{ll} C\rho ^{1+\theta _2} &{}{}\quad \text{ if } \, \theta _2<\theta _1,\\ C\rho ^{1+\theta _2}\ln \rho &{}{}\quad \text{ if } \, \theta _2=\theta _1, \end{array}\right. }\\ {}&\Vert \partial _{u}g_2(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}+\Vert \big (\partial _u^{\lambda _1+1}g_2\big )(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})} +\Vert \big (\partial _u^{\lambda _1+1+\alpha _0}g_2\big )(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\le C\rho ^{1+\theta _2}, \end{aligned} \end{aligned}$$

where $\alpha _0\in (0,1)$ is the Hölder exponent.

Proof

We divide the proof into five steps.

1. It follows from (4.62) that, for $\rho \in [0,\rho _{*}]$,

$$\begin{aligned} |b_1(\rho )|+\rho ^{1-2\theta _1}|b_1'(\rho )|+\rho ^{2-2\theta _1}|b_1''(\rho )| +|b_2(\rho )|+\rho | b_2'(\rho )|+\rho ^{2}|b_2''(\rho )|\le C. \end{aligned}$$

(4.90)

From (4.61) and (3.7), we have

$$\begin{aligned} |b_1(\rho )|+|\rho b_1'(\rho )|+|\rho ^{2}b_1''(\rho )|\le C\rho ^{\frac{1}{2}-\frac{\theta _2}{2\theta _1}} \qquad \,\, \text{ for }\, \rho \ge \rho ^{*}. \end{aligned}$$

(4.91)

Using (4.84)–(4.85) and (4.90)–(4.91), we obtain that, for $\rho \ge \rho ^{*}$,

$$\begin{aligned}&\Big \vert \int _{0}^{\rho }k(s)^{\lambda _1}k'(s)^{-\frac{1}{2}}a_1''(s)\,\text {d}s\Big \vert \le C\int _{0}^{\rho ^{*}}s^{-1+\theta _1}\text {d}s +C\int _{\rho _{*}}^{\rho } s^{-1-\theta _2}\,\text {d}s\le C,\\ {}&\Big \vert \int _{0}^{\rho }k(s)^{\lambda _1+1}k'(s)^{-\frac{1}{2}}b_1''(s)\,\text {d}s\Big \vert +\Big \vert \int _{0}^{\rho }s k(s)^{-\lambda _1}k'(s)^{\frac{1}{2}}a_1''(s)\,\text {d}s\Big \vert \le C\ln \rho , \end{aligned}$$

which, with (4.61), yields that $|b_2(\rho )|\le C\rho ^{\frac{1}{2}-\theta _2(1+\frac{1}{2\theta _1})}$ for $\rho \ge \rho ^{*}$. Moreover, by calculating the derivatives explicitly, we obtain

$$\begin{aligned} \rho |b_2'(\rho )|+\rho ^2|b_2''(\rho )|\le C\rho ^{\frac{1}{2}-\theta _2(1+\frac{1}{2\theta _1})}\qquad \,\, \text {for}\, \rho \ge \rho ^{*}. \end{aligned}$$

(4.92)

2. For the remainder term $g_2(\rho ,u)$, recalling from [11, Proof of Theorem 2.2], $g_2$ satisfies

$$\begin{aligned} \left\{ \begin{aligned} \displaystyle&\partial _{\rho \rho }g_2(\rho ,u)-k'(\rho )^2\partial _{uu}g_2(\rho ,u) =ub_2''(\rho )k(\rho )^{2\lambda _1+2}f_{\lambda _1+1}(\frac{u}{k(\rho )}) +\frac{P''(\rho )}{\rho }\partial _{u}g_1(\rho ,u),\\ \displaystyle&g_2(0,u)=0,\quad \partial _{\rho }g_2(0,u)=0, \end{aligned}\right. \end{aligned}$$

where $f_{\lambda _1}(y)=[1-y^2]_{+}^{\lambda _1}$. Similar to the arguments for Lemma 4.8, we obtain

$$\begin{aligned}&k'(\rho )g_2(\rho ,u)\nonumber \\ {}&\quad =\frac{1}{2\rho } \int _{0}^{\rho }d(s)k'(s)\Big (g_2(s,u+k(\rho )-k(s))+g_2(s,u-k(\rho )+k(s))\Big )\,\text {d}s\nonumber \\ {}&\qquad \, +\frac{1}{2\rho }\int _{0}^{\rho }sb_2''(s)k(s)^{2\lambda _1+2} \Big (\int _{u-k(\rho )+k(s)}^{u+k(\rho )-k(s)}y f_{\lambda _1+1}(\frac{y}{k(s)})\,\text {d}y\Big )\,\text {d}s\nonumber \\ {}&\qquad \, +\frac{1}{2\rho }\int _{0}^{\rho }P''(s)\Big (g_1(s,u+k(\rho )-k(s))-g_{1}(s,u-k(\rho )+k(s))\Big )\,\text {d}s, \end{aligned}$$

(4.93)

which yields

$$\begin{aligned}&\Vert k'(\rho )g_2(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\nonumber \\ {}&\quad =\frac{1}{\rho }\int _{0}^{\rho }d(s)\Vert k'(s)g_2(s,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\,\text {d}s +\frac{C}{\rho }\int _{0}^{\rho }s|b_2''(s)|k(s)^{2\lambda _1+4}\,\text {d}s\nonumber \\ {}&\qquad \, +\frac{C}{\rho }\int _{0}^{\rho }P''(s)\Vert g_1(s,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\,\text {d}s\nonumber \\ {}&\quad :=\frac{1}{\rho }\int _{0}^{\rho }d(s)\Vert k'(s)g_2(s,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\,\text {d}s+I_1+I_2. \end{aligned}$$

(4.94)

It follows from (3.2)–(3.3), (3.6)–(3.7), (4.90), (4.92), and Lemma 4.11 that, for $\rho \ge \rho ^{*}$

$$\begin{aligned} |I_1|&=\Big \vert \frac{C}{\rho }\Big (\int _{0}^{\rho _{*}}+\int _{\rho _{*}}^{\rho ^{*}}+\int _{\rho ^{*}}^{\rho }\Big ) s|b_2''(s)|k(s)^{2\lambda _1+4}\,\textrm{d}s\Big \vert \nonumber \\&\le \frac{C}{\rho }\Big (1+\int _{\rho ^{*}}^{\rho }s^{-\frac{1}{2}+2\theta _2+\frac{\theta _1}{2\theta _2}}\,\textrm{d}s\Big ) \le C\rho ^{-\frac{1}{2}+2\theta _2+\frac{\theta _1}{2\theta _2}}\le C\rho ^{2\theta _2}, \end{aligned}$$

(4.95)

$$\begin{aligned} \vert I_2|&=\Big \vert \frac{C}{\rho }\Big (\int _{0}^{\rho _{*}}+\int _{\rho _{*}}^{\rho ^{*}}+\int _{\rho ^{*}}^{\rho }\Big ) P''(s)\Vert g_1(s,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\,\textrm{d}s\Big \vert \nonumber \\&\le \frac{C}{\rho }\Big (1+\int _{\rho ^{*}}^{\rho }s^{2\theta _2-1}\Vert g_1(s,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\,\textrm{d}s\Big ) \le {\left\{ \begin{array}{ll} C\rho ^{2\theta _2}\quad &{}\text {if }\theta _2<\theta _1,\\ C\rho ^{2\theta _2}\ln \rho \,\,\, &{}\text {if }\theta _2=\theta _1. \end{array}\right. } \end{aligned}$$

(4.96)

Substituting (4.95)–(4.96) into (4.94), applying Lemma A.3 and Corollary A.4, and using (3.7), we obtain that, for $\rho \ge \rho ^{*}$,

$$\begin{aligned} \Vert g_2(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})} \le {\left\{ \begin{array}{ll} C\rho ^{1+\theta _2}\quad &{}\text {if}\, \theta _2<\theta _1,\\ C\rho ^{1+\theta _2}\ln \rho \,\,\, &{}\text {if}\, \theta _2=\theta _1. \end{array}\right. } \end{aligned}$$

3. Denoting $\widetilde{f}(s)=sf_{\lambda _1+1}(s)$ and applying $\partial _{u}$ to (4.93), we have

$$\begin{aligned}&k'(\rho )\partial _{u}g_2(\rho ,u)\nonumber \\&\quad =\frac{1}{2\rho }\int _{0}^{\rho }d(s)k'(s)\Big (\partial _{u}g_2(s,u+k(\rho )-k(s)) +\partial _{u}g_2(s,u-k(\rho )+k(s))\Big )\,\textrm{d}s\nonumber \\&\qquad \ +\frac{1}{2\rho }\int _{0}^{\rho }sb_2''(s)k(s)^{2\lambda _1+3} \Big (\widetilde{f}(\frac{u+k(\rho )-k(s)}{k(s)}) -\widetilde{f}(\frac{u-k(\rho )+k(s)}{k(s)})\Big )\,\textrm{d}s\nonumber \\&\qquad \ +\frac{1}{2\rho }\int _{0}^{\rho }P''(s) \Big (\partial _{u}g_1(s,u+k(\rho )-k(s))-\partial _{u}g_{1}(s,u-k(\rho )+k(s))\Big )\,\textrm{d}s. \end{aligned}$$

(4.97)

Since $|\widetilde{f}(s)|\le 1$, by similar arguments as in Step 2, we obtain

$$\begin{aligned} \Vert \partial _{u}g_2(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})}\le C\rho ^{1+\theta _2}\qquad \, \text {for}\, \rho \ge \rho ^{*}. \end{aligned}$$

4. Applying $\partial _{u}^{\lambda _1}$ to (4.97), we have

$$\begin{aligned}&k'(\rho )\partial _{u}^{\lambda _1+1}g_2(\rho ,u)\\&\quad =\frac{1}{2\rho }\int _{0}^{\rho }d(s)k'(s)\Big ((\partial _{u}^{\lambda _1+1}g_2)(s,u+k(\rho )-k(s)) +(\partial _{u}^{\lambda _1+1}g_2)(s,u-k(\rho )+k(s))\Big )\,\textrm{d}s\\&\qquad \ +\frac{1}{2\rho }\int _{0}^{\rho }sb_2''(s)k(s)^{\lambda _1+3} \Big (\widetilde{f}^{(\lambda _1)}(\frac{u+k(\rho )-k(s)}{k(s)}) -\widetilde{f}^{(\lambda _1)}(\frac{u-k(\rho )+k(s)}{k(s)})\Big )\,\textrm{d}s\\&\qquad \ +\frac{1}{2\rho }\int _{0}^{\rho }P''(s)\Big ((\partial _{u}^{\lambda _1+1}g_1)(s,u+k(\rho )-k(s)) -(\partial _{u}^{\lambda _1+1}g_{1})(s,u-k(\rho )+k(s))\Big )\,\textrm{d}s, \end{aligned}$$

where $\widetilde{f}^{(\lambda _1)}:=\partial _{s}^{\lambda _1}\widetilde{f}(s)$. Since $\widetilde{f}^{(\lambda _1)}$ is uniformly bounded, similar arguments as in Step 2 yield

$$\begin{aligned} \Vert \big (\partial _{u}^{\lambda _1+1}g_2\big )(\rho ,\cdot )\Vert _{L_{u}^{\infty }(\mathbb {R})} \le C\rho ^{1+\theta _2}\qquad \, \text {for}\, \rho \ge \rho ^{*}. \end{aligned}$$

5. Applying $\partial _{u}^{\lambda _1+\alpha _0}$ to (4.97), we have

$$\begin{aligned}&k'(\rho )\partial _{u}^{\lambda _1+1+\alpha _0}g_2(\rho ,u)\\&\quad =\frac{1}{2\rho }\int _{0}^{\rho }d(s)k'(s)(\partial _{u}^{\lambda _1+1+\alpha _0}g_2)(s,u+k(\rho )-k(s))\,\textrm{d}s\\&\qquad +\frac{1}{2\rho }\int _{0}^{\rho }d(s)k'(s)(\partial _{u}^{\lambda _1+1+\alpha _0}g_2)(s,u-k(\rho )+k(s))\,\textrm{d}s\\&\qquad +\frac{1}{2\rho }\int _{0}^{\rho }sb_2''(s)k(s)^{\lambda _1+3-\alpha _0} \Big (\widetilde{f}^{(\lambda _1)}(\frac{u+k(\rho )-k(s)}{k(s)}) -\widetilde{f}^{(\lambda _1)}(\frac{u-k(\rho )+k(s)}{k(s)})\Big )\,\textrm{d}s\\&\qquad +\frac{1}{2\rho }\int _{0}^{\rho }P''(s)(\partial _{u}^{\lambda _1+1+\alpha _0})g_1(s,u+k(\rho )-k(s)) \,\textrm{d}s\\&\qquad -\frac{1}{2\rho }\int _{0}^{\rho }P''(s)(\partial _{u}^{\lambda _1+1+\alpha _0})g_{1}(s,u-k(\rho )+k(s))\,\textrm{d}s. \end{aligned}$$

Noting that $\widetilde{f}^{(\lambda _1+\alpha _0)}(s)$ is uniformly bounded, by similar arguments as in Step 2, we have

$$\begin{aligned} \Vert \partial _{u}^{\lambda _1+1+\alpha _0}g_2(\rho ,\cdot ) \Vert _{L_{u}^{\infty }(\mathbb {R})}\le C\rho ^{1+\theta _2} \qquad \text {for}\, \rho \ge \rho ^{*}. \end{aligned}$$

This completes the proof. $\square$

The following lemma provides the explicit singularities of $\chi (\rho ,u-s)$ and $(\sigma -u\chi )(\rho ,u-s)$.

Lemma 4.14

The fractional derivatives $\partial _{u}^{\lambda _1+1}\chi$ and $\partial _{u}^{\lambda _1+1}(\sigma -u\chi )$ admit the expansions:

$$\begin{aligned}&\partial _{s}^{\lambda _1+1}\chi (\rho ,u-s)\nonumber \\&\quad =\sum \limits _{\pm }\Big (A_{1,\pm }(\rho )\,\delta (s-u\pm k(\rho ))+A_{2,\pm }(\rho )\,H(s-u\pm k(\rho ))\Big )\nonumber \\&\qquad \ +\sum \limits _{\pm }\Big (A_{3,\pm }(\rho )\,PV(s-u\pm k(\rho ))+A_{4,\pm }(\rho )\, Ci(s-u\pm k(\rho ))\Big )\nonumber \\&\qquad \ +r_{\chi }(\rho ,u-s), \end{aligned}$$

(4.98)

$$\begin{aligned}&\partial _{s}^{\lambda _1+1}(\sigma -u\chi )(\rho ,u-s)\nonumber \\&\quad =\sum \limits _{\pm }(s-u)\Big (B_{1,\pm }(\rho )\,\delta (s-u\pm k(\rho )) +B_{2,\pm }(\rho )\,H(s-u\pm k(\rho ))\Big )\nonumber \\&\qquad \ +\sum \limits _{\pm }(s-u)\Big (B_{3,\pm }(\rho )\,PV(s-u\pm k(\rho ))+B_{4,\pm }\,Ci(s-u\pm k(\rho ))\Big ) \nonumber \\&\qquad \ +\sum \limits _{\pm }\Big (B_{5,\pm }(\rho )\,H(s-u\pm k(\rho ))+B_{6,\pm }(\rho )\,Ci(s-u\pm k(\rho ))\Big ) \nonumber \\&\qquad \ +r_{\sigma }(\rho ,u-s), \end{aligned}$$

(4.99)

where $\delta$ is the Dirac measure, H is the Heaviside function, PV is the principle value distribution, and Ci is the Cosine integral:

$$\begin{aligned} Ci(s):=-\int _{|s|}^{\infty }\frac{\cos y}{y}\,\textrm{d}y =\log |s|+\int _{0}^{|s|}\frac{\cos y-1}{y}\,\textrm{d}y+C_0 \qquad \text{ for }\, s\in \mathbb {R}\end{aligned}$$

for some constant $C_0>0$. The remainder terms $r_{\chi }$ and $r_{\sigma }$ are Hölder continuous functions. Moreover, there exists a positive constant $C=C(\gamma _1,\gamma _2,\rho _{*},\rho ^{*})$ such that, for $\rho \ge \rho ^{*}$,

$$\begin{aligned}&\sum \limits _{j=1,\pm }^{4}|A_{j,\pm }(\rho )|+\sum \limits _{j=1,\pm }^6|B_{j,\pm }| \le C\rho ^{\frac{1}{2}-\frac{\theta _2}{2}},\\&\quad \Vert r_{\chi }(\rho ,\cdot )\Vert _{C^{\alpha _1}(\mathbb {R})} \le C\rho ,\quad \Vert r_{\sigma }(\rho ,\cdot )\Vert _{C^{\alpha _1}(\mathbb {R})}\le C\rho ^{1+\theta _2}, \end{aligned}$$

where $\alpha _1\in (0,\alpha _0]$ is the common Hölder exponent of $r_{\chi }$ and $r_{\sigma }$.

Proof

From [64, Lemma 6.4], we obtain (4.98)–(4.99), where the coefficients are given by

$$\begin{aligned}&A_{1, \pm }(\rho )=a_{1}(\rho ) k(\rho )^{\lambda _1} A_{1}^{\lambda _{1}}, \\&A_{2, \pm }(\rho )=\pm a_{1}(\rho ) k(\rho )^{\lambda _1-1} A_{3}^{\lambda _1}+ a_{2}(\rho ) k(\rho )^{\lambda _1+1} A_{1}^{\lambda _1+1},\\&A_{3, \pm }(\rho )=\pm a_{1}(\rho ) k(\rho )^{\lambda _1} A_{2}^{\lambda _1}, \\&A_{4, \pm }(\rho )=\pm a_{1}(\rho ) k(\rho )^{\lambda _1-1} A_{4}^{\lambda _1} \pm a_{2}(\rho ) k(\rho )^{\lambda _1+1} A_{2}^{\lambda _1+1},\\&r_{\chi }(\rho ,u-s)= a_1(\rho )k(\rho )^{\lambda _1-1}\tilde{q}(\frac{s-u}{k(\rho )}) +a_2(\rho )k(\rho )^{\lambda _1+1}\tilde{r}(\frac{s-u}{k(\rho )})\\&\qquad \qquad \qquad \qquad \, -A_{4}^{\lambda _1} k(\rho )^{\lambda _1-1}(\textrm{log}k(\rho ))^2+\partial _{s}^{\lambda _1+1}g_1(\rho ,u-s), \end{aligned}$$

where $A_{i}^{\lambda _1}, i=1,\ldots ,4$, are constants depending only on $\lambda _1$, and $\tilde{r}$ and $\tilde{q}$ are uniformly bounded Hölder continuous functions. Thus, using Lemma 4.11, we see that, for $\rho \ge \rho ^{*}$,

$$\begin{aligned}&|A_{i,\pm }(\rho )|\le C\rho ^{\frac{1}{2}-\frac{\theta _2}{2\theta _1}}\rho ^{\theta _2(\frac{1}{2\theta _1}-\frac{1}{2})} \le C\rho ^{\frac{1}{2}-\frac{1}{2}\theta _2}\qquad \,\, \text {for}\,\,\,\, i=1,3,\\&|A_{j,\pm }(\rho )|\le C\rho ^{\frac{1}{2}-\frac{3\theta _2}{2}}+C\rho ^{\frac{1}{2}-\frac{\theta _2}{2\theta _1}-\theta _2}\rho ^{\theta _2(\frac{1}{2\theta _1}+\frac{1}{2})} \le C\rho ^{\frac{1}{2}-\frac{\theta _2}{2}}\qquad \,\, \text {for}\,\,\,\, j=2,4.\\&\Vert r_{\chi }(\rho ,\cdot )\Vert _{C^{\alpha }(\mathbb {R})}:=\Vert r_{\chi }(\rho ,\cdot )\Vert _{L^{\infty }(\mathbb {R})}+[r_{\chi }(\rho ,\cdot )]_{C^{\alpha }(\mathbb {R})}\\&\qquad \qquad \qquad \qquad \le C\big (\rho ^{\frac{1}{2}-\frac{\theta _2}{2}}+\rho ^{\frac{\theta _2}{2\theta _1}-\frac{3}{2}\theta _2}|\ln \rho |^2+\rho \big ) \le C\rho . \end{aligned}$$

Similarly, we have

$$\begin{aligned}&B_{1,\pm }(\rho )=b_1(\rho )k(\rho )^{\lambda _1}A_{1}^{\lambda _1},\quad B_{2,\pm }(\rho )=\pm b_{1}(\rho )k(\rho )^{\lambda _1-1}A_3^{\lambda _1}+b_2(\rho )k(\rho )^{\lambda _1+1}A_{1}^{\lambda _1+1},\\ {}&B_{3,\pm }(\rho )=\pm b_1(\rho )k(\rho )^{\lambda _1}A_{2}^{\lambda _1},\quad B_{4,\pm }(\rho )=\pm b_1(\rho )k(\rho )^{\lambda _1-1}A_{4}^{\lambda _1}\pm b_2(\rho )k(\rho )^{\lambda _1+1}A_{2}^{\lambda _1+1},\\ {}&B_{5,\pm }(\rho )=(\lambda _1+1)b_1(\rho )k(\rho )^{\lambda _1}A_{1}^{\lambda _1},\quad B_{6,\pm }(\rho )=\pm (\lambda _1+1)b_1(\rho )k(\rho )^{\lambda _1}A_{2}^{\lambda _1},\\&r_{\sigma }(\rho ,u-s)\\ {}&\quad =(s-u)k(\rho )^{\lambda _1-1}\Big (b_1(\rho )\big (-A_4^{\lambda _1}(\text {log}k(\rho ))^2+\tilde{q}(\frac{s-u}{k(\rho )})\big )+b_2(\rho )k(\rho )^2\tilde{r}(\frac{s-u}{k(\rho )})\Big )\\ {}&\qquad +(\lambda _1+1)k(\rho )^{\lambda _1}\Big (b_1(\rho )\tilde{r}(\frac{s-u}{k(\rho )})+b_2(\rho )k(\rho )^{2}\tilde{\ell }(\frac{s-u}{k(\rho )})\Big )+\partial _{s}^{\lambda _1+1}g_2(\rho ,u-s), \end{aligned}$$

where $\tilde{\ell }$ is also a uniformly bounded Hölder continuous function. Using Lemma 4.13, we conclude that, for $\rho \ge \rho ^{*}$,

$$\begin{aligned}&|B_{i,\pm }(\rho )|\le C\rho ^{\frac{1}{2}-\frac{\theta _2}{2\theta _1}}\rho ^{\theta _2(\frac{1}{2\theta _1}-\frac{1}{2})} \le C\rho ^{\frac{1}{2}-\frac{\theta _2}{2}}\qquad \text {for }i=1,3,5,6,\\&|B_{j,\pm }(\rho )|\le C\rho ^{\frac{1}{2}-\frac{3\theta _2}{2}}+C\rho ^{\frac{1}{2}-(\frac{1}{2\theta _1}+1)\theta _2}\rho ^{\theta _2(\frac{1}{2\theta _1}+\frac{1}{2})} \le C\rho ^{\frac{1}{2}-\frac{\theta _2}{2}}\qquad \text {for }j=2,4,\\&\Vert r_{\sigma }(\rho ,\cdot )\Vert _{C^{\alpha }(\mathbb {R})}:=\Vert r_{\sigma }(\rho ,\cdot )\Vert _{L^{\infty }(\mathbb {R})}+[r_{\sigma }(\rho ,\cdot )]_{C^{\alpha }(\mathbb {R})}\\&\qquad \qquad \qquad \qquad \le C\big (\rho ^{\frac{1}{2}-\frac{\theta _2}{2}}|\ln \rho |^2+\rho ^{1+\theta _2}\big ) \le C\rho ^{1+\theta _2}. \end{aligned}$$

This completes the proof. $\square$

5 Uniform Estimates of Approximate Solutions

As in [10], we construct the approximate solutions via the following approximate free boundary problem for CNSPEs:

$$\begin{aligned} \left\{ \begin{aligned}&\rho _{t}+(\rho u)_r+\frac{2}{r}\rho u=0,\\&(\rho u)_t+(\rho u^2+P(\rho ))_{r}+\frac{2}{r}\rho u^2+\frac{\rho }{r^{2}}\int _{a}^{r}\rho (t,y)\,y^{2}\textrm{d}y =\varepsilon \Big (\rho (u_r+\frac{2}{r}u)\Big )_{r} -\frac{2\varepsilon }{r}\rho _{r}u, \end{aligned} \right. \end{aligned}$$

(5.1)

for $(t,r)\in \Omega _{T}$ with

$$\begin{aligned} \Omega _{T}=\{(t,r)\in [0,\infty )\times \mathbb {R}\,:\,a\le r\le b(t),\;0\le t\le T\}, \end{aligned}$$

(5.2)

where $\{r=b(t):\,0\le t\le T\}$ is a free boundary determined by

$$\begin{aligned} b'(t)=u(t,b(t))\quad \text {for}\, t>0,\qquad \,\,\, b(0)=b, \end{aligned}$$

(5.3)

and $a=b^{-1}$ with $b\gg 1$. On the free boundary $r=b(t)$, we impose the stress-free boundary condition:

$$\begin{aligned} \big (P(\rho )-\varepsilon \rho (u_{r}+\frac{2}{r}u)\big )(t,b(t))=0\qquad \text {for }t>0. \end{aligned}$$

(5.4)

On the fixed boundary $r=a=b^{-1}$, we impose the Dirichlet boundary condition:

$$\begin{aligned} u(t,r)\vert _{r=a}=0\qquad \text {for }t>0. \end{aligned}$$

(5.5)

The initial condition is

$$\begin{aligned} (\rho ,\rho u)\vert _{t=0}=(\rho _{0}^{\varepsilon ,b},\rho _{0}^{\varepsilon ,b}u_{0}^{\varepsilon ,b})\qquad \, \text {for }r\in [a,b]. \end{aligned}$$

(5.6)

5.1 Basic estimates

Denote

$$\begin{aligned}&E_{0}^{\varepsilon , b}:=\omega _3\int _{a}^{b} \rho _{0}^{\varepsilon , b}\Big (\frac{1}{2}\big |u_{0}^{\varepsilon , b}\big |^{2} +e(\rho _{0}^{\varepsilon , b})\Big ) \,r^{2}\textrm{d}r, \qquad E_{1}^{\varepsilon , b}:=\omega _{3}\varepsilon ^{2} \int _{a}^{b}\Big |\big (\sqrt{\rho _{0}^{\varepsilon , b}}\big )_{r}\Big |^{2}\, r^{2}\textrm{d} r. \end{aligned}$$

For given total energy $E_{0}^{\varepsilon ,b}>0$, the critical mass $M_\textrm{c}^{\varepsilon ,b}$ is defined in (2.5)–(2.8) by replacing $E_{0}$ with $E_{0}^{\varepsilon ,b}$.

For the approximate initial data $(\rho _{0}^{\varepsilon },m_{0}^{\varepsilon })$ imposed in (2.17) satisfying (2.9)–(2.10), using similar arguments in [10, Appendix A], we can construct a sequence of smooth functions $(\rho _{0}^{\varepsilon ,b},u_{0}^{\varepsilon ,b})$ defined on [a, b], which is compatible with the boundary conditions (5.4)–(5.5), such that

• (i) There exists a constant $C_{\varepsilon ,b}>0$ depending on $(\varepsilon ,b)$ so that, for all $\varepsilon \in (0,1]$ and $b>1$,

  $$\begin{aligned} 0<C_{\varepsilon ,b}^{-1}\le \rho _{0}^{\varepsilon ,b}(r)\le C_{\varepsilon ,b}<\infty . \end{aligned}$$

  (5.7)

• (ii) For all $\varepsilon \in (0,1]$ and $b>1$,

  $$\begin{aligned}&\int _{a}^{b}\rho _{0}^{\varepsilon ,b}(r)r^{2}dr=\frac{M}{\omega _3}, \qquad E_{0}^{\varepsilon ,b}\le C(1+E_{0}),\qquad E_{1}^{\varepsilon ,b}\le C(1+M)\varepsilon , \end{aligned}$$

  (5.8)
  $$\begin{aligned}&\rho _{0}^{\varepsilon ,b}(b)\cong b^{-(3-\alpha )}\qquad \,\, \text {with }\alpha :=\min \{\frac{1}{2},\frac{3(\gamma _1-1)}{\gamma _1}\}. \end{aligned}$$

  (5.9)

• (iii) For each fixed $\varepsilon \in (0,1]$, as $b\rightarrow \infty$, $(E_{0}^{\varepsilon ,b},E_{1}^{\varepsilon ,b})\rightarrow (E_0^{\varepsilon },E_{1}^{\varepsilon })$ and

  $$\begin{aligned}{} & {} (\rho _{0}^{\varepsilon ,b},\rho _{0}^{\varepsilon ,b}u_{0}^{\varepsilon ,b})\longrightarrow (\rho _0^{\varepsilon },m_{0}^{\varepsilon }) \qquad \text {in}\, L^{\tilde{q}}([a,b];r^{2}dr)\times L^1([a,b];r^{2}dr)\nonumber \\{} & {} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \, \text {for }\tilde{q}\in \{1,\gamma _{2}\}. \end{aligned}$$

  (5.10)

• (iv) For each fixed $\varepsilon \in (0,\varepsilon _0]$, there exists a large constant $\mathcal {B}(\varepsilon )>0$ such that

  $$\begin{aligned} M<M_\textrm{c}^{\varepsilon ,b}\qquad \text {for }b\ge \mathcal {B}(\varepsilon )\text { and }\gamma _2\in (\frac{6}{5},\frac{4}{3}], \end{aligned}$$

  (5.11)

  where $M_\textrm{c}^{\varepsilon ,b}$ is defined in (2.5)–(2.8) by replacing $E_{0}$ with $E_{0}^{\varepsilon ,b}$.

We point out that (5.9) is important for us to close the BD-type entropy estimate in Lemma 5.4 and to obtain the higher integrability of the density in Lemma 5.6 below.

Once the free boundary problem (5.1)–(5.6) is solved, we define the potential function $\Phi$ to be the solution of the Poisson equation:

$$\begin{aligned} \Delta \Phi =\rho \textbf{I}_{\Omega _{t}},\qquad \lim \limits _{|\textbf{x}|\rightarrow \infty }\Phi (\textbf{x})=0, \end{aligned}$$

with $\Omega _{t}:=\{\textbf{x}\in \mathbb {R}^3\,:\,a\le |\textbf{x}|\le b(t)\}$, for which $\rho$ has been extended to be zero outside $\Omega _{t}$. In fact, we can show that $\Phi (t, \textbf{x})=\Phi (t, r)$ with

$$\begin{aligned} \Phi _{r}(t, r)=\left\{ \begin{array}{ll} 0&{}\quad \text { for } 0 \le r \le a, \\ \frac{1}{r^{2}} \int _{a}^{r} \rho (t, y) \, y^{2}\textrm{d}y &{} \quad \text { for } a \le r \le b(t), \\ \frac{M}{\omega _{3}} \frac{1}{r^{2}}&{}\quad \text { for } r \ge b(t), \end{array}\right. \end{aligned}$$

(5.12)

so that $\Phi (t,r)$ can be recovered by integrating (5.12).

In this section, parameters $(\varepsilon ,b)$ are fixed with $\varepsilon \in (0,\varepsilon _0]$ and $b\ge \max \{\rho _{*}^{-\frac{\gamma _1}{3}},\mathcal {B}(\varepsilon )\}$ such that (5.11) holds and $\rho _{0}^{\varepsilon ,b}(b)\le \rho _{*}$. The global existence of smooth solutions of our approximate problem (5.1)–(5.6) whose initial data satisfy (5.7)–(5.11) and pressure satisfies (1.4)–(1.6) can be obtained by using similar arguments in [25, Sect. 3] with $\gamma _2\in (\frac{4}{3},\infty )$, or with $\gamma _2\in (\frac{6}{5},\frac{4}{3}]$ and $M<M_\textrm{c}^{\varepsilon ,b}(\gamma _2)$, so the details are omitted here for simplicity.

Noting that the upper and lower bounds of $\rho ^{\varepsilon ,b}$ in [25] depend on parameters $(\varepsilon , b)$, we now establish some uniform estimates, independent of b, such that the limit: $b\rightarrow \infty$ can be taken to obtain the global weak solutions of problem (1.10) and (2.17)–(2.18) in Sect. 6 below as approximate solutions of problem (1.1) and (1.13)–(1.14). Throughout this section, we drop the superscript in both the approximate solutions $(\rho ^{\varepsilon ,b},u^{\varepsilon ,b})(r)$ and the approximate initial data $(\rho _{0}^{\varepsilon ,b},u_{0}^{\varepsilon ,b})$ for simplicity.

For smooth solutions, it is convenient to analyze (5.1)–(5.6) in the Lagrangian coordinates. It follows from (5.3) that

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\int _{a}^{b(t)}\rho (t,r)\,r^{2}\textrm{d}r =(\rho u)(t,b(t))b(t)^{2}-\int _{a}^{b(t)}(\rho u r^{2})_{r}(t,r)\,\textrm{d}r=0, \end{aligned}$$

which implies

$$\begin{aligned} \int _{a}^{b(t)}\rho (t,r)\,r^{2}\textrm{d}r =\int _{a}^{b}\rho _{0}(r)\,r^{2}\textrm{d}r=\frac{M}{\omega _3} \qquad \, \text{ for } \text{ all } t\ge 0. \end{aligned}$$

(5.13)

For $r\in [a,b(t)]$ and $t\in [0,T]$, the Lagrangian coordinates $(\tau , x)$ are defined by

$$\begin{aligned} \tau =t, \qquad x(t,r)=\int _{a}^{r}\rho (t,y)\,y^{2}\textrm{d}y, \end{aligned}$$

which translate $[0,T]\times [a,b(t)]$ into a fixed domain $[0,T]\times [0,\frac{M}{\omega _3}]$. By direct calculation, we see that $\nabla _{(t,r)}x=(-\rho u r^{2},\rho r^{2})$, $\nabla _{(t,r)}\tau =(1,0)$, $\nabla _{(\tau ,x)}r=(u,\rho ^{-1}r^{-2})$, and $\nabla _{(\tau ,x)}t=(1,0)$. In the Lagrangian coordinates, the initial-boundary value problem (5.1)–(5.6) becomes

$$\begin{aligned} \left\{ \begin{aligned}&\rho _{\tau }+\rho ^2(r^{2}u)_{x}=0,\\&u_{\tau }+r^{2}P_{x}=-\frac{x}{r^2}+\varepsilon r^{2}(\rho ^2(r^{2}u)_{x})_{x}-2\varepsilon r\rho _{x} u, \end{aligned} \right. \end{aligned}$$

(5.14)

for $(\tau ,x)\in [0,T]\times [0,\frac{M}{\omega _3}]$, and

$$\begin{aligned} u(\tau , 0)=0,\quad (P-\varepsilon \rho ^2(r^{2}u)_{x})(\tau ,\frac{M}{\omega _{3}})=0\qquad \,\, \text {for }\tau \in [0,T], \end{aligned}$$

(5.15)

where $r=r(\tau ,x)$ is defined by $\,\frac{\textrm{d}}{\textrm{d}\tau }r(\tau ,x)=u(\tau ,x)$ for $(\tau ,x)\in [0,T]\times [0,\frac{M}{\omega _3}]$, and the fixed boundary $x=\frac{M}{\omega _3}$ corresponds to the free boundary: $b(\tau )=r(\tau ,\frac{M}{\omega _3})$ in the Eulerian coordinates.

Lemma 5.1

(Basic energy estimate). The smooth solution $(\rho , u)(t,r)$ of problem (5.1)–(5.6) satisfies

$$\begin{aligned} \begin{aligned}&\int _{a}^{b(t)}\Big (\frac{1}{2}\rho u^2+\rho e(\rho )\Big )\,r^{2}\textrm{d}r -\frac{1}{2}\int _{a}^{\infty }\frac{1}{r^{2}}\Big (\int _{a}^{r}\rho (t,z)\,z^{2}\textrm{d}z\Big )^2\,\textrm{d}r\\&\qquad +\varepsilon \int _{0}^{t}\int _{a}^{b(s)}\Big (\rho u_{r}^2+2\frac{\rho u^2}{r^2}\Big )\,r^{2}\textrm{d}r\textrm{d}s +2\varepsilon \int _{0}^{t}(\rho u^2)(s,b(s))b(s)\,\textrm{d}s\\&\quad =\int _{a}^{b}\Big (\frac{1}{2}\rho _{0}u_{0}^2+\rho _{0}e(\rho _0)\Big )\,r^{2}\textrm{d}r -\frac{1}{2}\int _{a}^{\infty }\frac{1}{r^{2}}\Big (\int _{a}^{r}\rho _{0}(t,z)z^{2}dz\Big )^2\,\textrm{d}r, \end{aligned} \end{aligned}$$

where $\rho (t, r)$ has been understood to be 0 for $r\in [0, a]\cup (b(t),\infty )$ in the second term of the left-hand side (LHS) and the second term of the right-hand side (RHS). In particular, there exists a positive constant $C(E_{0},M)$ depending only on the total initial energy $E_{0}$ and initial-mass M such that the following estimates hold for the two separate cases:

Case 1. $\displaystyle \gamma _2\in (\frac{6}{5},\frac{4}{3}]$ and $M<M_\textrm{c}^{\varepsilon ,b}$:

$$\begin{aligned}&\int _{a}^{b(t)} \rho \Big (\frac{1}{2}u^{2}+e(\rho )\Big )\,r^{2}\textrm{d} r +\varepsilon \int _{0}^{t}\int _{a}^{b(s)}\rho \Big (u_{r}^{2}+\frac{2u^{2}}{r^{2}}\Big )(t,r)\,r^{2}\textrm{d}r\textrm{d}s\nonumber \\&\quad +2 \varepsilon \int _{0}^{t}(\rho u^{2})(s, b(s))b(s)\, \textrm{d} s\le C(E_{0},M). \end{aligned}$$

(5.16)

Case 2. $\displaystyle \gamma _2>\frac{4}{3}$:

$$\begin{aligned}&\int _{a}^{b(t)} \frac{1}{2} \rho \left( u^{2}+e(\rho )\right) \, r^{2}\textrm{d} r +\varepsilon \int _{0}^{t} \int _{a}^{b(s)} \rho \Big (u_{r}^{2}+\frac{2u^{2}}{r^{2}}\Big )(t, r)\, r^{2}\textrm{d} r \textrm{d} s\nonumber \\&\quad +2 \varepsilon \int _{0}^{t}(\rho u^{2})(s,b(s)) b(s)\, \textrm{d} s \le C(E_{0},M). \end{aligned}$$

(5.17)

Proof

We divide the proof into three steps.

1. Using (2.3) and similar calculations as in the proof [10, Lemma 3.1], we have

$$\begin{aligned} \begin{aligned}&\int _{a}^{b(t)} \rho \Big (\frac{1}{2} u^{2}+e(\rho )\Big )\, r^{2}\textrm{d} r - \int _{a}^{b(t)}\Big (\int _{a}^{r} \rho (t, z)\, z^{2}\textrm{d} z\Big ) \rho \, r\textrm{d} r \\&\qquad +\varepsilon \int _{0}^{t} \int _{a}^{b(s)} \Big (\rho u_{r}^{2}+2 \rho \frac{u^{2}}{r^{2}}\Big )\,r^{2}\textrm{d} r \textrm{d} s +2 \varepsilon \int _{0}^{t}(\rho u^{2})(s, b(s))\, b(s)\textrm{d} s \\&\quad =\int _{a}^{b} \rho _{0}\Big (\frac{1}{2} u_{0}^{2}+e\left( \rho _{0}\right) \Big )\, r^{2}\textrm{d} r -\int _{a}^{b}\Big (\int _{a}^{r} \rho _{0}(z) z^{2} \,\textrm{d} z\Big ) \rho _{0}(r)\, r\textrm{d} r. \end{aligned} \end{aligned}$$

(5.18)

2. We now control the second term on the LHS of (5.18) and the second term on the RHS of (5.18) to close the estimates. By similar calculations as in [10, Lemma 3.1], one can obtain

$$\begin{aligned} \int _{a}^{b(t)}\Big (\int _{a}^{r} \rho \, z^{2}\textrm{d} z\Big ) \rho \, r \textrm{d} r =\frac{1}{2 \omega _{3}}\Vert \nabla \Phi \Vert _{L^{2}\left( \mathbb {R}^{3}\right) }^{2} =\frac{1}{2}\int _{a}^{\infty }\frac{1}{r^{2}}\Big (\int _{a}^{r}\rho \, z^{2}\textrm{d}z\Big )^{2} \,\textrm{d}r, \end{aligned}$$

(5.19)

where we have understood $\rho$ to be zero for $r\in [0,a)\cup (b(t),\infty )$ in (5.19).

3. Now we use the internal energy to control the gravitational potential term. First, we obtain from (3.12) that there exist two constants $C_{1},C_{2}>0$ depending only on $\rho ^{*}$ such that

$$\begin{aligned} \big |\rho e(\rho )-\frac{\kappa _2\rho ^{\gamma _2}}{\gamma _2-1}\big | \le C_{1}\rho ^{\max \{\gamma _{2}-\epsilon ,0\}}\,\,\,\, \text {for }\rho \ge \rho ^{*}, \quad \,\,\, \big |\rho e(\rho )-\frac{\kappa _2\rho ^{\gamma _2}}{\gamma _2-1}\big |\le C_{2}\rho ^{\gamma _2}\,\,\,\, \text {for }\rho \le \rho ^{*}. \end{aligned}$$

Thus, we have

$$\begin{aligned}&\Big \vert \int _{a}^{b(t)}\big (\rho e(\rho )-\frac{\kappa _2}{\gamma _2-1}\rho ^{\gamma _{2}}\big )\,r^2\textrm{d}r\Big \vert \nonumber \\&\quad =\int _{\rho (t,r)\ge K}\Big \vert \rho e(\rho )-\frac{\kappa _2}{\gamma _2-1}\int _{a}^{b(t)}\rho ^{\gamma _{2}}\Big \vert \,r^2\textrm{d}r\nonumber \\&\qquad \quad +\int _{\rho (t,r)\le K}\Big \vert \rho e(\rho )-\frac{\kappa _2}{\gamma _2-1}\int _{a}^{b(t)}\rho ^{\gamma _{2}}\Big \vert \,r^2\textrm{d}r\nonumber \\&\quad \le C_{1}K^{-\min \{\gamma _2,\epsilon \}}\int _{a}^{b(t)}\rho ^{\gamma _{2}}\,r^2\textrm{d}r+C_{2}\omega _{3}^{-1}K^{\gamma _{2}-1}\,M, \end{aligned}$$

(5.20)

where $K>\rho ^{*}$ is some large constant to be chosen later.

Multiplying (5.11) by $\Phi$ and integrating by parts yield

$$\begin{aligned} \Vert \nabla \Phi \Vert _{L^{2}(\mathbb {R}^{3})}^{2} \le \Vert \Phi \Vert _{L^{6}(\mathbb {R}^{3})}\Vert \rho \Vert _{L^{\frac{6}{5}}(\Omega _{t})} \le \sqrt{A_{3}}\Vert \nabla \Phi \Vert _{L^{2}(\mathbb {R}^{3})}\Vert \rho \Vert _{L^{\frac{6}{5}}(\Omega _{t})}, \end{aligned}$$

(5.21)

where we have used the positive constant $A_3:=\frac{4}{3}\omega _{4}^{-\frac{2}{3}}>0$ that is the sharp constant for the Sobolev inequality in $\mathbb {R}^3$ (see Lemma A.1). Then it follows from (5.19) and (5.21) that

$$\begin{aligned}&\int _{a}^{b(t)}\Big (\int _{a}^{r} \rho \, z^{2}\textrm{d} z\Big ) \rho \, r \textrm{d} r=\frac{1}{2 \omega _{3}}\Vert \nabla \Phi \Vert _{L^{2}(\mathbb {R}^{3})}^{2}\le \frac{2}{3\omega _{3}}\omega _{4}^{-\frac{2}{3}}\Vert \rho \Vert _{L^{\frac{6}{5}}(\Omega _{t})}^2\nonumber \\&\quad \le \frac{2}{3\omega _{3}}\omega _{4}^{-\frac{2}{3}}\Big (\int _{\Omega _{t}}\rho ^{\frac{6(\gamma _2-1)}{5\gamma _2-6}} \big (\beta \rho +\rho e(\rho )\big )^{-\frac{1}{5\gamma _2-6}}\,\textrm{d}\textbf{x}\Big )^{\frac{5\gamma _2-6}{3(\gamma _2-1)}} \Big (\int _{\Omega _{t}}\big (\beta \rho +\rho e(\rho )\big )\,\textrm{d}\textbf{x}\Big )^{\frac{1}{3(\gamma _2-1)}}\nonumber \\&\quad \le \frac{2}{3}\omega _{4}^{-\frac{2}{3}}\omega _{3}^{\frac{4-3\gamma _2}{3(\gamma _2-1)}} \Big (\int _{\Omega _{t}}C_{\max }(\beta )\rho \,\textrm{d}\textbf{x}\Big )^{\frac{5\gamma _2-6}{3(\gamma _2-1)}} \Big (\int _{a}^{b(t)}\big (\beta \rho +\rho e(\rho )\big )\,r^2\textrm{d}r\Big )^{\frac{1}{3(\gamma _2-1)}}\nonumber \\&\quad =B_{\beta }M^{\frac{5\gamma _2-6}{3(\gamma _2-1)}}\Big (\int _{a}^{b(t)}\big (\beta \rho +\rho e(\rho )\big )\,r^2\textrm{d}r\Big )^{\frac{1}{3(\gamma _2-1)}}, \end{aligned}$$

(5.22)

where $B_{\beta }$ is the constant defined in (2.8).

When $\gamma _2>\frac{4}{3}$, i.e., $\frac{1}{3(\gamma _2-1)}<1$, it follows from (5.22) by taking $\beta =1$ that

$$\begin{aligned}&\int _{a}^{b(t)} \rho e(\rho )\, r^{2}\textrm{d} r - \int _{a}^{b(t)}\Big (\int _{a}^{r} \rho \, z^{2}\textrm{d} z\Big ) \rho \, r\textrm{d} r \nonumber \\&\quad \ge \int _{a}^{b(t)} \rho e(\rho )\, r^{2}\textrm{d} r -B_{1}M^{\frac{5 \gamma _2-6}{3(\gamma _2-1)}}\Big (\left( \omega _{3}^{-1} M\right) ^{\frac{1}{3(\gamma _2-1)}}+\Big (\int _{a}^{b(t)}\rho e(\rho )\,r^{2}\textrm{d}r\Big )^{\frac{1}{3(\gamma _2-1)}}\Big )\nonumber \\&\quad \ge \frac{1}{2} \int _{a}^{b(t)} \rho e(\rho )\, r^{2}\textrm{d} r-C(M), \end{aligned}$$

(5.23)

which, with (5.18), yields (5.17).

When $\gamma _{2}=\frac{4}{3}$, i.e., $\frac{1}{3(\gamma _2-1)}=1$. It has been proved in [18, Theorem 3.1] that there exists an optimal constant $C_{\min }=6\kappa _2M_\textrm{ch}^{-\frac{3}{2}}$ such that

$$\begin{aligned} \int _{a}^{b(t)}\Big (\int _{a}^{r}\, \rho \, z^{2}\textrm{d} z\Big ) \rho \, r \textrm{d} r=\frac{1}{2 \omega _{3}}\Vert \nabla \Phi \Vert _{L^{2}(\mathbb {R}^{3})}^{2}&\le \frac{C_{\min }}{2\omega _{3}}\Vert \rho \Vert _{L^{1}(\Omega _{t})}^{\frac{2}{3}}\Vert \rho \Vert _{L^{\frac{4}{3}}(\Omega _{t})}^{\frac{4}{3}}\nonumber \\&=\frac{C_{\min }}{2}M^{\frac{2}{3}}\int _{a}^{b(t)}\rho ^{\frac{4}{3}}\,r^2\textrm{d}r, \end{aligned}$$

(5.24)

which, with (5.20), yields

$$\begin{aligned}&\int _{a}^{b(t)} \rho e(\rho )\, r^{2}\textrm{d} r - \int _{a}^{b(t)}\Big (\int _{a}^{r} \rho \, z^{2}\textrm{d} z\Big ) \rho \, r\textrm{d} r\nonumber \\&\quad \ge \Big (3\kappa _2-\frac{C_{\min }}{2}M^{\frac{2}{3}}-C_{1}K^{-\min \{\gamma _2,\epsilon \}}\Big )\int _{a}^{b(t)}\rho ^{\frac{4}{3}}\,r^2\textrm{d}r-C(M,K). \end{aligned}$$

(5.25)

Since $M<M_\textrm{ch}$, we can always choose $K>\rho ^{*}$ large enough such that

$$\begin{aligned} 3\kappa _2-\frac{C_{\min }}{2}M^{\frac{2}{3}}-C_{1}K^{-\min \{\gamma _2,\epsilon \}}>0. \end{aligned}$$

Then one can deduce (5.16) for $\gamma _2=\frac{4}{3}$ from (5.18), (5.25), and the fact that $\rho ^{\gamma _2}\ge C\rho e(\rho )$.

When $\gamma _{2}\in (\frac{6}{5},\frac{4}{3})$, we define

$$\begin{aligned} F(s;\beta )=s-B_{\beta }M^{\frac{5\gamma _2-6}{3(\gamma _2-1)}}\left( \omega _{3}^{-1}\beta M+s\right) ^{\frac{1}{3(\gamma _2-1)}}\qquad \,\, \text {for }s\ge 0\text { and any fixed }\beta >0. \end{aligned}$$

A direct calculation shows that

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\textrm{d}F(s;\beta )}{\textrm{d}s}=1-\frac{1}{3(\gamma _2-1)} B_{\beta }M^{\frac{5 \gamma _2-6}{3(\gamma _2-1)}} \big (\omega _{3}^{-1}\beta M+s\big )^{\frac{4-3 \gamma _2}{3(\gamma _2-1)}},\\&\frac{\textrm{d}^2F(s;\beta )}{\textrm{d}s^2}=-\frac{4-3 \gamma _2}{9(\gamma _2-1)^{2}} B_{\beta } M^{\frac{5\gamma _2-6}{3(\gamma _2-1)}} \big (\omega _{3}^{-1}\beta M+s\big )^{\frac{7-6\gamma _2}{3(\gamma _2-1)}}, \end{aligned}\right. \end{aligned}$$

which yields that $\frac{\textrm{d}^2F(s;\beta )}{\textrm{d}s^2}<0$ for $s>0$ since $\gamma _2<\frac{4}{3}$. Thus, $F(s;\beta )$ is concave with respect to $s>0$. We denote

$$\begin{aligned} s_{*}(\beta )=\Big (\frac{ B_{\beta }}{3(\gamma _2-1)}\Big )^{-\frac{3(\gamma _2-1)}{4-3 \gamma _2}} M^{-\frac{5 \gamma _2-6}{4-3\gamma _2}}-\omega _{3}^{-1}\beta M, \end{aligned}$$

(5.26)

which is the critical point of F(s) satisfying $\frac{\textrm{d}F(s;\beta )}{\textrm{d}s}(s_{*}(\beta ))=0$. The maximum of $F(s;\beta )$ with respect to $s>0$ is

$$\begin{aligned} F(s_{*}(\beta );\beta )=(4-3\gamma _2)\Big (\frac{B_{\beta }}{3(\gamma _2-1)}\Big )^{-\frac{3(\gamma _2-1)}{4-3 \gamma _2}} M^{-\frac{5 \gamma _2-6}{4-3 \gamma _2}}-\omega _{3}^{-1}\beta M. \end{aligned}$$

(5.27)

It follows from the definition of $M_\textrm{c}^{\varepsilon ,b}$ that, if $M<M_\textrm{c}^{\varepsilon ,b}$, there exists $\beta _{0}>0$ such that $M<M_\textrm{c}^{\varepsilon ,b}(\beta _0)$. Then, from (5.26)–(5.27), we have

$$\begin{aligned}&F(s_{*}(\beta _{0});\beta _{0})>\frac{E_{0}^{\varepsilon ,b}}{\omega _{3}}, \end{aligned}$$

(5.28)

$$\begin{aligned}&s_{*}(\beta _{0})>\Big (\frac{ B_{ \beta _0}}{3(\gamma _2-1)}\Big )^{-\frac{3(\gamma _2-1)}{4-3 \gamma _2}} \big (M_\textrm{c}^{\varepsilon , b}(\beta _{0})\big )^{-\frac{5 \gamma _2-6}{4-3 \gamma _2}} -\omega _{3}^{-1}\beta _{0}M_\textrm{c}^{\varepsilon ,b}(\beta _{0})\nonumber \\&\qquad \quad =\frac{1}{4-3\gamma _2} \big (E_{0}^{\varepsilon , b}+\omega _{3}^{-1}\beta _{0}M_\textrm{c}^{\varepsilon ,b}(\beta _{0})\big ) -\omega _{3}^{-1}\beta _{0}M_\textrm{c}^{\varepsilon ,b}(\beta _{0})> \frac{E_{0}^{\varepsilon ,b}}{\omega _{3}}, \end{aligned}$$

(5.29)

where we have used that $\frac{1}{4-3\gamma _2}>\frac{5}{2}>1$ for $\gamma _2\in (\frac{6}{5},\frac{4}{3})$. Then, combining (5.18) and (5.22) with (5.28)–(5.29), we obtain

$$\begin{aligned} \begin{aligned}&F(\int _{a}^{b(t)} \rho e(\rho )\, r^{2}\textrm{d} r;\beta _{0}) \le \frac{E_{0}^{\varepsilon ,b}}{\omega _{3}}<F(s_{*}(\beta _0);\beta _{0}),\\&\int _{a}^{b} \big (\rho _{0}e(\rho _{0})\big )(r)\, r^{2}\textrm{d} r \le \frac{E_{0}^{\varepsilon ,b}}{\omega _{3}}<s_{*}(\beta _0). \end{aligned} \end{aligned}$$

(5.30)

Hence, due to the continuity of $\int _{a}^{b(t)}\big (\rho e(\rho )\big )(t,r)\,r^{2}\textrm{d}r$ with respect to t, the strict inequality:

$$\begin{aligned} \int _{a}^{b(t)}\big (\rho e(\rho )\big )(t,r)\,r^{2}\textrm{d} r<s_{*}(\beta _0) \end{aligned}$$

(5.31)

must hold. Otherwise, there exists $t_0>0$ such that $\int _{a}^{b(t_0)}\big (\rho e(\rho )\big )(t_{0},r)\,r^{2}\textrm{d}r=s_{*}(\beta _0)$, which yields

$$\begin{aligned} F\big (\int _{a}^{b(t_{0})} \big (\rho e(\rho )\big )(t_{0},r)\, r^{2}\textrm{d} r;\beta _{0}\big )=F(s_{*}(\beta _0);\beta _{0})>\frac{E_{0}^{\varepsilon ,b}}{\omega _{3}}. \end{aligned}$$

This contradicts (5.30). Thus, we prove (5.31) under condition (5.11).

Therefore, under condition (5.11), it follows from (5.26) and (5.31) that

$$\begin{aligned}&F(\int _{a}^{b(t)}\rho e(\rho )\,r^{2}\textrm{d}r;\beta _{0})\nonumber \\&\quad \ge \int _{a}^{b(t)}\rho e(\rho )\,r^{2}\textrm{d}r \nonumber \\&\qquad -B_{\beta _{0}}M^{\frac{5\gamma _2-6}{3(\gamma _2-1)}} \big (s_{*}(\beta _0)+\omega _{3}^{-1}\beta _{0}M\big )^{\frac{4-3\gamma _2}{3(\gamma _2-1)}} \Big (\int _{a}^{b(t)}\rho e(\rho )\,r^{2}\textrm{d}r+\omega _{3}^{-1}\beta _{0}M\Big )\nonumber \\&\quad =(4-3\gamma _2)\int _{a}^{b(t)}\rho e(\rho )\,r^{2}\textrm{d}r-3(\gamma _2-1)\omega _{3}^{-1}\beta _{0}M^{\frac{5}{3}}. \end{aligned}$$

(5.32)

Combining (5.18) and (5.25) with (5.32), we conclude (5.16). $\square$

Corollary 5.2

Under the assumptions of Lemma 5.1 and noting (3.5),

$$\begin{aligned} \int _{a}^{b(t)}\rho ^{\gamma _2}(t,r)\,r^{2}\textrm{d}r \le C \int _{a}^{b(t)}\big (\rho +\rho e(\rho )\big )(t,r)\,r^{2}\textrm{d}r\le C(M,E_0) \qquad \text { for }\,t\ge 0. \end{aligned}$$

Corollary 5.3

Under the assumptions of Lemma 5.1, it follows from (5.12), (5.16)–(5.17), and (5.19) that, for $t\ge 0$ and $r\ge 0$,

$$\begin{aligned}&\left| r^{2} \Phi _{r}(t, r)\right| \le \frac{M}{\omega _{3}},\\&\int _{a}^{b(t)}\Big (\int _{a}^{r} \rho (t, y)\, y^{2}\textrm{d} y\Big ) \rho (t, r)\, r\textrm{d} r +\Vert \Phi (t)\Vert _{L^{6}(\mathbb {R}^{3})}+\Vert \nabla \Phi (t)\Vert _{L^{2}(\mathbb {R}^{3})} \le C(M, E_{0}). \end{aligned}$$

For later use, we analyze the boundary value of density $\rho$. Using (5.14)$_1$ and (5.15), we have

$$\begin{aligned} \rho _{\tau }(\tau ,\frac{M}{\omega _3})=-\frac{1}{\varepsilon }\,P(\tau ,\frac{M}{\omega _3})\le 0, \end{aligned}$$

(5.33)

which yields that $\rho (\tau , \frac{M}{\omega _3})\le \rho _{0}(\frac{M}{\omega _3})$. In the Eulerian coordinates, it is equivalent to

$$\begin{aligned} \rho (t,b(t))\le \rho _0(b). \end{aligned}$$

(5.34)

Moreover, noting (5.8) and $b\ge (\rho _{*})^{-\gamma _1/3}$, we see that $\rho (t,b(t))\le \rho _0(b)\le \rho _{*}$ for all $t\ge 0$. From (3.2)$_1$ and (5.33), there exists a positive constant $\tilde{C}$ depending only on $(\gamma _1, \kappa _1)$ such that $\rho _{\tau }(\tau ,\frac{M}{\omega _3})=-\frac{1}{\varepsilon }\, P(\tau ,\frac{M}{\omega _3}) \ge -\frac{\tilde{C}}{\varepsilon }\big (\rho (\tau ,\frac{M}{\omega _3})\big )^{\gamma _1}$, which implies

$$\begin{aligned} \rho (\tau ,\frac{M}{\omega _3})\ge \rho _{0}(\frac{M}{\omega _3}) \Big (1+\frac{\tilde{C}(\gamma _1-1)}{\varepsilon }\big (\rho _{0}(\frac{M}{\omega _3})\big )^{\gamma _1-1}\tau \Big )^{-\frac{1}{\gamma _1-1}}. \end{aligned}$$

Therefore, in the Eulerian coordinates,

$$\begin{aligned} \rho (t,b(t))\ge \rho _{0}(b)\Big (1+\frac{\tilde{C}(\gamma _1-1)}{\varepsilon } (\rho _{0}(b))^{\gamma _1-1}t\Big )^{-\frac{1}{\gamma _1-1}} \qquad \text{ for } \, t\ge 0. \end{aligned}$$

(5.35)

Lemma 5.4

(BD-type entropy estimate). Under the conditions of Lemma 5.1, for any given $T>0$,

$$\begin{aligned}&\varepsilon ^2\int _{a}^{b(t)}\big |\left( \sqrt{\rho }\right) _{r}\big |^2\,r^{2}\textrm{d}r +\varepsilon \int _{0}^{t}\int _{a}^{b(s)}\frac{P'(\rho )}{\rho }|\rho _{r}|^2\,r^{2}\textrm{d}r\textrm{d}s +\frac{1}{3}\,P(\rho (t,b(t)))\,b(t)^3\nonumber \\&\quad +\frac{1}{3\varepsilon }\int _{0}^t \big (P(\rho )P'(\rho )\big )(s,b(s))\,b(s)^3\,\textrm{d}s \le C(E_0,M,T)\qquad \,\, \text{ for } \, t\in [0,T]. \end{aligned}$$

(5.36)

Proof

We divide the proof into three steps.

1. Using (2.3) and similar calculations as in the proof [10, Lemma 3.3], we have

$$\begin{aligned}&\int _{a}^{b(t)}\Big (\frac{1}{2}\big (u+\varepsilon \frac{\rho _{r}}{\rho }\big )^2\rho +\rho e(\rho )\Big ) \,r^2\textrm{d}r -\int _{a}^{b(t)}\Big (\int _{a}^{r}\rho (t,y)\,y^{2}\textrm{d}y\Big )\rho \, r\textrm{d}r\\&\qquad +\varepsilon \int _{0}^{t}\int _{a}^{b(s)}\frac{P'(\rho )}{\rho }\rho _{r}^2\,r^{2}\textrm{d}r\textrm{d}s +\frac{1}{3}P(\rho (t,b(t)))\,b(t)^3\\&\qquad +\frac{1}{3\varepsilon }\int _{0}^{t}\big (P(\rho )P'(\rho )\big )(s,b(s))\,b(s)^3\textrm{d}s \\&\quad =\int _{a}^{b}\Big (\frac{1}{2}\big (u_0+\varepsilon \frac{\rho _{0,r}}{\rho }\big )^2+e(\rho _0)\Big )\rho _{0} \,r^{2}\textrm{d}r -\int _{a}^{b}\Big (\int _{a}^{r}\rho _{0}(y)\,y^{2}\textrm{d}y\Big )\rho _0(r)\,r\textrm{d}r\\&\qquad +\frac{1}{3}P(\rho _0\left( b\right) )b^3 +\varepsilon \int _{0}^{t}\int _{a}^{b(s)}\rho ^2\,r^{2}\textrm{d}r\textrm{d}s -\frac{M\varepsilon }{\omega _3}\int _{0}^{t}\rho (s,b(s))\,\textrm{d}s, \end{aligned}$$

which, with Lemma 5.1, yields

$$\begin{aligned}&\varepsilon ^2\int _{a}^{b(t)}|\left( \sqrt{\rho }\right) _{r}|^2\,r^{2}\textrm{d}r +\varepsilon \int _{0}^{t}\int _{a}^{b(s)}\frac{P'(\rho )}{\rho }|\rho _{r}|^2\,r^{2}\textrm{d}r\textrm{d}s\nonumber \\&\qquad +\frac{1}{3}P(\rho (t,b(t)))\,b(t)^3+\frac{1}{3\varepsilon }\int _{0}^t\big (P(\rho )P'(\rho )\big )(s,b(s))\,b(s)^3\textrm{d}s \nonumber \\&\quad \le C(E_0,M)+\frac{1}{3}P(\rho _0(b))b^3 +\varepsilon \int _{0}^{t}\int _{a}^{b(s)}\rho ^2\,r^{2}\textrm{d}r\textrm{d}s -\frac{M\varepsilon }{\omega _3}\int _{0}^{t}\rho (s,b(s))\,\textrm{d}s. \end{aligned}$$

(5.37)

2. For the second term on the RHS of (5.37), it follows from (5.8) and (3.2)$_1$ that

$$\begin{aligned} \frac{1}{3}P(\rho _0(b))b^3\le C. \end{aligned}$$

(5.38)

For the last term on the RHS of (5.37), using (5.34), we have

$$\begin{aligned} \Big |\frac{M\varepsilon }{\omega _{3}} \int _{0}^{t} \rho (s, b(s))\,\textrm{d}s\Big | \le C(M) \rho _{0}(b) T \le C(M, T). \end{aligned}$$

(5.39)

3. To close the estimates, we need to control the third term on the RHS of (5.37), that is,

$$\begin{aligned} \varepsilon \int _{0}^{t}\int _{a}^{b(s)}\rho ^2\,r^{2}\textrm{d}r\textrm{d}s=\frac{\varepsilon }{\omega _3}\int _{0}^{t}\Vert \rho (s,\cdot )\Vert _{L^2(\Omega _{s})}^2\,\textrm{d}s. \end{aligned}$$

We divide the estimate of the above term into the following two cases:

Case 1. For $\gamma _2\ge 2$, it follows from Corollary 5.2 that

$$\begin{aligned} \varepsilon \int _{0}^{t}\int _{a}^{b(s)}\rho ^2\,r^{2}\textrm{d}r\textrm{d}s \le \varepsilon \int _{0}^{t}\int _{a}^{b(s)}(\rho +\rho ^{\gamma _2})\,r^{2}\textrm{d}r\textrm{d}s\le C(E_0,M,T). \end{aligned}$$

(5.40)

Case 2. For $\displaystyle \gamma _2\in (\frac{6}{5},2)$, then $3\gamma _2>2$. A direct calculation shows that

$$\begin{aligned} \varepsilon \int _{0}^{t}\int _{a}^{b(s)}\rho ^2\textbf{I}_{\{\rho \le 2\rho ^{*}\}}\,r^2\textrm{d}r\textrm{d}s \le 2\varepsilon \rho ^{*}\int _{0}^{t}\int _{a}^{b(s)}\rho \, r^{2}\textrm{d}r\textrm{d}s\le C(M,\rho ^{*}). \end{aligned}$$

(5.41)

Denote $\sqrt{F(\rho )}:=\int _{0}^{\rho }\sqrt{\frac{P'(s)}{s}}\,\textrm{d}s$. Then it follows from (3.3)$_2$ that

$$\begin{aligned} \sqrt{F(\rho )}&\ge \big (1-2^{-\frac{\gamma _2}{2}}\big ) \frac{2\sqrt{(1-\mathfrak {a}_0)\kappa _2\gamma _2}}{\gamma _2}\rho ^{\frac{\gamma _2}{2}} :=C(\gamma _2)^{-\frac{\gamma _2}{2\overline{\vartheta }}}\rho ^{\frac{\gamma _2}{2}}\qquad \text {for }\rho \in [2\rho ^{*},\infty ), \end{aligned}$$

which, with Corollary 5.2, implies that, for $\overline{\vartheta }=\frac{3(2-\gamma _2)}{4}$,

$$\begin{aligned} \Vert \rho \textbf{I}_{\{\rho \ge 2\rho ^{*}\}}\Vert _{L^2(\Omega _{t})}&\le \Vert \rho \textbf{I}_{\{\rho \ge 2\rho ^{*}\}}\Vert _{L^{3\gamma _2}(\Omega _{t})}^{\overline{\vartheta }} \Vert \rho \textbf{I}_{\{\rho \ge 2\rho ^{*}\}}\Vert _{L^{\gamma _2}(\Omega _{t})}^{1-\overline{\vartheta }}\nonumber \\&\le C(\gamma _2)\Vert \sqrt{F(\rho )}\Vert _{L^{6}(\Omega _{t})}^{\frac{2\overline{\vartheta }}{\gamma _2}}\Vert \rho \Vert _{L^{\gamma _2}(\Omega _{t})}^{1-\overline{\vartheta }}. \end{aligned}$$

(5.42)

For $B_{R}({\textbf {0}})\subset \mathbb {R}^3$, the following Sobolev’s inequality holds:

$$\begin{aligned} \Vert f\Vert _{L^{6}(B_{R}(\textbf{0}))} \le C\big (\Vert \nabla f\Vert _{L^{2}(B_{R}(\textbf{0}))}+ R^{-1}\Vert f\Vert _{L^{2}(B_{R}(\textbf{0}))}\big ). \end{aligned}$$

(5.43)

It follows from (5.13) and Corollary 5.2 that

$$\begin{aligned} \begin{aligned} \frac{M}{\omega _{3}}&=\int _{a}^{b(t)} \rho (t, r)\, r^{2}\textrm{d} r \le \Big (\int _{a}^{b(t)} \rho ^{\gamma _2}\,r^{2} \textrm{d} r\Big )^{\frac{1}{\gamma _2}} \Big (\int _{a}^{b(t)} r^{2} \textrm{d} r\Big )^{1-\frac{1}{\gamma _2}}\\&\le C b(t)^{\frac{3(\gamma _2-1)}{\gamma _2}} \Big (\int _{a}^{b(t)} \rho ^{\gamma _2}\, r^{2} \textrm{d} r\Big )^{\frac{1}{\gamma _2}}, \end{aligned} \end{aligned}$$

which yields

$$\begin{aligned} b(t)^{-1} \le C M^{-\frac{\gamma _2}{3(\gamma _2-1)}}\Big (\int _{a}^{b(t)} \rho ^{\gamma _2}\, r^{2}\textrm{d} r\Big )^{\frac{1}{3(\gamma _2-1)}} \le C\left( M, E_{0}\right) . \end{aligned}$$

(5.44)

Using (3.2)$_2$–(3.3)$_2$ leads to $F(\rho )\le C(\rho +\rho ^{\gamma _2})$, which, with (5.43)–(5.44) and Corollary 5.2, implies

$$\begin{aligned} \big \Vert \sqrt{F(\rho )}\big \Vert _{L^{6}(\Omega _{t})}&\le C\left( \big \Vert \nabla (\sqrt{F(\rho )})\big \Vert _{L^2(\Omega _{t})} +b(t)^{-1}\big \Vert \sqrt{F(\rho )}\big \Vert _{L^2(\Omega _{t})}\right) \nonumber \\&\le C\Big (\int _{a}^{b(t)}\frac{P'(\rho )}{\rho }|\rho _{r}|^2\,r^{2}\textrm{d}r\Big )^{\frac{1}{2}} +C(M,E_0)\Big (\int _a^{b(t)}F(\rho )\,r^{2}\textrm{d}r\textrm{d}t\Big )^{\frac{1}{2}}\nonumber \\&\le C(M,E_0)\Big (1+\big (\int _{a}^{b(t)}\frac{P'(\rho )}{\rho }|\rho _{r}|^2\,r^{2}\textrm{d}r\big )^{\frac{1}{2}}\Big ). \end{aligned}$$

(5.45)

Substituting (5.45) into (5.42), we obtain

$$\begin{aligned} \varepsilon \int _{0}^{t}\int _{a}^{b(s)}\rho ^2r^{2}\textbf{I}_{\{\rho \ge 2\rho ^{*}\}}\,\textrm{d}r\textrm{d}s&\le C(M,E_0,T)\varepsilon \Big (1+\big (\int _{0}^{t}\int _{a}^{b(s)}\frac{P'(\rho )}{\rho }|\rho _{r}|^2\,r^{2} \textrm{d}r\textrm{d}s\big )^{\frac{2\overline{\vartheta }}{\gamma _2}}\Big )\nonumber \\&\le C(M,E_0,T)+\frac{\varepsilon }{2}\int _{0}^{t}\int _{a}^{b(s)}\frac{P'(\rho )}{\rho }|\rho _{r}|^2\,r^{2}\textrm{d}r\textrm{d}s, \end{aligned}$$

(5.46)

where we have used $\frac{2\bar{\vartheta }}{\gamma _2}\in (0,1)$ for $\gamma _2>\frac{6}{5}$. Finally, substituting (5.38)–(5.41) and (5.46) into (5.37), we conclude (5.36). $\square$

In order to take the limit: $b\rightarrow \infty$, we need to make sure that domain $\Omega _{T}$ can be expanded to $[0,T]\times \mathbb {R}_{+}$ for fixed $\varepsilon >0$: $\lim \limits _{b\rightarrow \infty }b(t)=\infty$.

Lemma 5.5

(Expanding of domain $\Omega _{T}$). Given $T>0$ and $\varepsilon \in (0,\varepsilon _0]$, there exists $C_1(M,E_0,T,\varepsilon )>0$ such that, if $b\ge C_1(M,E_0,T,\varepsilon )$,

$$\begin{aligned} b(t)\ge \frac{1}{2}b\qquad \,\, \text {for }\, t\in [0,T]. \end{aligned}$$

(5.47)

Proof

Noting $b(0)=b$ and the continuity of b(t), we first make the a priori assumption:

$$\begin{aligned} b(t)\ge \frac{1}{2}b. \end{aligned}$$

(5.48)

Integrating (5.3) over [0, t] yields

$$\begin{aligned} b(t)=b+\int _{0}^tu(s,b(s))\,\textrm{d}s. \end{aligned}$$

(5.49)

It follows from (5.35), (5.48), and Lemma 5.1 that

$$\begin{aligned} \int _{0}^{t}|u(s,b(s))|\,\textrm{d}s&\le \frac{C}{\sqrt{\varepsilon }}\Big (\int _{0}^{t}\varepsilon (\rho u^2 r)(s,b(s))\,\textrm{d}s\Big )^{\frac{1}{2}} \Big (\int _{0}^{t}\frac{1}{\rho (s,b(s))b(s)}\,\textrm{d}s\Big )^{\frac{1}{2}}\nonumber \\&\le C(M,E_0)\varepsilon ^{-\frac{1}{2}}\Big (\int _{0}^{t}\frac{(1+\tilde{C}(\gamma _1-1)\varepsilon ^{-1}\rho _{*}^{\gamma _1-1}s)^{\frac{1}{\gamma _1-1}}}{\rho _0(b)b} \,\textrm{d}s\Big )^{\frac{1}{2}}\nonumber \\&\le C(M,E_0,T,\rho _{*},\gamma _1,\gamma _2,\varepsilon )\rho _{0}(b)^{-\frac{1}{2}}b^{-\frac{1}{2}}. \end{aligned}$$

(5.50)

We take $C_{1}(M,E_0,T,\varepsilon ):= \max \big \{\rho _{*}^{-\frac{\gamma _1}{3}}, (4C(M,E_0,T,\rho _{*},\gamma _1,\gamma _2,\varepsilon ))^{\frac{2}{\alpha }}, \mathcal {B}(\varepsilon )\big \},$ which, with (5.8) and (5.50), implies that

$$\begin{aligned} \Big \vert \int _{0}^{t}u(s,b(s))\,\textrm{d}s\Big \vert \le \int _{0}^{t}|u(s,b(s))|\,\textrm{d}s\le \frac{1}{4}b, \end{aligned}$$

(5.51)

provided that $b\ge C_1(M,E_0,T,\varepsilon )$. Combining (5.51) with (5.49), we have

$$\begin{aligned} b(t)\ge \frac{3}{4}b. \end{aligned}$$

(5.52)

Thus, we have closed the a priori assumption (5.48). Finally, using (5.52) and the continuity argument, we can conclude (5.47). $\square$

5.2 Higher integrability of the density and the velocity

As implied in [13], the higher integrabilities of the density and the velocity are important for the $L^p$ compensated compactness framework. However, for the general pressure law, due to the lack of an explicit formula for the entropy kernel, for the special entropy pair $(\eta ^{\psi },q^{\psi })$ by taking the test function $\psi =\frac{1}{2} s|s|$ in (2.15)–(2.16), we can not obtain that $q^{\psi } \gtrsim \rho |u|^3 + \rho ^{\gamma +\theta }$ in general. To derive the higher integrability of the velocity, we use the special entropy pair constructed in Lemma 4.1, at the cost of the higher integrability of the density over domain $[0,T]\times [d, b(t)]$ for some $d>0$. Since $b(t)\rightarrow \infty$ as $b\rightarrow \infty$, we indeed need the higher integrability of the density on the unbounded domain. We point out that this is different from the case of [10] in which only the higher integrability on the bounded domain $[0,T]\times [d,D]$ for any given $0<d<D<\infty$ is needed.

Lemma 5.6

(Higher integrability on the density). Let $(\rho ,u)$ be a smooth solution of (5.1)–(5.6). Then, under the assumption of Lemma 5.1, for any given $d>2b^{-1}>0$,

$$\begin{aligned} \int _{0}^{T}\int _{d}^{b(t)}\rho P(\rho )\,r^{2}\textrm{d}r\textrm{d}t\le C(d,M,E_0,T). \end{aligned}$$

(5.53)

Proof

Let $\omega (r)$ be a smooth function with ${\text {supp}}\omega \subset (\frac{d}{2},\infty )$ and $\omega (r)=1$ for $r\in [d,\infty )$. Multiplying (5.1)$_2$ by $w(y)y^{2}$, we have

$$\begin{aligned}&(y^{2}\rho u \omega )_{t}+(y^{2}\rho u^2 \omega )_{y}+(y^{2}P(\rho )\omega )_{y} -\omega _{y}\big (y^{2}\rho u^2+y^2P(\rho )\big ) +\rho \omega \int _{a}^{r}\rho \, z^{2}\textrm{d}z\nonumber \\&\quad =2yP(\rho )\omega +\varepsilon (y^{2}\rho u_{y}\omega )_{y}-\varepsilon \omega _{y}y^{2}\rho u_{y}-2\varepsilon \rho u\,\omega . \end{aligned}$$

(5.54)

Integrating (5.54) with respect to y from $\frac{d}{2}$ to r and then multiplying the equation by $\rho (t,r)$ yield

$$\begin{aligned}&r^{2}\rho (t,r)P(\rho (t,r))\omega (r)\nonumber \\&\quad =-\rho \frac{\textrm{d}}{\textrm{d}t}\int _{\frac{d}{2}}^{r}\rho u\,\omega \, y^2\textrm{d}y -r^{2}\rho ^2 u^2\omega (r)+\rho \int _{\frac{d}{2}}^{r}\omega _{y}\rho u\,y^2\textrm{d}y\nonumber \\&\qquad +\rho \int _{\frac{d}{2}}^{r}\omega _{y}P(\rho )\,y^2\textrm{d}y+2\rho \int _{\frac{d}{2}}^{r}P(\rho )\,\omega \,y\textrm{d}y -\rho \int _{\frac{d}{2}}^{r}\rho \omega \Big (\int _{a}^{y}\rho \, z^{2}\textrm{d}z\Big )\,\textrm{d}r \nonumber \\&\qquad +\varepsilon r^{2}\rho ^2 u_{r}\omega (r)-\varepsilon \rho \int _{\frac{d}{2}}^{r}\omega _y\rho u_{y}\,y^2\textrm{d}y -2\varepsilon \rho \int _{\frac{d}{2}}^{r}\rho u\,\omega \, \textrm{d}y. \end{aligned}$$

(5.55)

Using (5.1)$_1$, we have

$$\begin{aligned} \rho \frac{\textrm{d}}{\textrm{d}t}\int _{\frac{d}{2}}^{r}\rho u\omega \,y^2\textrm{d}y&=\Big (\rho \int _{\frac{d}{2}}^{r}\rho u\omega \,y^2\textrm{d}y\Big )_{t} +\Big (\rho u\int _{\frac{d}{2}}^{r}\rho u\omega \,y^2\textrm{d}y\Big )_{r}\\&\quad -\rho ^2u^2\omega (r)r^2+\frac{2}{r}\rho u\int _{\frac{d}{2}}^{r}\rho u\omega \,y^2\textrm{d}y, \end{aligned}$$

which, with (5.55), yields that

$$\begin{aligned}&r^{2}\rho (t,r)P(\rho (t,r))\omega (r)\nonumber \\&\quad =-\Big (\rho \int _{\frac{d}{2}}^{r}\rho u\omega \,y^2\textrm{d}y\Big )_{t} -\Big (\rho u\int _{\frac{d}{2}}^{r}\rho u\omega \,y^2\textrm{d}y\Big )_{r}\nonumber \\&\qquad -\frac{2}{r}\rho u\int _{\frac{d}{2}}^{r}\rho u\omega \,y^2\textrm{d}y +\rho \int _{\frac{d}{2}}^{r}\omega _{y}\rho u^2\,y^2\textrm{d}y\nonumber \\&\qquad +\rho \int _{\frac{d}{2}}^{r}\omega _{y}P(\rho )\,y^2\textrm{d}y +2\rho \int _{\frac{d}{2}}^{r}P(\rho )\omega \,y\textrm{d}y +\varepsilon \rho ^2 u_{r}\omega (r)r^2-\varepsilon \rho \int _{\frac{d}{2}}^{r}\rho u_{y}\omega _y\,y^{2}\textrm{d}y\nonumber \\&\qquad -2\varepsilon \rho \int _{\frac{d}{2}}^{r}\rho u\omega \,\textrm{d}y-\rho \int _{\frac{d}{2}}^{r}\rho \omega \, \Big (\int _{a}^{y}\rho \, z^{2}\textrm{d}z\Big )\,\textrm{d}y. \end{aligned}$$

(5.56)

Multiplying (5.56) by $\omega (r)$ leads to

$$\begin{aligned}&r^{2}\rho (t,r)P(\rho (t,r))\omega ^2(r)\nonumber \\&\quad =-\Big (\rho \omega (r)\int _{\frac{d}{2}}^{r}\rho u\omega (y)\,y^2\textrm{d}y\Big )_{t} -\Big (\rho u\omega (r)\int _{\frac{d}{2}}^{r}\rho u\omega (y)\,y^2\textrm{d}y\Big )_{r} +\omega _{r}\rho u\int _{\frac{d}{2}}^{r}\rho u\omega \,y^2\textrm{d}y\nonumber \\&\qquad -\frac{2}{r}\rho u\omega (r)\int _{\frac{d}{2}}^{r}\rho u\omega \,y^2\textrm{d}y +\rho \omega (r)\int _{\frac{d}{2}}^{r}\rho u\omega _{y}\,y^2\textrm{d}y +\rho \omega (r)\int _{\frac{d}{2}}^{r}P(\rho )\,\omega _{y}\,y^2\textrm{d}y \nonumber \\&\qquad +2\rho \omega (r)\int _{\frac{d}{2}}^{r}P(\rho )\omega \,y\textrm{d}y -\rho \omega (r)\int _{\frac{d}{2}}^{r}\rho \omega \Big (\int _{a}^{y}\rho \, z^{2}\textrm{d}z\Big )\,\textrm{d}y -\varepsilon \rho \omega (r)\int _{\frac{d}{2}}^{r}\rho u_{y}\,\omega _{y}\,y^2\textrm{d}y \nonumber \\&\qquad -2\varepsilon \rho \omega (r)\int _{\frac{d}{2}}^{r}\rho u\,\omega \,\textrm{d}y +\varepsilon r^{2}\rho ^2 u_{r}\omega ^2(r) := \sum \limits _{i=1}^{11}I_{i}. \end{aligned}$$

(5.57)

Using Lemma 5.1 and (5.13), we have

$$\begin{aligned} \begin{aligned}&\Big |\int _{\frac{d}{2}}^{r}\big ((\rho u+P(\rho ))\omega (y)+\varepsilon \rho u_{y}\omega _y\big )\,y^2\textrm{d}y\Big |\\&\quad \le C\int _{a}^{b(t)}\big (\rho u^2+\rho +\rho ^{\gamma _2}\big )\omega (y)\,y^2\textrm{d}y\\&\qquad +\varepsilon \int _{a}^{b(t)}\rho (u_{y}^2+1)|\omega _y|\,y^{2}\textrm{d}y \le C(M,E_0,\Vert \omega \Vert _{C^1}), \end{aligned} \end{aligned}$$

which yields

$$\begin{aligned}&\Big |\int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}I_{i}\,\textrm{d}r\textrm{d}t\Big |\le C(M,E_0,T,\Vert \omega \Vert _{C^1})(d^{-2}+d^{-4}) \quad \,\, \text {for }\, i=3,4,\cdots ,10, \end{aligned}$$

(5.58)

$$\begin{aligned}&\Big |\int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}I_1\,\textrm{d}r\textrm{d}t\Big | =\Big | \int _{\frac{d}{2}}^{b(t)}\rho (T,r)\omega (r) \Big (\int _{\frac{d}{2}}^{r}y^{2}\rho (T,y)u(T,y)\omega (y)\,\textrm{d}y\Big )\textrm{d}r\Big |\nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad +\Big |\int _{\frac{d}{2}}^{b(t)}\rho (0,r)\omega (r) \Big (\int _{\frac{d}{2}}^{r}y^{2}\rho (0,y)u(0,y)\omega (y)\,\textrm{d}y\Big )\textrm{d}r\Big |\nonumber \\&\qquad \qquad \qquad \qquad \qquad \le C(M,E_0,\Vert \omega \Vert _{C^1})d^{-2}. \end{aligned}$$

(5.59)

For $I_2$, using (5.8), (5.34), (5.51), and $b\gg 1$, we have

$$\begin{aligned} \Big |\int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}I_{2}\,\text {d}r\text {d}t\Big |&=\Big |\int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}\Big (\rho u\omega (r)\int _{\frac{d}{2}}^{r}\rho u\omega (y)\,y^2\text {d}y\Big )_{r}\,\text {d}r\text {d}t\Big |\nonumber \\ {}&\le \Big |\int _{0}^{T}(\rho u)(t,b(t))\Big (\int _{\frac{d}{2}}^{b(t)}\rho u\omega (y)\,y^2\text {d}y\Big )\,\text {d}t\Big |\nonumber \\ {}&\le C(E_0,M)b^{-3+\frac{1}{2}}b\le C(E_0,M). \end{aligned}$$

(5.60)

For $I_{11}$, we obtain

$$\begin{aligned} \Big | \int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}I_{11}\,\textrm{d}r\textrm{d}t\Big |&=\varepsilon \Big \vert \int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}\rho ^2 u_{r}\omega ^2\,r^2\textrm{d}r\textrm{d}t\Big \vert \nonumber \\&\le \varepsilon \int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}\rho ^3\omega ^2\,r^{2}\textrm{d}r\textrm{d}t+C(M,E_0,T,\Vert \omega \Vert _{C^1}). \end{aligned}$$

(5.61)

We divide the estimate of $\int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}\varepsilon \rho ^3\omega ^2\,r^{2}\textrm{d}r\textrm{d}t$ into two cases:

Case 1. $\gamma _2\in (\frac{6}{5},2)$: For $t\in [0,T]$, denoting $A(t):=\{r\in [\frac{d}{2},b(t)]\,:\, \rho (t,r)\ge \rho ^{*}\},$ then it follows from (5.13) that $|A(t)|\le C(d,\rho ^{*})M$. For any $r\in A(t)$, let $r_{0}$ be the closest point to r so that $\rho (t,r_{0})=\rho ^{*}$ with $|r-r_{0}|\le |A(t)|\le C(d,\rho ^{*})M$. Then, for any smooth function $f(\rho )$,

$$\begin{aligned} \sup _{r\in A(t)}\big (f(\rho (t,r))\omega ^2(r)\big )&\le f(\rho (t,r_0))\omega ^2(r_0) +\Big |\int _{r_0}^{r}\partial _y\big (f(\rho (t,y))\omega ^{2}(y)\big )\,\textrm{d}y\Big |\\&\le C(\Vert \omega \Vert _{C^1})|f(\rho ^{*})| +\int _{A(t)}\big |\partial _y\big (f(\rho (t,y))\omega ^{2}(y)\big )\big |\,\textrm{d}y. \end{aligned}$$

Recalling (3.3) and (3.5), we notice that $P(\rho )\cong \rho ^{\gamma _2}$ and $e(\rho )\cong \rho ^{\gamma _2-1}$ for any $r\in A(t)$. Then

$$\begin{aligned}&\varepsilon \int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}\rho ^3\omega ^2\,r^{2}\textrm{d}r\textrm{d}t\nonumber \\&\quad = \varepsilon \int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}\rho ^3\textbf{I}_{\{\rho \le \rho ^{*}\}}\omega ^2\,r^2\textrm{d}r\textrm{d}t + \varepsilon \int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}\rho ^3 \textbf{I}_{\{\rho \ge \rho ^{*}\}}\omega ^2\,r^{2}\textrm{d}r\textrm{d}t\nonumber \\&\quad \le C(M,E_0,\rho ^{*},T)+ C(M,E_0)\, \varepsilon \int _{0}^{T}\Big (\int _{\frac{d}{2}}^{b(t)}\rho e(\rho )\, r^2 dr\Big ) \sup _{r\in A(t)} \Big (\frac{\rho ^2}{e(\rho )}\omega ^2\Big )\textrm{d}t\nonumber \\&\quad \le C(M,E_0,\rho ^{*},T)+ C(M,E_0)\, \varepsilon \int _{0}^{T}\int _{A(t)}\Big \vert \Big (\frac{\rho ^2}{e(\rho )}\omega ^2\Big )_{r}(t,r)\Big \vert \,\textrm{d}r\textrm{d}t\nonumber \\&\quad \le C(M,E_0)\,\varepsilon \int _{0}^{T}\int _{A(t)}\Big (\big (\frac{2\rho }{e(\rho )}-\frac{P(\rho )}{e(\rho )^2}\big ) |\rho _{r}|\omega ^2+\frac{\rho ^2}{e(\rho )}\omega |\omega _{r}|\Big )\,\textrm{d}r\textrm{d}t\nonumber \\&\qquad +C(M,E_0,\rho ^{*},T). \end{aligned}$$

(5.62)

A direct calculation shows that

$$\begin{aligned}&\int _{0}^{T}\int _{A(t)}\varepsilon \Big (\frac{2\rho }{e(\rho )}-\frac{P(\rho )}{e(\rho )^2}\Big ) |\rho _{r}|\,\omega ^2\,\textrm{d}r\textrm{d}t\nonumber \\&\quad \le \!\int _{0}^{T}\int _{A(t)}\varepsilon \frac{P'(\rho )}{\rho }|\rho _{r}|^2\omega ^2\,r^{2}\textrm{d}r\textrm{d}t \!+\!\int _{0}^{T}\int _{A(t)}\varepsilon \Big (\frac{2\rho }{e(\rho )}-\frac{P(\rho )}{e(\rho )^2}\Big )^2\frac{\rho }{P'(\rho )}\omega ^2\,r^{-2}\,\textrm{d}r\textrm{d}t \nonumber \\&\quad \le C(M,E_0,T)+\int _{0}^{T}\int _{A(t)}\varepsilon \rho ^{6-3\gamma _2}\omega ^2\,r^{-2}\,\textrm{d}r\textrm{d}t\nonumber \\&\quad \le C(M,E_0,T)+\varepsilon C(M,E_{0})^{-1}\int _{0}^{T}\int _{A(t)}\rho ^3\omega ^2\,r^{2}\,\textrm{d}r\textrm{d}t\nonumber \\&\qquad +C(M,E_0)\varepsilon \int _{0}^{T}\int _{A(t)}(r^{2})^{-\frac{3-\gamma _2}{\gamma _2-1}}\omega ^2\,\textrm{d}r\textrm{d}t\nonumber \\&\quad \le C(M,E_0,T)+\varepsilon C(M,E_0)^{-1}\int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}\rho ^3\omega ^2\,r^{2}\textrm{d}r\textrm{d}t. \end{aligned}$$

(5.63)

$$\begin{aligned}&\int _{0}^{T}\int _{A(t)}\varepsilon \frac{\rho ^2}{e(\rho )}\omega |\omega _{r}|\,\textrm{d}r\textrm{d}t \le \int _{0}^{T}\Big (\varepsilon \sup _{r\in A(t)}(\rho \omega )(t,r)\int _{A(t)}\frac{\rho }{e(\rho )}|\omega _{r}|\,\textrm{d}r\Big )\,\textrm{d}t\nonumber \\&\quad \le C(\rho ^{*})\int _{0}^{T}\Big (\varepsilon \sup _{r\in A_{t}}(\rho \omega )(t,r)\int _{A(t)}\rho |\omega _{r}|\,\textrm{d}r\Big )\,\textrm{d}t\nonumber \\&\quad \le C(\rho ^{*},M,\Vert \omega \Vert _{C^1})d^{-2}\int _{0}^{T}\varepsilon \sup _{r\in A(t)}(\rho \omega )(t,r)\,\textrm{d}t\nonumber \\&\quad \le C(\rho ^{*},M,\Vert \omega \Vert _{C^1},T)d^{-2}+C(\rho ^{*},M,\Vert \omega \Vert _{C^1})d^{-2}\int _{0}^{T}\int _{A(t)}\varepsilon \big (|\rho _{r}|\omega +\rho |\omega _{r}|\big )\,\textrm{d}r\textrm{d}t\nonumber \\&\quad \le C(\rho ^{*},M,\Vert \omega \Vert _{C^1},T)d^{-2}\Big (1+\int _{0}^{T}\int _{A(t)}\varepsilon \Big (\frac{P'(\rho )}{\rho }|\rho _{r}|^2\omega +\rho |\omega _{r}|+\rho ^{2-\gamma _2}\omega \Big )\,\textrm{d}r\textrm{d}t\Big )\nonumber \\&\quad \le C(\rho ^{*},M,\Vert \omega \Vert _{C^1},T)d^{-2}+C(\rho ^{*},M,E_0,T,\Vert \omega \Vert _{C^1})d^{-4}. \end{aligned}$$

(5.64)

Combining (5.62)–(5.64), we obtain that, for $\gamma _2\in (\frac{6}{5},2)$,

$$\begin{aligned} \varepsilon \int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}\rho ^3\omega ^2\,r^{2}\textrm{d}r\textrm{d}t \le C(M,E_0,\rho ^{*},T,\Vert \omega \Vert _{C^1}) (1+d^{-4}). \end{aligned}$$

(5.65)

Case 2. $\gamma _2\in [2,3)$: Using (5.13) and the same argument as for (5.64), we have

$$\begin{aligned} \varepsilon \int _{0}^{T}\int _{\frac{d}{2}}^{b(t)}\rho ^3\omega ^2\,r^{2}\textrm{d}r\textrm{d}t&\le C(M)\int _{0}^{T}\varepsilon \sup _{r\in [\frac{d}{2},b(t)]}(\rho ^2\omega )(t,r)\,\textrm{d}t\nonumber \\&\le C(M,\rho ^{*},\Vert \omega \Vert _{C^1},T)+C(M)\int _{0}^{T}\varepsilon \sup _{r\in A(t)}(\rho ^2\omega )(t,r)\,\textrm{d}t\nonumber \\&\le C(M,\rho ^{*},\Vert \omega \Vert _{C^1},T)+C(M,E_0,\rho ^{*},\Vert \omega \Vert _{C^1},T)d^{-2}. \end{aligned}$$

(5.66)

Finally, integrating (5.57) over $[0,T]\times [\frac{d}{2},b(t)]$ and using (5.58)–(5.61) and (5.65)–(5.66), we conclude (5.53). $\square$

Corollary 5.7

It follows from (3.3) and Lemma 5.6 that

$$\begin{aligned} \int _{0}^{T}\int _{d}^{b(t)}\rho ^{\gamma _2+1}(t,r)\,r^{2}\textrm{d}r\textrm{d}t&\le C\int _{0}^{T}\int _{d}^{b(t)}\big (\rho +\rho P(\rho )\big )(t,r)\,r^{2}\textrm{d}r\textrm{d}t\nonumber \\&\le C(d, M, E_0,T). \end{aligned}$$

(5.67)

In order to use the $L^p$ compensated compactness framework, we still need to obtain the higher integrability of the velocity (see [13]). With the help of Lemma 5.6, we use the special entropy pair constructed in Lemma 4.1 to achieve this.

Lemma 5.8

(Higher integrability of the velocity). Let $(\rho ,u)$ be the smooth solution of (5.1)–(5.6). Then, under the assumption of Lemma 5.1,

$$\begin{aligned} \int _{0}^{T}\int _{d}^{D}(\rho |u|^3)(t,r)\,r^{2}\textrm{d}r\textrm{d}t\le C(d, D, \rho ^{*}, M, E_0, T) \qquad \,\text{ for } \text{ any }\, (d, D)\Subset [a,b(t)]. \end{aligned}$$

Proof

Considering $(5.1)_1\times \hat{\eta }_{\rho }r^{2}+(5.1)_2\times \hat{\eta }_{m}r^{2}$, we can obtain

$$\begin{aligned}&(\hat{\eta }r^{2})_{t}+(\hat{q}r^{2})_{r}+2r\big (-\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_m\big )\nonumber \\&\quad =\varepsilon \,r^{2}\Big ((\rho u_{r})_{r}+2\rho \big (\frac{u}{r}\big )_{r}\Big )\hat{\eta }_{m} -\rho \int _{a}^{r}\rho \, y^{2}\textrm{d}y\,\hat{\eta }_{m}. \end{aligned}$$

(5.68)

Using (5.3), a direct calculation yields

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}\int _{r}^{b(t)}\hat{\eta }\,y^{2}\textrm{d}y =(u\hat{\eta })(t,b(t))\,b(t)^{2}+\int _{r}^{b(t)}\partial _t\hat{\eta }(t,y)\,y^{2}\textrm{d}y. \end{aligned}$$

(5.69)

Integrating (5.68) over [r, b(t)) and using (5.69), we have

$$\begin{aligned} \hat{q}(t,r)\,r^{2}&=-\varepsilon \int _{r}^{b(t)}\hat{\eta }_{m}(t,y)(\rho u_{y}y^{2})_{y}\,\textrm{d}y +2\varepsilon \int _{r}^{b(t)}\hat{\eta }_{m}(t,y)\,\rho u\,\textrm{d}y\nonumber \\&\qquad +\Big (\int _{r}^{b(t)}\hat{\eta }(t,y)\,y^{2}\textrm{d}y\Big )_{t} +(\hat{q}-u\hat{\eta })(t,b(t))\,b(t)^{2}\nonumber \\&\qquad +2\int _{r}^{b(t)}\big (-\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_{m}\big )\,y\textrm{d}y +\int _{r}^{b(t)}\Big (\int _{a}^{y}\rho \, z^2 \textrm{d}z\Big )\rho \, \hat{\eta }_{m}\,\textrm{d}y. \end{aligned}$$

(5.70)

We now control the terms on the RHS of (5.70). For the third term on the RHS of (5.70), it follows from (5.9), (5.34), (5.44), and Lemmas 4.1, 5.1, and 5.4–5.5 that

$$\begin{aligned}&\int _{0}^{T}\big |(\hat{q}-u\hat{\eta })(t,b(t))\big |b(t)^{2}\,\textrm{d}t\nonumber \\&\quad \le C\int _{0}^{T}\left( (\rho (t,b(t)))^{\gamma _1+\theta _1}+(\rho ^{\gamma _1}|u|)(t,b(t))\right) b(t)^{2}\,\textrm{d}t \nonumber \\&\quad \le C\Big (\int _{0}^{T}\varepsilon (\rho |u|^2)(t,b(t))b(t)\,\textrm{d}t\Big )^{\frac{1}{2}} \Big (\int _{0}^{T}\frac{1}{\varepsilon }(\rho (t,b(t)))^{2\gamma _1-1}b(t)^3\,\textrm{d}t\Big )^{\frac{1}{2}} \nonumber \\&\qquad + C(M,E_0,T)\int _{0}^{T} (\rho (t,b(t)))^{\theta _1} b(t)^{-1}\,\textrm{d}t \le C(M,E_0,T). \end{aligned}$$

(5.71)

For the first term on the RHS of (5.70), integrating by parts yields

$$\begin{aligned} \varepsilon \int _{r}^{b(t)}\hat{\eta }_{m}(t,y)\,(\rho u_{y}y^{2})_{y}\,\textrm{d}y&=\varepsilon \hat{\eta }_{m}(t,b(t))\,(\rho u_{r})(t,b(t))\,b(t)^{2}\nonumber \\&\qquad -\varepsilon \hat{\eta }_{m}(t,r)\,(\rho u_{r})(t,r)\,r^{2}\nonumber \\&\qquad -\varepsilon \int _{r}^{b(t)}\rho u_{y}\big (\hat{\eta }_{mu}u_{y}+\hat{\eta }_{m\rho }\rho _{y}\big )\,y^2\textrm{d}y. \end{aligned}$$

(5.72)

It follows from (5.4) and Lemma 4.1 that

$$\begin{aligned}&\vert \varepsilon \hat{\eta }_{m}(t,b(t))\,(\rho u_{r})(t,b(t))\,b(t)^{2}\vert \\&\quad =\Big \vert \hat{\eta }_{m}(t,b(t))\Big (\varepsilon \rho \big (u_{r}+\frac{2}{r}u\big )(t,b(t)) -2\varepsilon b(t)^{-1}\,(\rho u)(t,b(t))\Big )b(t)^{2}\Big \vert \\&\quad =\Big \vert \hat{\eta }_{m}(t,b(t))\, p(\rho )(t,b(t))\, b(t)^{2}-2\varepsilon \hat{\eta }_{m}(t,b(t))\,(\rho u)(t,b(t))\,b(t)\Big \vert \\&\quad \le C\big ((\rho ^{\gamma _1}|u|)(t,b(t))+(\rho (t,b(t)))^{\gamma _1+\theta _1}\big )b(t)^{2}\\&\qquad \quad +C\varepsilon \big ((\rho |u|^2)(t,b(t))+(\rho (t,b(t)))^{\gamma _1}\big )b(t). \end{aligned}$$

which, with similar arguments as in (5.71), yields

$$\begin{aligned} \int _{0}^{T}\left| \varepsilon \hat{\eta }_{m}(t,b(t))\, (\rho u_{r})(t,b(t))\, b^{2}(t)\right| \textrm{d}t\le C(M, E_0,T). \end{aligned}$$

(5.73)

Hence, using (5.36), (5.72)–(5.73), and Lemma 4.1, we have

$$\begin{aligned}&\int _{0}^{T}\int _{d}^{D}\Big \vert \varepsilon \int _{r}^{b(t)}\hat{\eta }_m(\rho u_yy^{2})_{y}\,\textrm{d}y\Big \vert \, \textrm{d}r\textrm{d}t\nonumber \\&\quad \le \int _{0}^{T}\int _{d}^{D}\vert \varepsilon \hat{\eta }_{m}(t,b(t))\,(\rho u_{r})(t,b(t))\vert \,b(t)^{2}\,\textrm{d}r\textrm{d}t\nonumber \\&\qquad \quad +\int _{0}^{T}\int _{d}^{D}\vert \varepsilon \hat{\eta }_{m}(t,r)\,(\rho u_{r})(t,r)\vert \,r^2\textrm{d}r\textrm{d}t\nonumber \\&\qquad \quad +\varepsilon \int _{0}^{T}\int _{d}^{D}\Big \vert \int _{r}^{b(t)}\rho |u_{y}|\big (\hat{\eta }_{mu}|u_{y}| +\hat{\eta }_{m\rho }|\rho _{y}|\big )\,y^2\textrm{d}y\Big \vert \,\textrm{d}r\textrm{d}t \nonumber \\&\quad \le C(D,M,E_0,T)+C\int _{0}^{T}\int _{d}^{D}\varepsilon |\rho u_{r}|\big (|u|+\rho ^{\theta (\rho )}\big )\,r^{2}\textrm{d}r\textrm{d}t\nonumber \\&\qquad \quad +C(D)\int _{0}^{T}\int _{d}^{D}\int _{r}^{b(t)} \varepsilon \rho |u_{y}|\big (|u_{y}|+\rho ^{\theta (\rho )-1}|\rho _{y}|\big )\,y^{2}\textrm{d}y\textrm{d}r\textrm{d}t\nonumber \\&\quad \le C(D,M,E_0,T)+C\int _{0}^{T}\int _{d}^{D}\varepsilon \big (\rho |u|^2+\rho |u_{r}|^2+\rho ^{\gamma (\rho )}\big )\,r^{2}\textrm{d}r\textrm{d}t\nonumber \\&\qquad \quad +C(D)\int _{0}^{T}\int _{d}^{b(t)}\varepsilon \rho |u_{y}|^2\,y^{2}\textrm{d}y\textrm{d}t +C(D)\int _{0}^{T}\int _{d}^{b(t)}\varepsilon \rho ^{\gamma (\rho )-2}|\rho _{y}|^2\,y^{2}\textrm{d}y\textrm{d}t\nonumber \\&\quad \le C(D, M, E_0, T). \end{aligned}$$

(5.74)

For the second term, third term, and sixth term on the RHS of (5.70), using (3.4)–(3.5) and Lemmas 4.1 and 5.1, we obtain

$$\begin{aligned}&\Big \vert \int _{0}^{T}\int _{d}^{D}\Big (2\varepsilon \int _{r}^{b(t)}\hat{\eta }_{m}(t,y)\rho u\,\textrm{d}y\Big ) \,\textrm{d}r\textrm{d}t\Big \vert \nonumber \\&\quad \le C(d)\int _{0}^{T}\int _{d}^{D}\int _{r}^{b(t)}\varepsilon \big (|u|+\rho ^{\theta (\rho )}\big )\,\rho |u| \,y^2\textrm{d}y\textrm{d}r\textrm{d}t\nonumber \\&\quad \le C(d, D)\int _{0}^{T}\int _{d}^{b(t)}\varepsilon \big (\rho |u|^2+\rho +\rho e(\rho )\big )\,y^{2}\textrm{d}y\textrm{d}t \le C(d, D, M, E_0, T), \end{aligned}$$

(5.75)

$$\begin{aligned}&\Big \vert \int _{0}^{T}\int _{d}^{D}\Big (\int _{r}^{b(t)}\hat{\eta }(t,y)\,y^{2}\textrm{d}y\Big )_{t}\,\textrm{d}r\textrm{d}t\Big \vert \nonumber \\&\quad \le \int _{d}^{D}\int _{r}^{b(t)} |\hat{\eta }(T,y)|\, y^{2}\textrm{d}y\textrm{d}r + \int _{d}^{D}\int _{r}^{b} |\hat{\eta }(0,y)|\, y^{2}\textrm{d}y\textrm{d}r\nonumber \\&\quad \le C\sup _{t\in [0,T]}\int _{d}^{D}\int _{r}^{b(t)} \big (\rho ^{\gamma (\rho )}+\rho |u|^2\big )\, y^{2}\textrm{d}y\textrm{d}r\nonumber \\&\quad \le C\sup _{t\in [0,T]}\int _{d}^{D}\int _{r}^{b(t)} \big (\rho e(\rho )+\rho +\rho |u|^2\big )\,y^{2}\textrm{d}y\textrm{d}r \le C(D, M, E_0, T), \end{aligned}$$

(5.76)

$$\begin{aligned}&\Big \vert \int _{0}^T\int _{d}^{D}\Big (\int _{r}^{b(t)}\Big (\int _{a}^{y}\rho \, z^{2}\textrm{d}z\Big ) \rho \, \hat{\eta }_{m}\,\textrm{d}y\Big )\,\textrm{d}r\textrm{d}t\Big \vert \nonumber \\&\quad \le \int _{0}^{T}\int _{d}^{D}\Big \vert \int _{r}^{b(t)}\Big (\int _{a}^{y}\rho \, z^{2}\textrm{d}z\Big )\rho \, \hat{\eta }_{m}\,\textrm{d}y\Big \vert \,\textrm{d}r\textrm{d}t\nonumber \\&\quad \le C(d,D,M)\int _{0}^{T}\int _{d}^{b(t)}\rho \big (|u|+\rho ^{\gamma (\rho )}\big )\,r^2\textrm{d}r\textrm{d}t\nonumber \\&\quad \le C(d,D,M)\int _{0}^{T}\int _{d}^{b(t)}\big (\rho |u|^2+\rho +\rho e(\rho )\big )\,r^{2}\textrm{d}r\textrm{d}t\le C(d, D, M, E_0, T). \end{aligned}$$

(5.77)

For the fifth term on the RHS of (5.70), we note from (3.6)–(3.7) and Lemma 4.1 that

$$\begin{aligned}&-\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_{m}=0\qquad \,\, \text{ if }\, |u|\ge k(\rho ), \end{aligned}$$

(5.78)

$$\begin{aligned}&\vert -\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_{m}\vert \le C\rho ^{\gamma (\rho )+\theta (\rho )}\le C\big (\rho +\rho ^{\gamma _2+\theta _2}\big )\qquad \,\, \text{ if }\, |u|\le k(\rho ). \end{aligned}$$

(5.79)

Then it follows from (5.78)–(5.79) and Corollary 5.7 that

$$\begin{aligned}&\int _{0}^{T}\int _{d}^{D}\Big |\int _{r}^{b(t)}\big (-\hat{q}+\rho u\hat{\eta }_{\rho }+\rho u^2\hat{\eta }_{m}\big )\,y\textrm{d}y\Big | \,\textrm{d}r\textrm{d}t\nonumber \\&\quad \le C(d, D)\int _{0}^{T}\int _{d}^{b(t)} \big (\rho +\rho ^{\gamma _2+\theta _2}\big )\,y^{2}\textrm{d}y\textrm{d}t \le C(d, D, M, E_0,T), \end{aligned}$$

(5.80)

where we have used $\theta _2\in (0,1)$ since $\gamma _2\in (\frac{6}{5},3)$. Combining (5.70)–(5.71), (5.74)–(5.77), and (5.80), we obtain that $\int _{0}^{T}\int _{d}^{D}\hat{q}\,r^{2}\textrm{d}r\textrm{d}t\le C(d, D, M, E_0,T)$, which, along with (5.67) and Lemma 4.1, gives

$$\begin{aligned}&\int _{0}^{T}\int _{[d, D]\cap \{r: |u|\ge k(\rho )\}}\rho |u|^3\,r^{2}\textrm{d}r\textrm{d}t\nonumber \\&\quad \le 2\int _{0}^{T}\int _{[d, D]\cap \{r: |u|\ge k(\rho )\}}\hat{q}\,r^{2}\textrm{d}r\textrm{d}t\nonumber \\&\quad =2\int _{0}^{T}\int _{d }^{D}\hat{q}\,r^{2}\textrm{d}r\textrm{d}t -2\int _{0}^{T}\int _{[d, D]\cap \{r: |u|<k(\rho )\}}\hat{q}\, r^{2}\textrm{d}r\textrm{d}t\nonumber \\&\quad \le C(d, D, M, E_0, T)+C\int _{0}^{T}\int _{d}^{D}(\rho +\rho ^{\gamma _2+1})\,r^{2}\textrm{d}r\textrm{d}t \nonumber \\&\quad \le C(d, D, M, E_0, T). \end{aligned}$$

(5.81)

On the other hand, we have

$$\begin{aligned}&\int _{0}^{T}\int _{[d, D]\cap \{r:\,|u|\le k(\rho )\}}\rho |u|^3\,r^{2}\textrm{d}r\textrm{d}t\nonumber \\&\quad \le C\int _{0}^{T}\int _{d}^{D}\rho ^{\gamma (\rho )+\theta (\rho )}\,r^{2}\textrm{d}r\textrm{d}t \nonumber \\&\quad \le C\int _{0}^{T}\int _{d}^{D}\big (\rho +\rho p(\rho )\big )\,r^{2}\textrm{d}r\textrm{d}t \le C(M, E_0,T). \end{aligned}$$

(5.82)

Combining (5.81) with (5.82), we obtain that $\int _{0}^{T}\int _{d}^{D}\rho |u|^3\,r^{2}\textrm{d}r\textrm{d}t\le C(d, D, M, E_0, T)$. This completes the proof of Lemma 5.8. $\square$

6 Existence of Global Weak Solutions of CNSPEs

In this section, for fixed $\varepsilon >0$, we take the limit: $b \rightarrow \infty$ to obtain the global existence of solutions of the Cauchy problem for (1.10). Meanwhile, some uniform estimates in Theorem 2.1 are obtained. To take the limit, some careful attention is required, since the weak solutions may involve the vacuum. We use similar compactness arguments as in [10, 17] to handle the limit: $b \rightarrow \infty$. Throughout this section, we denote the smooth solutions of (5.1)–(5.6) as $(\rho ^{\varepsilon , b}, u^{\varepsilon , b})$ for simplicity.

First of all, we extend our solutions $(\rho ^{\varepsilon , b}, u^{\varepsilon , b})$ to be zero on $([0, T] \times [0, \infty )) \backslash \Omega _{T}$. It follows from Lemma 5.5 that

$$\begin{aligned} \lim _{b \rightarrow \infty } \min _{t \in [0, T]} b(t)=\infty , \end{aligned}$$

(6.1)

which implies that domain $[0, T] \times [a, b(t)]$ expands to $[0, T] \times (0, \infty )$ as $b \rightarrow \infty$. That is, for any set $K \Subset (0, \infty )$, when $b \gg 1, K \Subset (a, b(t))$ for all $t \in [0, T]$. Now we define

$$\begin{aligned} (\rho ^{\varepsilon ,b},\mathcal {M}^{\varepsilon ,b},\Phi ^{\varepsilon ,b})(t,\textbf{x}) :=(\rho ^{\varepsilon ,b}(t,r),m^{\varepsilon ,b}(t,r)\frac{\textbf{x}}{r},\Phi ^{\varepsilon ,b}(t,r)) \qquad \text {for }r=|\textbf{x}|, \end{aligned}$$

where $m^{\varepsilon ,b}:=\rho ^{\varepsilon ,b}u^{\varepsilon ,b}$. Then it is direct to check that the corresponding functions $(\rho ^{\varepsilon ,b},\mathcal {M}^{\varepsilon ,b}, \Phi ^{\varepsilon ,b})(t,\textbf{x})$ are classical solutions of

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}\rho ^{\varepsilon ,b}+ \nabla \cdot \mathcal {M}^{\varepsilon ,b}=0,\\&\partial _{t}\mathcal {M}^{\varepsilon ,b}+ \nabla \cdot \Big (\frac{\mathcal {M}^{\varepsilon ,b}\otimes \mathcal {M}^{\varepsilon ,b}}{\rho ^{\varepsilon ,b}}\Big )+\nabla P(\rho ^{\varepsilon ,b})+\rho ^{\varepsilon ,b}\nabla \Phi ^{\varepsilon ,b}=\varepsilon \nabla \cdot \Big (\rho ^{\varepsilon ,b}D\big (\frac{\mathcal {M}^{\varepsilon ,b}}{\rho ^{\varepsilon ,b}}\big )\Big ),\\&\Delta \Phi ^{\varepsilon ,b}=\rho ^{\varepsilon ,b}, \end{aligned} \right. \end{aligned}$$

for $(t,\textbf{x})\in [0,\infty )\times \Omega _{t}$ with $\mathcal {M}^{\varepsilon ,b}\vert _{\partial B_{a}(\textbf{0})}=0$.

Based on the estimates obtained in Sect. 5, by the same arguments as in [10, Sect. 4], we have

Lemma 6.1

For fixed $\varepsilon >0$, as $b\rightarrow \infty$ (up to a subsequence), there exists a vector function $(\rho ^{\varepsilon }, m^\varepsilon )(t,r))$ such that

• (i) $(\sqrt{\rho ^{\varepsilon ,b}},\rho ^{\varepsilon ,b})\rightarrow (\sqrt{\rho ^{\varepsilon }}, \rho ^{\varepsilon })$ a.e. and strongly in $C(0,T;L_{\textrm{loc}}^p)$ for any $p\in [1,\infty )$, where $L_{\textrm{loc}}^p$ denotes $L^p(K)$ for any compact set $K\Subset (0,\infty )$. In particular, $\rho ^{\varepsilon }\ge 0$ a.e. on $\mathbb {R}_{+}^2$.

• (ii) The pressure function sequence $P(\rho ^{\varepsilon , b})$ is uniformly bounded in $L^{\infty }(0, T; L_{\textrm{loc}}^{p}(\mathbb {R}))$ for all $p \in [1, \infty ]$, and

  $$\begin{aligned} P(\rho ^{\varepsilon , b}) \longrightarrow P(\rho ^{\varepsilon }) \quad \text { strongly in } L^{p}(0, T; L_{\textrm{loc }}^{p}(\mathbb {R}))\qquad \text {for } p\in [1, \infty ). \end{aligned}$$

• (iii) The momentum function sequence $m^{\varepsilon , b}$ converges strongly in $L^{2}(0, T; L_{\textrm{loc}}^{p}(\mathbb {R}))$ to $m^{\varepsilon }$ for all $p \in [1, \infty )$. In particular,

  $$\begin{aligned} m^{\varepsilon , b}(t,r)=(\rho ^{\varepsilon , b} u^{\varepsilon , b})(t,r) \longrightarrow m^{\varepsilon }(t, r) \qquad { a.e.}\,\text { in }[0, T] \times (0, \infty ). \end{aligned}$$

• (iv) $m^{\varepsilon }(t, r)=0$ a.e. on $\{(t, r)\,:\,\rho ^{\varepsilon }(t, r)=0\}$. Furthermore, there exists a function $u^{\varepsilon }(t, r)$ such that $m^{\varepsilon }(t, r)=\rho ^{\varepsilon }(t, r) u^{\varepsilon }(t, r)$ a.e., $u^{\varepsilon }(t, r)=0$ a.e. on $\{(t, r)\,:\,\rho ^{\varepsilon }(t, r)=0\}$, and

  $$\begin{aligned} \begin{aligned} m^{\varepsilon , b}&\longrightarrow m^{\varepsilon }=\rho ^{\varepsilon } u^{\varepsilon } \qquad \text { strongly in}\, L^{2}\left( 0, T ; L_{\textrm{loc}}^{p}(\mathbb {R})\right) \,\text { for }\, p \in [1, \infty ), \\ \frac{m^{\varepsilon , b}}{\sqrt{\rho ^{\varepsilon , b}}}&\longrightarrow \frac{m^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} =\sqrt{\rho ^{\varepsilon }} u^{\varepsilon } \qquad \text { strongly in}\, L^{2}(0, T ; L_{\textrm{loc}}^{2}(\mathbb {R})). \end{aligned} \end{aligned}$$

Let $(\rho ^{\varepsilon },m^{\varepsilon })(t,r)$ be the limit function obtained above. Using (5.13), (6.1), Lemmas 5.1–5.6, 5.8, and 6.1, Corollaries 5.2–5.3 and 5.7, Fatou’s lemma, and the lower semicontinuity, we conclude the proof of (2.22)–(2.25).

Now we show the convergence of the potential functions $\Phi ^{\varepsilon ,b}$. Using the similar arguments as in [10, Lemma 4.6], we have

Lemma 6.2

For fixed $\varepsilon >0$, there exists a function $\Phi ^{\varepsilon }(t,\textbf{x})=\Phi ^{\varepsilon }(t,r)$ such that, as $b\rightarrow \infty$ (up to a subsequence),

$$\begin{aligned}&\Phi ^{\varepsilon ,b} \,{\rightharpoonup }\, \Phi ^{\varepsilon }\quad \text {weak-star in}\, L^{\infty }(0,T;H_{\textrm{loc}}^{1}(\mathbb {R}^3))\, \text { and weakly in}\, L^2{(0,T;H_{\textrm{loc}}^{1}(\mathbb {R}^3))}, \end{aligned}$$

(6.2)

$$\begin{aligned}&\Phi _{r}^{\varepsilon ,b}(t,r)r^{2}\,\rightarrow \,\Phi _{r}^{\varepsilon }(t,r)r^{2}=\int _{0}^{r}\rho ^{\varepsilon }(t,y)\,y^{2}\textrm{d}y \qquad \text {in}\, C_{\textrm{loc}}([0,T]\times [0,\infty )), \end{aligned}$$

(6.3)

$$\begin{aligned}&\Vert \Phi ^{\varepsilon }(t)\Vert _{L^{6}(\mathbb {R}^{3})}+\Vert \nabla \Phi ^{\varepsilon }(t)\Vert _{L^{2}(\mathbb {R}^{3})} \le C(M, E_{0}) \qquad \text {for }\, t \ge 0. \end{aligned}$$

(6.4)

Moreover, since $\gamma _2>\frac{6}{5}$,

$$\begin{aligned} \int _{0}^{\infty }\big |\big (\Phi _{r}^{\varepsilon , b}-\Phi _{r}^{\varepsilon }\big )(t, r)\big |^{2} \,r^{2}\textrm{d} r \rightarrow 0 \qquad \text { as}\, b \rightarrow \infty \,(\text { up to a subsequence}). \end{aligned}$$

(6.5)

Using (6.5), Fatou’s lemma, and Lemmas 5.1 and 6.1, we obtain the following energy inequality:

$$\begin{aligned}&\int _{0}^{\infty }\Big (\frac{1}{2}\Big |\frac{m^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}}\Big |^{2} +\rho ^{\varepsilon } e(\rho ^{\varepsilon })\Big )(t, r) \,r^{2} \textrm{d} r -\frac{1}{2} \int _{0}^{\infty }|\Phi ^{\varepsilon }(t, r)|^{2} \,r^{2}\textrm{d} r\nonumber \\&\quad \le \int _{0}^{\infty }\Big (\frac{1}{2}\Big |\frac{m_{0}^{\varepsilon }}{\sqrt{\rho _{0}^{\varepsilon }}}\Big |^{2} +\rho _{0}^{\varepsilon } e(\rho _{0}^{\varepsilon })\Big )(r)\, r^{2}\textrm{d} r -\frac{1}{2} \int _{0}^{\infty }|\Phi _{0}^{\varepsilon }(r)|^{2}\,r^{2}\textrm{d} r. \end{aligned}$$

(6.6)

We denote

$$\begin{aligned} (\rho ^{\varepsilon },\mathcal {M}^{\varepsilon }, \Phi ^{\varepsilon })(t,\textbf{x}):=(\rho ^{\varepsilon }(t,r),m^{\varepsilon }(t,r)\frac{\textbf{x}}{r},\Phi ^{\varepsilon }(t,r)). \end{aligned}$$

Then (2.20) follows directly from (6.6). Moreover, we can prove that $(\rho ^{\varepsilon },\mathcal {M}^{\varepsilon }, \Phi ^{\varepsilon })$ is a global weak solution of the Cauchy problem (1.10) and (2.17)–(2.18) in the sense of Definition 2.1. In fact, by the same arguments in [10, Remark 4.7 and Lemmas 4.9–4.11], we have

Lemma 6.3

Let $0 \le t_{1}<t_{2} \le T$, and let $\zeta (t, \textbf{x}) \in C_{0}^{1}([0, T] \times \mathbb {R}^{3})$ be any smooth function with compact support. Then

$$\begin{aligned} \int _{\mathbb {R}^{3}} \rho ^{\varepsilon }(t_{2}, \textbf{x}) \zeta (t_{2}, \textbf{x})\,\textrm{d} \textbf{x} =\int _{\mathbb {R}^{3}} \rho ^{\varepsilon }(t_{1},\textbf{x}) \zeta (t_{1},\textbf{x})\, \textrm{d} \textbf{x} +\int _{t_{1}}^{t_{2}} \int _{\mathbb {R}^{3}}(\rho ^{\varepsilon } \zeta _{t} +\mathcal {M}^{\varepsilon } \cdot \nabla \zeta )\,\textrm{d} \textbf{x}\textrm{d} t. \end{aligned}$$

(6.7)

Moreover, (2.21) holds, and the total mass is conserved:

$$\begin{aligned} \int _{\mathbb {R}^{3}} \rho ^{\varepsilon }(t, \textbf{x})\, \textrm{d} \textbf{x} =\int _{\mathbb {R}^{3}} \rho _{0}^{\varepsilon }(\textbf{x})\, \textrm{d} \textbf{x}=M \qquad \, \text {for }\, t \ge 0. \end{aligned}$$

(6.8)

Lemma 6.4

Let $\Psi (t, \textbf{x}) \in (C_{0}^{2}([0, T] \times \mathbb {R}^{3}))^{3}$ be any smooth function with compact support so that $\Psi (T, \textbf{x})=0$. Then

$$\begin{aligned}&\int _{\mathbb {R}_{+}^{4}}\Big \{\mathcal {M}^{\varepsilon } \cdot \partial _{t} \Psi +\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} \cdot \big (\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} \cdot \nabla \big ) \Psi +P(\rho ^{\varepsilon }) \nabla \cdot \Psi -\rho ^{\varepsilon }\nabla \Phi ^{\varepsilon }\cdot \Psi \Big \}\,\textrm{d} \textbf{x} \textrm{d} t \nonumber \\&\qquad \quad +\int _{\mathbb {R}^{3}} \mathcal {M}_{0}^{\varepsilon }(\textbf{x}) \cdot \Psi (0, \textbf{x}) \,\textrm{d} \textbf{x} \nonumber \\&\quad =-\varepsilon \int _{\mathbb {R}^{4}_+}\Big \{\frac{1}{2} \mathcal {M}^{\varepsilon } \cdot \big (\Delta \Psi +\nabla (\nabla \cdot \Psi )\big )+\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} \cdot \big (\nabla \sqrt{\rho ^{\varepsilon }} \cdot \nabla \big ) \Psi \nonumber \\&\qquad \qquad \qquad \qquad +\nabla \sqrt{\rho ^{\varepsilon }} \cdot \big (\frac{\mathcal {M}^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} \cdot \nabla \big ) \Psi \Big \}\,\textrm{d} \textbf{x} \textrm{d} t\nonumber \\&\quad =\sqrt{\varepsilon } \int _{\mathbb {R}_{+}^{4}} \sqrt{\rho ^{\varepsilon }}\Big \{V^{\varepsilon } \frac{\textbf{x} \otimes \textbf{x}}{r^{2}}+\frac{\sqrt{\varepsilon }}{r} \frac{m^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}}\Big (I_{3 \times 3}-\frac{\textbf{x} \otimes \textbf{x}}{r^{2}}\Big )\Big \}: \nabla \Psi \, \textrm{d} \textbf{x} \textrm{d} t \end{aligned}$$

(6.9)

with $V^{\varepsilon }(t, \textbf{x}) \in L^{2}(0, T; L^{2}(\mathbb {R}^{3}))$ as a function satisfying

$$\begin{aligned} \int _{0}^{T} \int _{\mathbb {R}^{3}}\left| V^{\varepsilon }(t, \textbf{x})\right| ^{2} \,\textrm{d} \textbf{x}\textrm{d} t \le C(E_{0}, M), \end{aligned}$$

where $C\left( E_{0}, M\right) >0$ is a constant independent of $T>0$.

Lemma 6.5

It follows from (6.3) that $\Phi ^{\varepsilon }$ satisfies Poisson’s equation in the classical sense except for the origin: $(t,{\textbf {x}})\in [0,\infty )\times (\mathbb {R}^{3}\backslash \{\varvec{0}\})$. Moreover, for any smooth function $\xi (\textbf{x})\in C_{0}^{1}(\mathbb {R}^3)$ with compact support,

$$\begin{aligned} \int _{\mathbb {R}^{3}} \nabla \Phi ^{\varepsilon }(t, \textbf{x}) \cdot \nabla \xi (\textbf{x})\, \textrm{d} \textbf{x} =- \int _{\mathbb {R}^{3}} \rho ^{\varepsilon }(t, \textbf{x})\, \xi (\textbf{x}) \,\textrm{d} \textbf{x} \qquad \text {for } t \ge 0. \end{aligned}$$

(6.10)

7 $W_{\textrm{loc}}^{-1,p}$–Compactness of Weak Entropy Dissipation Measures

In this section, using the estimates of the weak entropy pairs obtained in Lemmas 4.5, 4.7, and 4.10, we establish the compactness of weak entropy dissipation measures: $\partial _{t}\eta ^{\psi }(\rho ^\varepsilon ,m^\varepsilon )+\partial _{r}q^{\psi }(\rho ^\varepsilon ,m^\varepsilon )$ for each weak entropy pair $(\eta ^{\psi },q^{\psi })$. Unfortunately, we fail to obtain the same $H_{\textrm{loc}}^{-1}$-compactness as in [10, 17], since we only obtain that $q^{\varepsilon }$ is uniformly bounded in $L_{\textrm{loc}}^{2}$ from Lemma 4.10 and Corollary 5.7. Instead, using similar arguments as in [10, Lemma 4.2], we can obtain the compactness in $W_{\textrm{loc}}^{-1,p}$ for any $p\in [1,2)$.

Lemma 7.1

(Compactness of the entropy dissipation measures). Let $(\eta ^{\psi },q^{\psi })$ be a weak entropy pair defined in (4.63) for any smooth and compactly supported function $\psi (s)$ on $\mathbb {R}$. Then, for $\varepsilon \in (0,\varepsilon _0]$,

$$\begin{aligned} \partial _{t}\eta ^{\psi }(\rho ^{\varepsilon },m^{\varepsilon })+\partial _{r}q^{\psi }(\rho ^{\varepsilon },m^{\varepsilon })\qquad \text {is compact in }W_{\textrm{loc}}^{-1,p}(\mathbb {R}_{+}^2) \quad \text{ for } \text{ any }\, p\in [1,2). \end{aligned}$$

(7.1)

Proof

To establish (7.1), we first need to study the equation: $\partial _{t}\eta ^{\psi }(\rho ^{\varepsilon },m^{\varepsilon })+\partial _{r}q(\rho ^{\varepsilon },m^{\varepsilon })$ in the distributional sense, which is more complicated than that in [13, 14]. For simplicity, we denote $(\eta ^{\varepsilon ,b},q^{\varepsilon ,b})=(\eta ^{\psi }(\rho ^{\varepsilon ,b},m^{\varepsilon ,b}),q^{\psi }(\rho ^{\varepsilon ,b},m^{\varepsilon ,b}))$ and $(\eta ^{\varepsilon },q^{\varepsilon })=(\eta ^{\psi }(\rho ^{\varepsilon },m^{\varepsilon }),q^{\psi }(\rho ^{\varepsilon },m^{\varepsilon }))$. We divide it into four steps.

1. Considering $(5.1)_1\times \eta _{\rho }^{\varepsilon ,b}+(5.1)_2\times \eta _{m}^{\varepsilon ,b}$, we obtain

$$\begin{aligned}&\partial _{t} \eta (\rho ^{\varepsilon , b}, m^{\varepsilon , b})+\partial _{r} q(\rho ^{\varepsilon , b}, m^{\varepsilon , b}) +\frac{2}{r} m^{\varepsilon , b}\big (\eta _{\rho }^{\varepsilon , b}+u^{\varepsilon , b} \eta _{m}^{\varepsilon , b}\big )\nonumber \\&\quad =-\eta _{m}^{\varepsilon , b} \frac{\rho ^{\varepsilon , b}}{r^{2}} \int _{0}^{r} \rho ^{\varepsilon , b}(t, y)\,y^{2}\textrm{d}y +\varepsilon \eta _{m}^{\varepsilon , b}\Big \{\big (\rho ^{\varepsilon , b}(u_{r}^{\varepsilon , b} +\frac{2}{r} u^{\varepsilon , b})\big )_{r}-\frac{2}{r} \rho _{r}^{\varepsilon , b} u^{\varepsilon , b}\Big \}, \end{aligned}$$

(7.2)

where $\rho ^{\varepsilon ,b}$ is understood to be zero in domain $[0,T]\times [0,a)$ so that $\int _{a}^{r}\rho ^{\varepsilon ,b}(t,z)\,z^{2}\textrm{d}z$ can be written as $\int _{0}^{r}\rho ^{\varepsilon ,b}(t,z)\,z^{2}\textrm{d}z$ in the potential term. Let $\phi (t, r) \in C_{0}^{\infty }\left( \mathbb {R}_{+}^{2}\right)$ and $b \gg 1$ so that ${\text {supp}}\phi (t, \cdot ) \in (a, b(t))$. Multiplying (7.2) by $\phi$ and integrating by parts yield

$$\begin{aligned}&\int _{\mathbb {R}_{+}^{2}}(\partial _{t} \eta ^{\varepsilon , b}+\partial _{r} q^{\varepsilon , b}) \phi \,\textrm{d} r \textrm{d} t\nonumber \\&\quad =-\int _{\mathbb {R}_{+}^{2}} \frac{2}{r} m^{\varepsilon , b}(\eta _{\rho }^{\varepsilon , b} +u^{\varepsilon , b} \eta _{m}^{\varepsilon , b}) \phi \,\textrm{d} r \textrm{d} t -\varepsilon \int _{\mathbb {R}_{+}^{2}} \rho ^{\varepsilon , b}(\eta _{m}^{\varepsilon , b})_{r} \big (u_{r}^{\varepsilon , b}+\frac{2}{r} u^{\varepsilon , b}\big ) \phi \,\textrm{d} r\textrm{d} t \nonumber \\&\qquad \quad -\varepsilon \int _{\mathbb {R}_{+}^{2}} \rho ^{\varepsilon , b} \eta _{m}^{\varepsilon , b} \big (u_{r}^{\varepsilon , b}+\frac{2}{r} u^{\varepsilon , b}\big ) \phi _{r} \,\textrm{d} r \textrm{d} t -\varepsilon \int _{\mathbb {R}_{+}^{2}} \frac{2}{r} \eta _{m}^{\varepsilon , b} \rho _{r}^{\varepsilon , b} u^{\varepsilon , b} \phi \, \textrm{d} r \textrm{d} t\nonumber \\&\qquad \quad - \int _{\mathbb {R}_{+}^{2}} \frac{\rho ^{\varepsilon , b}}{r^{2}}\eta _{m}^{\varepsilon , b} \Big (\int _{0}^{r} \rho ^{\varepsilon , b}(t, y)\,y^{2}\textrm{d} y\Big ) \phi \, \textrm{d} r \textrm{d} t :=\sum _{i=1}^{5} I_{i}^{\varepsilon , b}. \end{aligned}$$

(7.3)

2. From Lemmas  4.2 and 6.1, it is clear to see that

$$\begin{aligned} \eta ^{\varepsilon ,b}\longrightarrow \eta ^{\varepsilon }\qquad { a.e.}\,\text { in }\{(t,r)\,:\rho ^{\varepsilon }\ne 0\}\quad \text {as }b\rightarrow \infty . \end{aligned}$$

(7.4)

In $\{(t,r):\,\rho ^{\varepsilon }(t,r)=0\}$, it follows from Lemmas 4.5 and 4.7 that

$$\begin{aligned} |\eta ^{\varepsilon ,b}|\le C\rho ^{\varepsilon ,b}\longrightarrow 0=\eta ^{\varepsilon }\qquad \text {as}\, b\rightarrow \infty . \end{aligned}$$

(7.5)

Combining (7.4)–(7.5), we obtain

$$\begin{aligned} \eta ^{\varepsilon ,b}\longrightarrow \eta ^{\varepsilon }\qquad \, { a.e.}\,\text { as}\, b\rightarrow \infty . \end{aligned}$$

(7.6)

Similarly, it follows from Lemmas 4.3, 4.10, and 6.1 that

$$\begin{aligned} q^{\varepsilon ,b}\longrightarrow q^{\varepsilon }\qquad { a.e.}\,\text { as}\, b\rightarrow \infty . \end{aligned}$$

(7.7)

For $\gamma _2 \in (1,3)$ and any subset $K\Subset (0,\infty )$, it follows from Lemmas 4.5, 4.7, 4.10, and Corollary 5.7 that

$$\begin{aligned} \int _{0}^{T} \int _{K}\big (|\eta ^{\varepsilon , b}|^{\gamma _2+1} +|q^{\varepsilon , b}|^{2}\big )\,\textrm{d} r \textrm{d} t&\le C_{\psi }(K)\int _{0}^{T} \int _{K} \big (1+ |\rho ^{\varepsilon , b}|^{\gamma _2+1}\Big )\, \textrm{d} r \textrm{d} t\\&\le C_{\psi }(K, M, E_{0}, T), \end{aligned}$$

which implies that $(\eta ^{\varepsilon , b}, q^{\varepsilon , b})$ is uniformly bounded in $L_{\textrm{loc}}^{2}(\mathbb {R}_{+}^{2})$. This, with (7.6)–(7.7), yields that, up to a subsequence,

$$\begin{aligned} (\eta ^{\varepsilon , b}, q^{\varepsilon , b}) \rightharpoonup (\eta ^{\varepsilon }, q^{\varepsilon }) \qquad \text {in}\, L_{\textrm{loc}}^{2}(\mathbb {R}_{+}^{2})\, \text {as}\, b \rightarrow \infty . \end{aligned}$$

Thus, for any $\phi \in C_{0}^{1}(\mathbb {R}_{+}^{2})$, as $b \rightarrow \infty$ (up to a subsequence),

$$\begin{aligned} \int _{\mathbb {R}_{+}^{2}}\big (\partial _{t} \eta ^{\varepsilon , b}+\partial _{r} q^{\varepsilon , b}\big ) \phi \,\textrm{d} r \textrm{d} t&=-\int _{\mathbb {R}_{+}^{2}}\big (\eta ^{\varepsilon , b} \partial _{t} \phi +q^{\varepsilon , b} \partial _{r} \phi \big ) \,\textrm{d} r\textrm{d} t\nonumber \\&\longrightarrow -\int _{\mathbb {R}_{+}^{2}}\big (\eta ^{\varepsilon } \partial _{t} \phi +q^{\varepsilon } \partial _{r} \phi \big ) \,\textrm{d} r \textrm{d} t. \end{aligned}$$

(7.8)

Furthermore, $(\eta ^{\varepsilon }, q^{\varepsilon })$ is uniformly bounded in $L_{\textrm{loc }}^{2}(\mathbb {R}_{+}^{2})$, which implies that

$$\begin{aligned} \partial _{t} \eta ^{\varepsilon }+\partial _{r} q^{\varepsilon } \quad \text { is uniformly bounded in } \varepsilon >0 \text { in } W_{\text {loc }}^{-1, 2}(\mathbb {R}_{+}^{2}). \end{aligned}$$

(7.9)

Since $q^{\varepsilon ,b}$ is only uniformly bounded in $L_{\textrm{loc}}^{2}(\mathbb {R}_{+}^2)$ in view of Lemma 4.10 and Corollary 5.7, we cannot conclude that $\partial _{t}\eta ^{\varepsilon }+\partial _{r}q^{\varepsilon }$ is uniformly bounded in $\varepsilon >0$ in $W_{\textrm{loc}}^{-1,p}(\mathbb {R}_{+}^2)$ with $p>2$, which is different from [10].

3. For the terms on the RHS of (7.3), using Lemmas 4.5, 4.7, and 4.10, and similar calculations as in [10, Lemma 5.11], we obtain that

$$\begin{aligned} \begin{aligned}&I_{1}^{\varepsilon , b} \longrightarrow -\int _{\mathbb {R}_{+}^{2}} \frac{2}{r} m^{\varepsilon }(\eta _{\rho }^{\varepsilon } +u^{\varepsilon } \eta _{m}^{\varepsilon }) \phi \,\textrm{d} r \textrm{d} t \qquad \text {up to a subsequence as}\, b \rightarrow \infty , \\&\int _{0}^{T} \int _{K}\frac{2}{r}\big | m^{\varepsilon }(\eta _{\rho }^{\varepsilon } +u^{\varepsilon } \eta _{m}^{\varepsilon })\big |^{\frac{7}{6}}\, \textrm{d} r \textrm{d} t \le C_{\psi }(K, M, E_{0}, T), \end{aligned} \end{aligned}$$

(7.10)

and there exist local bounded Radon measures $(\mu _{1}^{\varepsilon }, \mu _{2}^{\varepsilon }, \mu _{3}^{\varepsilon })$ on $\mathbb {R}_{+}^{2}$ such that, as $b \rightarrow \infty$ (up to a subsequence),

$$\begin{aligned}&-(\varepsilon \rho ^{\varepsilon , b}(\eta _{m}^{\varepsilon , b})_{r}(u_{r}^{\varepsilon , b}+\frac{2}{r} u^{\varepsilon , b}), \,\frac{2\varepsilon }{r} \eta _{m}^{\varepsilon , b}\rho _{r}^{\varepsilon , b} u^{\varepsilon , b},\, \kappa \eta _{m}^{\varepsilon , b} \frac{\rho ^{\varepsilon , b}}{r^{n-1}} \int _{0}^{r} \rho ^{\varepsilon , b}(t, z) \,z^{2}\textrm{d} z) \\&\rightharpoonup (\mu _{1}^{\varepsilon }, \mu _{2}^{\varepsilon }, \mu _{3}^{\varepsilon }). \end{aligned}$$

In addition, for $i=1,2,3$,

$$\begin{aligned} \mu _{i}^{\varepsilon }((0, T) \times V) \le C_{\psi }(K, T, E_{0}) \qquad \,\, \text{ for } \text{ any } \text{ open } \text{ subset }\, V \subset K. \end{aligned}$$

(7.11)

Then, up to a subsequence, we have

$$\begin{aligned} I_{2}^{\varepsilon , b}+I_{4}^{\varepsilon , b}+I_{5}^{\varepsilon ,b} \longrightarrow \langle \,\mu _{1}^{\varepsilon } +\mu _{2}^{\varepsilon }+\mu _{3}^{\varepsilon },\,\phi \rangle \qquad \text { as}\, b \rightarrow \infty . \end{aligned}$$

(7.12)

Moreover, there exists a function $f^{\varepsilon }$ such that, as $b \rightarrow \infty$ (up to a subsequence),

$$\begin{aligned} \begin{aligned}&-\sqrt{\varepsilon } \rho ^{\varepsilon , b} \eta _{m}^{\varepsilon , b}\big (u_{r}^{\varepsilon , b} +\frac{2}{r} u^{\varepsilon , b}\big ) \rightharpoonup f^{\varepsilon } \qquad \text { weakly in}\, L_{\textrm{loc}}^{\frac{4}{3}}(\mathbb {R}_{+}^{2}),\\&\int _{0}^{T} \int _{K}\left| f^{\varepsilon }\right| ^{\frac{4}{3}} \,\textrm{d} r \textrm{d} t \le C_{\psi }(K, M, E_{0}, T). \end{aligned} \end{aligned}$$

(7.13)

Then it follows from (7.13) that

$$\begin{aligned} I_{3}^{\varepsilon , b} \longrightarrow \sqrt{\varepsilon } \int _{0}^{T} \int _{K} f^{\varepsilon } \phi _{r} \,\textrm{d} r \textrm{d} t \qquad \text { as}\, b \rightarrow \infty \, (\text {up to a subsequence}). \end{aligned}$$

(7.14)

4. Taking $b \rightarrow \infty$ (up to a subsequence) on both sides of (7.3), it follows from (7.8), (7.10), (7.12), and (7.14) that

$$\begin{aligned} \partial _{t} \eta ^{\varepsilon }+\partial _{r} q^{\varepsilon }=-\frac{2}{r} \rho ^{\varepsilon } u^{\varepsilon }(\eta _{\rho }^{\varepsilon }+u^{\varepsilon } \eta _{m}^{\varepsilon })+\mu _{1}^{\varepsilon }+\mu _{2}^{\varepsilon }+\mu _{3}^{\varepsilon }-\sqrt{\varepsilon } f_{r}^{\varepsilon } \end{aligned}$$

in the distributional sense. It follows from (7.10)–(7.11) that

$$\begin{aligned} -\frac{2}{r} \rho ^{\varepsilon } u^{\varepsilon }\left( \eta _{\rho }^{\varepsilon }+u^{\varepsilon } \eta _{m}^{\varepsilon }\right) +\mu _{1}^{\varepsilon }+\mu _{2}^{\varepsilon }+\mu _{3}^{\varepsilon } \end{aligned}$$

(7.15)

is a locally uniformly bounded Radon measure sequence. From (7.13), we know that

$$\begin{aligned} \sqrt{\varepsilon } f_{r}^{\varepsilon } \longrightarrow 0 \qquad \text { in}\, W_{{\text {loc}}}^{-1, \frac{4}{3}}(\mathbb {R}_{+}^{2})\,\text { as}\, \varepsilon \rightarrow 0. \end{aligned}$$

(7.16)

Then it follows from (7.15)–(7.16) that the sequence:

$$\begin{aligned} \partial _{t} \eta ^{\varepsilon }+\partial _{r} q^{\varepsilon } \quad \text { is confined in a compact subset of}\, W_{\text {loc }}^{-1, p_{2}}(\mathbb {R}_{+}^{2})\,\text { for some}\, p_{2} \in (1,2), \end{aligned}$$

(7.17)

which also implies that

$$\begin{aligned} \partial _{t}\eta ^{\varepsilon }+\partial _{r}q^{\varepsilon }\qquad \text {is compact in}\, W_{\textrm{loc}}^{-1,p}(\mathbb {R}_{+}^2)\,\text { with}\, 1\le p\le p_{2}. \end{aligned}$$

(7.18)

On the other hand, the interpolation compactness theorem (cf. [8, 22]) indicates that, for $p_{2}>1, p_{1} \in \left( p_{2}, \infty \right]$, and $p_{0} \in \left[ p_{2}, p_{1}\right)$,

$$\begin{aligned}&\big (\text {compact set of }W_{\textrm{loc}}^{-1, p_{2}}(\mathbb {R}_{+}^{2})\big ) \cap \big (\text {bounded set of }W_{\textrm{loc}}^{-1, p_{1}}(\mathbb {R}_{+}^{2})\big )\\&\quad \subset \big (\text {compact set of }W_{\textrm{loc}}^{-1, p_{0}}(\mathbb {R}_{+}^{2})\big ), \end{aligned}$$

which is a generalization of Murat’s lemma in [58, 67]. Combining this theorem for $1<p_{2}<2$ and $p_{1}=2$ with the facts in (7.9) and (7.17), we conclude that

$$\begin{aligned} \partial _{t}\eta ^{\varepsilon }+\partial _{r}q^{\varepsilon }\qquad \text {is compact in}\, W_{\textrm{loc}}^{-1,p}(\mathbb {R}_{+}^2)\,\text { with}\, p_2\le p<2. \end{aligned}$$

(7.19)

Combining (7.19) with (7.18), we conclude (7.1). $\square$

8 $L^p$ Compensated Compactness Framework

In this section, with the help of our understanding of the singularities of the entropy kernel and entropy flux kernel obtained in Lemma 4.14, we now establish the $L^p$ compensated compactness framework and complete the proof of Theorem 2.2. The key ingredient is to prove the reduction of the Young measure. The arguments are similar to [63, Sect. 4] and [64, Sect. 7], based on [11, 13], so we only sketch the proof for self-containedness.

We denote the upper half-plane by

$$\begin{aligned} \mathbb {H}:=\{(\rho ,u)\in \mathbb {R}^2:\, \rho >0\} \end{aligned}$$

and consider the following subset of continuous functions:

$$\begin{aligned} \overline{C}(\mathbb {H}):=\left\{ \phi \in C(\overline{\mathbb {H}})\, : \,\, \begin{array}{l} \phi (\rho ,u) \,\text { is constant on the vacuum states }\, \{\rho =0\} \, \text { and} \\ \text {the map: } (\rho ,u) \mapsto \lim \limits _{s \rightarrow \infty } \phi (s \rho , s u) \text { belongs to } C(\mathbb {S}^{1} \cap \overline{\mathbb {H}}) \end{array}\right\} , \end{aligned}$$

where $\mathbb {S}^{1} \subset \mathbb {R}^{2}$ is the unit circle. Since $\overline{C}(\mathbb {H})$ is a complete sub-ring of the continuous functions on $\mathbb {H}$ containing the constant functions, there exists a compactification $\overline{\mathcal {H}}$ of $\mathbb {H}$ such that $C(\overline{\mathcal {H}})$ is isometrically isomorphic to $\overline{C}(\mathbb {H})$ (cf. [43]), written $C(\overline{\mathcal {H}})\cong \overline{C}(\mathbb {H})$. The topology of $\overline{\mathcal {H}}$ is the weak-star topology induced by $C(\overline{\mathcal {H}})$, i.e., a sequence $\{v_{n}\}_{n \in \mathbb {N}}$ in $\overline{\mathcal {H}}$ converges to $v \in \overline{\mathcal {H}}$ if $|\varphi (v_{n})-\varphi (v)| \rightarrow 0$ for all $\varphi \in C(\overline{\mathcal {H}})$, which is separable and metrizable (cf. [43]). Denote by V the weak-star closure of the vacuum states $\{(\rho ,u) \in \mathbb {R}^2:\, \rho =0\}$ and define $\mathcal {H}:=\mathbb {H}\cup V$. In view of the functions that lie in $\overline{C}(\mathbb {H})$, the topology of $\overline{\mathcal {H}}$ does not distinguish points in V. Since $\overline{\mathcal {H}}$ is homeomorphic to a compact metric space, we may apply the fundamental theorem of Young measures in Alberti-Müller [1, Theorem 2.4].

Lemma 8.1

([1, Theorem 2.4]). Given any sequence of measurable functions $\left( \rho ^{\varepsilon }, \rho ^{\varepsilon }u^{\varepsilon }\right) : \mathbb {R}_{+}^{2} \rightarrow \overline{\mathbb {H}}$, there exists a subsequence (still denoted) $(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })$ generating a Young measure $\nu _{(t, r)} \in {\text {Prob}}(\overline{\mathcal {H}})$ in the sense that, for any $\phi \in \overline{C}(\mathbb {H})$,

$$\begin{aligned} \phi (\rho ^{\varepsilon }(t, r), u^{\varepsilon }(t, r))\, \overset{*}{\rightharpoonup }\, \int _{\overline{\mathcal {H}}} \iota (\phi )(\rho ,u) \,\textrm{d} \nu _{(t, r)}(\rho , u) \qquad \text { in}\, L^{\infty }(\mathbb {R}_{+}^{2}), \end{aligned}$$

where $\iota :\overline{C}(\mathbb {H})\rightarrow C(\overline{\mathcal {H}})$ is an isometrically isomorphism. Moreover, sequence $(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })$ converges to $(\rho , \rho u):\,\mathbb {R}_{+}^2\rightarrow \overline{\mathcal {H}}$ a.e. if and only if

$$\begin{aligned} \nu _{(t,r)}(\rho ,u)=\delta _{(\rho (t,r),m(t,r))} \qquad \text {a.e.}\, (t,r)\in \mathbb {R}_{+}^2, \end{aligned}$$

in the phase coordinates $(\rho ,m)$ with $m=\rho u$.

From now on, we often use the same letter $\nu _{(t,r)}$ for an element of $\big (\overline{C}(\mathbb {H})\big )^{*}$ or $\big (C(\overline{\mathcal {H}})\big )^{*}$, and use the same letter for $\iota (\phi )$ and $\phi$ for simplicity, when no confusion arises.

The following lemma shows that the Young measure $\nu _{(t,r)}$, generated by the sequence of measurable approximate solutions $(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })$ satisfying the assumptions of Theorem 2.2, is only supported on the interior of $\mathcal {H}$. Moreover, the Young measure $\nu _{(t,r)}$ can be extended to a larger class of test functions than just $\overline{C}(\mathbb {H})$. This is proved in [13, Proposition 5.1]; also see [43, Proposition 2.3].

Lemma 8.2

([13, Proposition 5.1]). The following statements hold:

• (i) For the Young measure $\nu _{(t,r)}$ generated by the sequence of measurable approximate solutions $(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })$ satisfying the assumptions in Theorem 2.2,

  $$\begin{aligned} (t,r)\mapsto \int _{\mathbb {H}}(\rho ^{\gamma _2+1}+\rho |u|^3)\,\textrm{d}\nu _{(t,r)}(\rho ,u)\in L_{\textrm{loc}}^1(\mathbb {R}_{+}^2) . \end{aligned}$$

• (ii) Let $\phi (\rho ,u)$ be a function such that

  • (a) $\phi$ is continuous on $\overline{\mathbb {H}}$ and $\phi =0$ on $\partial \mathbb {H}$,

  • (b) there exists a constant $\mathfrak {a}>0$ such that ${\text {supp}}\phi \subset \{u+k(\rho )\ge -\mathfrak {a},u-k(\rho )\le \mathfrak {a}\}$,

  • (c) $|\phi (\rho ,u)|\le \rho ^{\beta (\gamma _2+1)}$ for all $(\rho ,u)$ with large $\rho$ and some $\beta \in (0,1)$.

  Then $\phi$ is $\nu _{(t,r)}$–integrable for $(t,r)\in \mathbb {R}_{+}^2$ a.e. and

  $$\begin{aligned} \phi (\rho ^{\varepsilon }(t,r), u^{\varepsilon }(t,r))\rightharpoonup \int _{\mathbb {H}}\phi (\rho ,u) \,\textrm{d}\nu _{(t,r)}(\rho ,u)\qquad \text {in }\;L_{\textrm{loc}}^1(\mathbb {R}_{+}^2). \end{aligned}$$

• (iii) $\nu _{(t,r)}\in {\text {Prob}}(\mathcal {H})$ for $(t,r)\in \mathbb {R}_{+}^2$ a.e., that is, $\, \nu _{(t,r)}\big (\overline{\mathcal {H}}\backslash (\mathbb {H}\cup V)\big )=0.$

We now prove the commutation relation. Since we only have the $W_{\textrm{loc}}^{-1,p}$–compactness of the entropy dissipation measures for $p\in [1,2)$, the classical div-curl lemma in [58] fails to obtain the commutation relation. Thus, we adopt an improved version of the div-curl lemma.

Lemma 8.3

([19, Theorem]). Let $\Omega \subset \mathbb {R}^n$ be an open bounded set, and $p,q\in (1,\infty )$ with $\frac{1}{p}+\frac{1}{q}=1$. Let $\textbf{v}^{\varepsilon }$ and $\textbf{w}^{\varepsilon }$ are sequences of vector fields such that

$$\begin{aligned} \textbf{v}^{\varepsilon }\rightharpoonup \textbf{v} \,\, \text { in }L^p(\Omega ;\mathbb {R}^n), \quad \textbf{w}^{\varepsilon }\rightharpoonup \textbf{w} \,\, \text { in }L^q(\Omega ;\mathbb {R}^n)\qquad \text { as }\varepsilon \rightarrow 0. \end{aligned}$$

Suppose that $\textbf{v}^{\varepsilon }\cdot \textbf{w}^{\varepsilon }$ is equi-integrable uniformly in $\varepsilon$, and

$$\begin{aligned} \begin{aligned}&{\text {div}} \textbf{v}^{\varepsilon } \quad \,\,\, \text { is}\,\text {(pre-)compact in }\,W^{-1,1}(\Omega ),\\&{\text {curl}} \textbf{w}^{\varepsilon } \quad \text { is}\, (\text {pre-})\text {compact in }W^{-1,1}(\Omega ;\mathbb {R}^{n\times n}). \end{aligned} \end{aligned}$$

Then $\textbf{v}^{\varepsilon }\cdot \textbf{w}^{\varepsilon }\,\rightharpoonup \,\textbf{v}\cdot \textbf{w}$ in $\mathcal {D}^{\prime }(\Omega )$.

Lemma 8.4

(Commutation relation). Let $\{(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })\}_{\varepsilon >0}$ be the measurable approximate solutions satisfying the assumptions of Theorem 2.2, and let $\nu _{(t,r)}$ be a Young measure generated by the family $\{(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })\}_{\varepsilon >0}$ in Lemma 8.2. Then

$$\begin{aligned} \overline{\chi (s_1)\sigma (s_2)-\chi (s_2)\sigma (s_1)}=\overline{\chi (s_1)}\;\overline{\sigma (s_2)}-\overline{\chi (s_2)}\;\overline{\sigma (s_1)} \end{aligned}$$

(8.1)

for all $s_1,s_2\in \mathbb {R}$, where $\displaystyle \overline{f}:=\int f\, \textrm{d}\nu _{(t,r)}$, $\chi (s_i)=\chi (\cdot , \cdot -s_i)$, and $\sigma (s_i)=\sigma (\cdot ,\cdot -s_i)$.

Proof

For any $\psi \in C_0^2(\mathbb {R})$, it follows from Lemmas 4.5, 4.7, and 4.10 that

$$\begin{aligned} |\eta ^{\psi }(\rho ,m)|\le C_{\psi }\rho ,\qquad |q^{\psi }(\rho ,m)|\le C_{\psi }\big (\rho +\rho ^{1+\theta _2}\big ). \end{aligned}$$

(8.2)

It is clear that the support of $(\eta ^{\psi }, q^{\psi })$ is contained in $\left\{ k(\rho )+u \ge -L, u-k(\rho ) \le L\right\}$ for some $L>0$ depending only on ${\text {supp}}\,\psi$. For any $\psi _{1}, \psi _{2} \in C_{0}^{2}(\mathbb {R})$, we consider the sequences of vector fields:

$$\begin{aligned} \textbf{v}^{\varepsilon }=(\eta ^{\psi _{1}}(\rho ^{\varepsilon }, \rho ^{\varepsilon } u^{\varepsilon }), q^{\psi _{1}}(\rho ^{\varepsilon }, \rho ^{\varepsilon } u^{\varepsilon })), \qquad \textbf{w}^{\varepsilon }=(q^{\psi _{2}}(\rho ^{\varepsilon }, \rho ^{\varepsilon } u^{\varepsilon }), -\eta ^{\psi _{2}}(\rho ^{\varepsilon }, \rho ^{\varepsilon } u^{\varepsilon })). \end{aligned}$$

Noting $\rho ^{\varepsilon }\in L_{\textrm{loc}}^{1+\gamma _2}(\mathbb {R}_{+}^2)$ and (8.2), we see that both $\textbf{v}^{\varepsilon }$ and $\textbf{w}^{\varepsilon }$ are uniformly bounded sequences in $L_{\textrm{loc}}^2(\mathbb {R}_{+}^2)$. Moreover, by Lemma 8.2 and the uniqueness of weak limits, we obtain

$$\begin{aligned} \textbf{v}^{\varepsilon } \rightharpoonup (\overline{\eta ^{\psi _{1}}}, \overline{q^{\psi _{1}}})\,\,\, \text { in}\, L_{\textrm{loc}}^{2}(\mathbb {R}_{+}^2), \qquad \textbf{w}^{\varepsilon } \rightharpoonup (\overline{q^{\psi _{2}}},-\overline{\eta ^{\psi _{2}}}) \,\,\,\text { in}\,L_{\textrm{loc}}^{2}(\mathbb {R}_{+}^2). \end{aligned}$$

By direct calculation, we see that

$$\begin{aligned} \begin{aligned}&{\text {div}} \textbf{v}^{\varepsilon } =\partial _{t}\eta ^{\psi _1}(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })+\partial _{r}q^{\psi _1}(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon }) \quad \,\,\,\text {are compact in }W_{\textrm{loc}}^{-1,1}(\mathbb {R}_{+}^2),\\&{\text {curl}} \textbf{w}^{\varepsilon }=\partial _{t}\eta ^{\psi _2}(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon }) +\partial _{r}q^{\psi _2}(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })\quad \text {are compact in }W_{\textrm{loc}}^{-1,1}(\mathbb {R}_{+}^2). \end{aligned} \end{aligned}$$

Using (8.2), we obtain that $\vert \textbf{v}^{\varepsilon }\cdot \textbf{w}^{\varepsilon }\vert \le C\big ((\rho ^{\varepsilon })^2+(\rho ^{\varepsilon })^{2+\theta _2}\big )$ for $\rho >0$ which, with (2.25), yields that $\textbf{v}^{\varepsilon }\cdot \textbf{w}^{\varepsilon }\in L_{\textrm{loc}}^{\frac{1+\gamma _2}{2+\theta _2}}(\mathbb {R}_{+}^2)$ uniformly in $\varepsilon$. Thus, $\textbf{v}^{\varepsilon }\cdot \textbf{w}^{\varepsilon }$ is equi-integrable uniformly in $\varepsilon$ since $\frac{1+\gamma _2}{2+\theta _2}>1$. It follows from Lemma 8.3 that

$$\begin{aligned} \textbf{v}^{\varepsilon } \cdot \textbf{w}^{\varepsilon }\, \rightarrow \, \overline{\eta ^{\psi _{1}}}\;\overline{q^{\psi _{2}}}-\overline{\eta ^{\psi _{2}}}\; \overline{q^{\psi _{1}}} \qquad \text {in the sense of distributions in}\, \mathbb {R}_{+}^2. \end{aligned}$$

(8.3)

On the other hand, using (8.2) and Lemma 8.2, we find that

$$\begin{aligned} \textbf{v}^{\varepsilon } \cdot \textbf{w}^{\varepsilon } \,\rightarrow \, \overline{\eta ^{\psi _{1}} q^{\psi _{2}}-\eta ^{\psi _{2}} q^{\psi _{1}}} \qquad \text { in}\, L_{\textrm{loc}}^{1}(\mathbb {R}_{+}^2), \end{aligned}$$

which, with (8.3), yields that

$$\begin{aligned} \overline{\eta ^{\psi _{1}} q^{\psi _{2}}-\eta ^{\psi _{2}} q^{\psi _{1}}}=\overline{\eta ^{\psi _{1}}} \;\overline{q^{\psi _{2}}}-\overline{\eta ^{\psi _{2}}} \;\overline{q^{\psi _{1}}}. \end{aligned}$$

(8.4)

It follows from (8.4) and Fubini’s theorem that

$$\begin{aligned}&\int _{{\mathbb {R}}^2} \Big (\overline{\chi (s_{1}) \sigma (s_{2})-\chi (s_{2}) \sigma (s_{1})} - \overline{\chi (s_{1})}\;\overline{\sigma (s_{2})}+\overline{\chi (s_{2})}\;\overline{\sigma (s_{1})}\Big ) \psi _{1}(s_{1}) \psi _{2}(s_{2}) \,\text {d} s_{1} \text {d} s_{2}=0. \end{aligned}$$

Since $\psi _{1}, \psi _{2} \in C_{0}^{2}(\mathbb {R})$ are arbitrary, we conclude

$$\begin{aligned} \overline{\chi (s_{1})\sigma (s_{2})-\chi (s_{2}) \sigma (s_{1})}=\overline{\chi (s_{1})} \;\overline{\sigma (s_{2})}-\overline{\chi (s_{2})} \;\overline{\sigma (s_{1})}\qquad \text{ for } \text{ any } s_1, s_2\in \mathbb {R}. \end{aligned}$$

This completes the proof. $\square$

Theorem 8.5

(Reduction of the Young measure). Let $\nu _{(t,r)}\in \textrm{Prob}(\mathcal {H})$ be the Young measure generated by sequence $\{(\rho ^{\varepsilon },\rho ^{\varepsilon }u^{\varepsilon })\}_{\varepsilon >0}$ in Lemma 8.2. Then either $\nu _{(t,r)}$ is contained in V or the support of $\nu _{(t,r)}$ is a single point in $\mathbb {H}$.

Proof

The proof is similar to [11, 48, 63, 64]. Since the estimates of the entropy kernel and entropy flux kernel are different, we sketch the proof for self-containedness.

Taking $s_1,s_2,s_3\in \mathbb {R}$ and multiplying (8.1) by $\overline{\chi (s_3)}$, one obtains

$$\begin{aligned} \overline{\chi (s_3)}\;\overline{\chi (s_1)\,\sigma (s_2)-\chi (s_2)\,\sigma (s_1)} =\overline{\chi (s_3)}\;\overline{\chi (s_1)}\;\overline{\sigma (s_2)}-\overline{\chi (s_3)}\;\overline{\chi (s_2)}\;\overline{\sigma (s_1)}. \end{aligned}$$

Cyclically permuting index $s_j$ and adding the resultant equations together, we have

$$\begin{aligned}&\overline{\chi (s_{1})}\; \overline{\chi (s_{2}) \sigma (s_{3})-\chi (s_{3}) \sigma (s_{2})}\\&\quad = \overline{\chi (s_{3})}\; \overline{\chi (s_{2}) \sigma (s_{1})-\chi (s_{1}) \sigma (s_{2})} -\overline{\chi (s_{2})}\; \overline{\chi (s_{3}) \sigma (s_{1})-\chi (s_{1}) \sigma (s_{3})} . \end{aligned}$$

Applying the fractional derivative operators $P_{2}:=\partial _{s_{2}}^{\lambda _1+1}$ and $P_{3}:=\partial _{s_{3}}^{\lambda _1+1}$ in the sense of distributions to obtain

$$\begin{aligned}&\overline{\chi (s_1)}\;\overline{P_2\chi (s_2)\,P_3\sigma (s_3)-P_3\chi (s_3)\,P_2\sigma (s_2)}\nonumber \\&\quad =\overline{P_3\chi (s_3)}\;\overline{P_2\chi (s_2)\,\sigma (s_1)-\chi (s_1)\,P_2\sigma (s_2)}\nonumber \\&\qquad -\overline{P_2\chi (s_2)}\;\overline{P_3\chi (s_3)\,\sigma (s_1)-\chi (s_1)\,P_3\sigma (s_3)}, \end{aligned}$$

(8.5)

where, for example, distribution $\overline{P_2\chi (s_2)}$ is defined by

$$\begin{aligned} \langle \overline{P_2\chi (s_2)},\psi \rangle =-\int _{\mathbb {R}}\overline{\partial _{s_2}^{\lambda _1}\chi (s_2)} \,\psi '(s_2)\,\textrm{d}s_2\qquad \, \text {for }\, \psi \in \mathcal {D}(\mathbb {R}). \end{aligned}$$

We take two standard but different functions $\phi _{2}, \phi _{3} \in C_{0}^{\infty }(-1,1)$ such that $\displaystyle \int _{\mathbb {R}} \phi _{j}(s_{j})\,\textrm{d} s_{j}=1$ with $\phi _{j} \ge 0$ for $j=2,3$. For $\tau >0$, denote $\phi _{j}^{\tau }(s_{j}):=\frac{1}{\tau }\phi _{j}(\frac{s_{j}}{\tau })$. As indicated in [43], we can always choose $\phi _2$ and $\phi _3$ such that

$$\begin{aligned} Y(\phi _{2}, \phi _{3})=\int _{-\infty }^{\infty } \int _{-\infty }^{s_{2}} \big (\phi _{2}(s_{2}) \phi _{3}(s_{3}) -\phi _{2}(s_{3}) \phi _{3}(s_{2})\big ) \,\textrm{d} s_{3} \textrm{d}s_{2}>0. \end{aligned}$$

(8.6)

Multiplying (8.5) by $\phi _{2}^{\tau }(s_1-s_2)\phi _3^{\tau }(s_1-s_3)$ and integrating the resultant equation with respect to $(s_2, s_3)$ yield

$$\begin{aligned}&\overline{\chi (s_1)}\;\overline{P_2\chi _{2}^{\tau }\,P_3\sigma _3^{\tau }-P_3\chi _3^{\tau }\,P_2\sigma _{2}^{\tau }}\nonumber \\&\quad =\overline{P_3\chi _{3}^{\tau }}\;\overline{P_2\chi _2^{\tau }\,\sigma _1-\chi _1\,P_2\sigma _2^{\tau }} -\overline{P_2\chi _2^{\tau }}\;\overline{P_3\chi _3^{\tau }\,\sigma _1-\chi _1\,P_3\sigma _3^{\tau }}, \end{aligned}$$

(8.7)

where we have used the notion: $\overline{P_{j} \chi _{j}^{\tau }}=\overline{P_{j} \chi _{j}} * \phi _{j}^{\tau }(s_{1}) =\int _{\mathbb {R}} \overline{\partial _{s_{j}}^{\lambda } \chi (s_{j})} \frac{1}{\tau ^{2}} \phi _{j}^{\prime }(\frac{s_{1}-s_{j}}{\tau }) \,\textrm{d}s_{j}$ for $j=2,3$.

Multiplying (8.7) by $\psi (s_1)\in \mathcal {D}(\mathbb {R})$, integrating the resultant equation with respect to $s_1$, then taking limit $\tau \rightarrow 0$ and applying Lemmas 8.8–8.9 below, we obtain

$$\begin{aligned} Y(\phi _{2}, \phi _{3}) \int _{\mathcal {H}} Z(\rho ) \sum _{\pm }(K^{\pm })^{2}\, \overline{\chi (u \pm k(\rho ))}\, \psi (u \pm k(\rho )) \,\textrm{d} \nu _{(t,r)}(\rho , u)=0. \end{aligned}$$

(8.8)

Noting that $Z(\rho )>0$ for $\rho >0$ from Lemma 8.7 below, $Y(\phi _2,\phi _3)> 0$ from (8.6), and $\psi (s)$ is an arbitrary test function, we deduce from (8.8) that

$$\begin{aligned} \int _{\mathcal {H}} Z(\rho )\,\overline{\chi (u\pm k(\rho ))}\,\textrm{d} \nu _{(t,r)}(\rho , u)=0. \end{aligned}$$

(8.9)

We define $\mathbb {S}=\{s\in \mathbb {R}:\,\overline{\chi (s)}>0\}$. It follows from [63] that $\mathbb {S}$ admits the representation:

$$\begin{aligned} \mathbb {S}=\left\{ s\in \mathbb {R}:\, u-k(\rho )<s<u+k(\rho )\,\, \text { with}\, (\rho ,u)\in {\text {supp}}\nu _{(t,r)}\right\} . \end{aligned}$$

For the case: $\mathbb {S}=\emptyset$, it is clear that $\overline{\chi (s)}=0$ for all $s\in \mathbb {R}$ so that ${\text {supp}}\nu _{(t,r)}\subset V$, since $\chi (s)>0$ for all $\rho >0$ and $s\in (u-k(\rho ), u+k(\rho ))$.

For the case: $\mathbb {S}\ne \emptyset$, it follows from (8.15) below that $s\mapsto \overline{\chi (s)}$ is a continuous map. Then $\mathbb {S}$ is an open set so that $\mathbb {S}$ is at most a countable union of open intervals. Thus, we may write

for at most countably many numbers $\zeta _k:=u_{k}-k(\rho _{k})$ and $\xi _k:=u_{k}+k(\rho _{k})$ with $(\rho _{k},u_{k})\in \textrm{supp}\nu _{(t,r)}$ in the extended real line $\mathbb {R}\cup \{\pm \infty \}$ such that $\zeta _k<\xi _k\le \zeta _{k+1}$ for all k. For later use, we denote the Riemann invariants $z(\rho ,u):=u-k(\rho )$ and $w(\rho ,u):=u+k(\rho )$. Thus, noting that $\textrm{supp}\,\chi (s)=\{(\rho ,u)\,:\,z(\rho ,u)\le s\le w(\rho ,u)\}$, we obtain

If $\zeta _{k}$ and $\xi _{k}$ are both finite, due to the fact that $k(\rho )$ is a strictly monotone increasing and unbounded function of $\rho$, it is clear that $\left\{ (\rho , u) \,:\, \zeta _{k} \le z(\rho , u)\le w(\rho , u) \le \xi _{k}\right\}$ is bounded. Now we deduce from (8.9) that

$$\begin{aligned} {\text {supp}} \nu _{(t,r)} \cap \{(\rho , u) \in \mathbb {H}:\, \zeta _{k}< z(\rho ,u)<w(\rho , u)<\xi _{k}\} =\emptyset \qquad \text {for all } k. \end{aligned}$$

Thus, the support of measure $\nu _{(t,r)}$ must be contained in the vacuum set V and at most a countable union of points $P_k(\rho _{k},\,u_{k})$:

Therefore, we may write

$$\begin{aligned} \nu _{(t,r)}=\nu _{V}+\sum _{k} \alpha _{k} \delta _{P_{k}} \qquad \, \text{ for } \text{ all } \alpha _{k} \in [0,1] \end{aligned}$$

(8.10)

with measure $\nu _{V}$ supported on the vacuum set V. For later use, we denote

$$\begin{aligned} \chi (P_{k},s):=\chi (\rho _{k}, u_{k}, s), \quad \sigma (P_{k}, s):=\sigma (\rho _{k}, u_{k}, s) \qquad \,\, \text {for }\, s\in \mathbb {R}. \end{aligned}$$

We claim that, if $\chi (P_{k},s)>0$, then $\chi (P_{k'},s)=0$ for all $k\ne k'$. Indeed, recall that ${\text {supp}}\chi (s)=\{(\rho ,u)\,:\,z(\rho ,u)\le s\le w(\rho ,u)\}$ and that $\chi (\rho , u, s)>0$ if and only if $z(\rho ,u)<s<w(\rho ,u)$. If $\chi (P_{k},s)>0$, then $\zeta _{k}<s<\xi _{k}$. If, in addition, $\chi (P_{k'},s)>0$ for some $k\ne k'$, it must hold that $\zeta _{k'}< s < \xi _{k'}$. However, since $\xi _{k-1}\le \zeta _{k}< \xi _{k}\le \zeta _{k+1}$, this is impossible for any $P_{k'}$ with $k' \ne k$.

Thus, taking $s_{1}, s_{2} \in \mathbb {R}$ such that $\chi (P_{k},s_{1})\chi (P_{k},s_{2})>0$, we deduce from the commutation relation (8.1) and (8.10) that

$$\begin{aligned} (\alpha _{k}-\alpha _{k}^{2}) \big (\chi (P_{k}, s_{1}) \sigma (P_{k}, s_{2})-\chi (P_{k}, s_{2}) \sigma (P_{k}, s_{1})\big )=0. \end{aligned}$$

Now, choosing $s_{1}$ and $s_{2}$ such that the second factor in this expression is non-zero, we obtain that $\alpha _k=0$ or 1 for all k. This completes the proof. $\square$

Combining Theorem 8.5 with Lemma 8.1, we conclude that $(\rho ^{\varepsilon },m^{\varepsilon })$ converges to $(\rho ,m)$ almost everywhere. Moreover, noting that $|m^{\varepsilon }|^{\frac{3(\gamma _{2}+1)}{\gamma _{2}+3}}\le C\big ((\rho ^{\varepsilon })^{\gamma _{2}+1}+\rho ^{\varepsilon }|u^{\varepsilon }|^3\big )$ for any $T,d,D>0$, we have

$$\begin{aligned} \int _{0}^{T}\int _{d}^{D}|m^{\varepsilon }|^{\frac{3(\gamma _{2}+1)}{\gamma _{2}+3}}\,\textrm{d}r\textrm{d}t \le C\int _{0}^{T}\int _{d}^{D}\big ((\rho ^{\varepsilon })^{\gamma _{2}+1}+\rho ^{\varepsilon }|u^{\varepsilon }|^3\big )\,\textrm{d}r\textrm{d}t\le C(d,D,T), \end{aligned}$$

which implies that $m^{\varepsilon }$ is uniformly bounded in $L_{\textrm{loc}}^{\frac{3(\gamma _{2}+1)}{\gamma _{2}+3}}(\mathbb {R}_{+}^2)$ with respect to $\varepsilon$. This implies that (2.36) holds. Therefore, the proof of Theorem 2.2 is complete.

Now, we are going to prove the auxiliary lemmas, Lemmas 8.8–8.9, which are used in the proof of Theorem 8.5. We first recall two useful lemmas in [11, 43].

Lemma 8.6

([43, Lemmas 3.8–3.9]). Let $\mathfrak {R} \in C_{\textrm{loc }}^{\alpha }(\mathbb {R})$ be a Hölder continuous function for some $\alpha \in (0,1)$, and let $g \in C_{0}^{\alpha }(\mathbb {R})$ be a Hölder continuous function with compact support. Assume $L_{0}>2$ such that ${\text {supp}} g \subset B_{L_0-2}(0)$.

• (i) For any pair of distributions $T_{2}, T_{3} \in \mathcal {D}^{\prime }(\mathbb {R})$ from the following collection:

  $$\begin{aligned} (T_{2}, T_{3})=(\delta , Q_{3}), \,\, (PV, Q_{3}), \,\, (Q_{2}, Q_{3}) \end{aligned}$$

  with $Q_{2}, Q_{3} \in \{H, Ci, {\mathfrak {R}}\}$, there exists a constant $C>0$ independent of $(\rho , u)$ such that

  $$\begin{aligned}&\sup \limits _{\tau \in (0,1)}\Big \vert \int _{-\infty }^{\infty } g(s_{1})\Big \{\big (T_{2}(s_{2}-u \pm k(\rho )) T_{3}(s_{3}-u \pm k(\rho ))\\&\qquad \qquad \qquad \qquad \qquad \quad -T_{2}(s_{3}-u\pm k(\rho ))T_{3}(s_{2}-u\pm k(\rho ))\big ) * \phi _{2}^{\tau } * \phi _{3}^{\tau }\Big \} \left( s_{1}\right) \,\textrm{d} s_{1}\Big \vert \\&\quad \le C\Vert g\Vert _{C^{ \alpha }(\mathbb {R})} \big (1+\Vert \mathfrak {R}\Vert _{C^{ \alpha }(\overline{B_{L_0}(0)})}\big )^{2}. \end{aligned}$$

• (ii) For any pair of distributions from

  $$\begin{aligned} (T_{2}, T_{3})=(\delta , \delta ), \,\,(PV, PV),\,\, (Q_{2}, Q_{3}),\,\, (\delta , PV), \,\, (PV, Q_{3}), \end{aligned}$$

  with $Q_{2}, Q_{3} \in \{H, Ci, \mathfrak {R}\}$, there exists $C>0$ independent of $(\rho ,u)$ such that

  $$\begin{aligned}&\sup \limits _{\tau \in (0,1)} \Big \vert \int _{-\infty }^{\infty }\Big \{\big ((s_{2}-s_{3}) T_{2}(s_{2} -u \pm k(\rho ))T_{3}(s_{3}-u \pm k(\rho ))\big )* \phi _{2}^{\tau } * \phi _{3}^{\tau }\Big \}(s_{1}) \,\textrm{d} s_{1} \Big \vert \\&\quad \le C\Vert g\Vert _{C^{ \alpha }(\mathbb {R})}\big (1+\Vert \mathfrak {R}\Vert _{C^{ \alpha }(\overline{B_{L_0}(0)})}\big )^{2}. \end{aligned}$$

Motivated by [11], it follows from Lemmas 4.2–4.3, (1.4), and a direct calculation that

$$\begin{aligned} D(\rho )&:=a_1(\rho )b_1(\rho )-2k(\rho )^2(a_1(\rho )b_2(\rho )-a_2(\rho )b_1(\rho ))\nonumber \\&=\frac{M_{\lambda _1}^2}{2(\lambda _1+1)}k(\rho )^{-2\lambda _1}k'(\rho )^{-2} \big (k'(\rho )+(\rho k'(\rho ))'\big )>0\qquad \, \text {for }\, \rho >0. \end{aligned}$$

(8.11)

Lemma 8.7

([11, Lemmas 4.2–4.3]). The mollified fractional derivatives of the entropy kernel and the entropy flux kernel satisfy the following convergence properties:

• (i) When $0\le \rho <\infty$,

  $$\begin{aligned} P_{2} \chi _{2}^{\tau } P_{3} \sigma _{3}^{\tau }-P_{3} \chi _{3}^{\tau } P_{2} \sigma _{2}^{\tau } \longrightarrow Y(\phi _{2}, \phi _{3}) Z(\rho ) \sum _{\pm }(K^{\pm })^{2} \delta _{s_{1}=u \pm k(\rho )} \end{aligned}$$

  as $\tau \rightarrow 0$ weakly-star in measures in $s_{1}$ and locally uniformly in $(\rho , u)$, where $Y(\phi _2,\phi _3)$ satisfies (8.6), $Z(\rho ):=(\lambda _1+1) M_{\lambda }^{-2} k(\rho )^{2 \lambda } D(\rho )>0$ with $D(\rho )$ defined in (8.11), and $K^{\pm }\ne 0$ are some constants.

• (ii) For $j=2,3$, $\chi _1\,P_{j}\sigma _{j}^{\tau }-\sigma _1\,P_{j}\chi _{j}^{\tau }$ are Hölder continuous in $(\rho ,u,s_1)$, uniformly in $\tau$, and there exists a Hölder continuous function $X=X(\rho ,u,s_1)$, independent of the mollifying sequence $\phi _j$, such that

  $$\begin{aligned} \chi (s_1) P_{j} \sigma _{j}^{\tau }-P_{j} \chi _{j}^{\tau } \sigma (s_1) \longrightarrow X(\rho , u, s_{1}) \qquad \text {as}\, \tau \rightarrow 0 \end{aligned}$$

  uniformly in $\left( \rho , u, s_{1}\right)$ on the sets on which $\rho$ is bounded.

Lemma 8.8

For any test function $\psi \in \mathcal {D}(\mathbb {R})$,

$$\begin{aligned}&\lim \limits _{\tau \rightarrow 0}\int _{\mathbb {R}}\overline{\chi (s_1)}\;\overline{P_2\chi _{2}^{\tau }P_3\sigma _3^{\tau } -P_3\chi _3^{\tau }P_2\sigma _2^{\tau }}(s_1)\psi (s_1)\,\textrm{d}s_1\nonumber \\&\quad =Y(\phi _2,\phi _3)\int _{\mathcal {H}}Z(\rho ) \sum \limits _{\pm }(K^{\pm })^2\,\overline{\chi (u\pm k(\rho ))}\,\psi (u\pm k(\rho ))\,\textrm{d}\nu _{(t,r)}(\rho ,u), \end{aligned}$$

(8.12)

where $Y(\phi _2,\phi _3)$ is defined by (8.6) and $Z(\rho )$ is given in Lemma 8.7.

Proof

It follows from Lemma 8.7 that, when $\rho$ is bounded,

$$\begin{aligned} P_{2} \chi _{2}^{\tau } P_{3} \sigma _{3}^{\tau }-P_{3} \chi _{3}^{\tau } P_{2} \sigma _{2}^{\tau } \rightarrow Y(\phi _{2}, \phi _{3}) Z(\rho ) \sum _{\pm }\left( K^{\pm }\right) ^{2} \delta _{s_{1}=u \pm k(\rho )}\qquad \text {as }\tau \rightarrow 0 \end{aligned}$$

locally uniform in $(\rho ,u)$ and hence pointwise for all $(\rho ,u)$. Therefore, we have

$$\begin{aligned}&\lim _{\tau \rightarrow 0} \int _{-\infty }^{\infty } \overline{\chi (s_{1})}\langle \,\nu _{(t,r)},\, (P_{2} \chi _{2}^{\tau } P_{3} \sigma _{3}^{\tau }-P_{3} \chi _{3}^{\tau } P_{2} \sigma _{2}^{\tau }) {\textbf{I}}_{\{\rho \le \rho ^{*}\}}\rangle \psi (s_{1})\,\textrm{d} s_{1}\nonumber \\&\quad =\lim _{\tau \rightarrow 0}\langle \,\nu _{(t,r)},\, \int _{-\infty }^{\infty } \overline{\chi (s_{1})}(P_{2} \chi _{2}^{\tau } P_{3} \sigma _{3}^{\tau }-P_{3} \chi _{3}^{\tau } P_{2} \sigma _{2}^{\tau }) \psi (s_{1}) \,\textrm{d} s_{1}{\textbf{I}}_{\{\rho \le \rho ^{*}\}}\rangle \nonumber \\&\quad =\langle \,\nu _{(t,r)}, \, Y(\phi _{2}, \phi _{3}) Z(\rho ) \sum _{\pm }(K^{\pm })^{2} \overline{\chi (u \pm k(\rho ))} \psi (u \pm k(\rho )) {\textbf{I}}_{\{\rho \le \rho ^{*}\}}\rangle \nonumber \\&\quad =Y(\phi _{2}, \phi _{3}) \sum _{\pm }(K^{\pm })^{2}\langle \,\nu _{(t,r)}, \, Z(\rho ) \overline{\chi (u \pm k(\rho ))} \psi (u \pm k(\rho )){\textbf{I}}_{\{\rho \le \rho ^{*}\}} \rangle . \end{aligned}$$

(8.13)

For $\rho \ge \rho ^{*}$, we notice that

$$\begin{aligned} P_2\chi _{2}^{\tau }\,P_3\sigma _3^{\tau }-P_3\chi _{3}^{\tau }\, P_2\sigma _2^{\tau } =P_2\chi _{2}^{\tau }\,P_3(\sigma _3^{\tau }-u\chi _{3}^{\tau })-P_3\chi _{3}^{\tau }\,P_2(\sigma _{2}^{\tau }-u\chi _{2}^{\tau }). \end{aligned}$$

(8.14)

Using Lemma 4.14, we see that (8.14) consists of a sum of terms of the form:

$$\begin{aligned} A_{i,\pm }(\rho )B_{j,\pm }(\rho )(s_{3}-s_{2})T_2(s_2-u\pm k(\rho ))T_3(s_3-u\pm k(\rho )) \end{aligned}$$

with $T_{2}, T_{3} \in \{\delta , \textrm{PV},H, \textrm{Ci}\}$, the terms of the form:

$$\begin{aligned}{} & {} A_{i,\pm }(\rho )B_{j,\pm }(\rho ) \big (T_2(s_2-u\pm k(\rho ))T_3(s_3-u\pm k(\rho ))\\{} & {} \qquad \qquad \qquad \qquad \quad -T_2(s_3-u\pm k(\rho ))T_3(s_2-u\pm k(\rho ))\big ), \end{aligned}$$

and the terms of the form:

$$\begin{aligned}{} & {} A_{i,\pm }(\rho )B_{j,\pm }(\rho )(s_{3}-u)\big (T_2(s_2-u\pm k(\rho ))T_3(s_3-u\pm k(\rho ))\\{} & {} \qquad \qquad \qquad \qquad \qquad \qquad \quad -T_2(s_3-u\pm k(\rho ))T_3(s_2-u\pm k(\rho ))\big ) \end{aligned}$$

with $T_{2} \in \{\delta , H, \textrm{PV}, \textrm{Ci}, r_{\chi }\}$ and $T_{3} \in \{H, \textrm{Ci}, r_{\sigma }\}$. We emphasize that, in the last two cases, when $T_2,T_3\in \{r_{\chi },r_{\sigma }\}$, $A_{i,\pm }(\rho )B_{j,\pm }(\rho )$ or $A_{i,\pm }(\rho )B_{j,\pm }(\rho )(s_{k}-u)$ should be replaced by 1.

Before applying Lemma 8.6, we now show that $\overline{\chi (s)}$ is Hölder continuous. In fact, it follows from Corollary 4.12 and Lemma 8.2 that, for any $s,s'\in \mathbb {R}$ and $\alpha \in (0,\min \{\lambda _1,1\}]$,

$$\begin{aligned} \sup _{s,s'\in \mathbb {R}}\frac{|\overline{\chi (s)}-\overline{\chi (s')}|}{|s-s'|^{\alpha }} =\int _{\mathcal {H}}\frac{|\chi (s)-\chi (s')|}{|s-s'|^{\alpha }}\,\textrm{d}\nu _{(t,r)}\le \int _{\mathcal {H}}C\big (1+\rho |\ln \rho |\big )\,\textrm{d}\nu _{(t,r)}<\infty , \end{aligned}$$

(8.15)

which implies that $\overline{\chi (s)}$ is Hölder continuous. Hence, using Lemma 8.6 and the fact that $|s_{j}-u|\le k(\rho )$ for $j=2,3$, we obtain

$$\begin{aligned} \begin{aligned}&\Big \vert \int _{-\infty }^{\infty }\overline{\chi (s_{1})}(P_{2} \chi _{2}^{\tau } P_{3} \sigma _{3}^{\tau } -P_{3} \chi _{3}^{\tau } P_{2} \sigma _{2}^{\tau }) \psi (s_{1}) \,\textrm{d}s_{1}{\textbf{I}}_{\{\rho \ge \rho ^{*}\}}\Big \vert \\&\quad \le C\max _{j, k, \pm }\Big \{|A_{j, \pm } k(\rho )|\big (|B_{k, \pm }|+ \Vert r_{\sigma }(\rho ,\cdot )\Vert _{C^{\alpha _1}(\overline{B_{L_0}})}\big ),\nonumber \\&\, \Vert r_{\chi }(\rho ,\cdot )\Vert _{C^{ \alpha _1}(\overline{B_{L_0}})} \big (|B_{j, \pm }k(\rho )|+\Vert r_{\sigma }(\rho ,\cdot )\Vert _{C^{ \alpha _1}(\overline{B_{L_0}})}\big )\Big \}\\&\quad \le C\big (1+\rho ^{2+\theta _2}\big ) = C\big (\rho ^{\beta (\gamma _2+1)}+1\big )\qquad \, \text {for }\, \rho \ge \rho ^{*}, \end{aligned} \end{aligned}$$

with $L_{0}:=|{\text {supp}}\psi |+2$ and $\beta =\frac{\theta _2+2}{\gamma _2+1}\in (0,1)$. Thus, using Lemmas 8.2 and 8.7, and Lebesgue’s dominated convergence theorem, we obtain

$$\begin{aligned} \begin{aligned}&\lim _{\tau \rightarrow 0} \int _{-\infty }^{\infty } \overline{\chi (s_{1})} \langle \,\nu _{(t,r)},\,(P_{2} \chi _{2}^{\tau }\, P_{3} \sigma _{3}^{\tau }-P_{3} \chi _{3}^{\tau }\, P_{2} \sigma _{2}^{\tau }) {\textbf{I}}_{\{\rho \ge \rho ^{*}\}}\rangle \,\psi (s_{1})\,\textrm{d}s_{1} \\&\quad =Y(\phi _{2}, \phi _{3}) \sum _{\pm }(K^{\pm })^{2}\langle \nu _{(t,r)},\, Z(\rho )\, \overline{\chi (u \pm k(\rho ))} \,\psi (u \pm k(\rho )){\textbf{I}}_{\{\rho \ge \rho ^{*}\}}\rangle , \end{aligned} \end{aligned}$$

which, with (8.13), yields (8.12). This completes the proof. $\square$

Lemma 8.9

For any test function $\psi \in \mathcal {D}(\mathbb {R})$,

$$\begin{aligned}&\lim \limits _{\tau \rightarrow 0}\int _{\mathbb {R}} \Big (\overline{P_3\chi _3^{\tau }}\;\,\overline{P_2\chi _2^{\tau }\,\sigma (s_1)-\chi (s_1)\,P_2\sigma _2^{\tau }}\nonumber \\&\qquad \qquad \quad -\overline{P_2\chi _2^{\tau }}\;\,\overline{P_3\chi _3^{\tau }\,\sigma (s_1) +\chi (s_1)\,P_3\sigma _3^{\tau }}\Big )\,\psi (s_1)\,\textrm{d}s_1=0. \end{aligned}$$

(8.16)

Proof

Fix $(\rho ,u)\in \mathbb {H}$. It follows from Lemma 8.7 that

$$\begin{aligned} \big (\chi (s_1)\,P_3\sigma _3^{\tau }-P_3\chi _3^{\tau }\,\sigma (s_1)\big )(\rho ,u,s_1) \longrightarrow X(\rho ,u,s_1)\qquad \text {uniformly in }s_1\text { as }\tau \rightarrow 0. \end{aligned}$$

It is clear that

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}} \overline{P_{2} \chi _{2}^{\tau }}\,\big (\chi (s_1)\, P_{3} \sigma _{3}^{\tau } -P_{3} \chi _{3}^{\tau } \, \sigma (s_1)\big )\, \psi (s_{1}) \,\textrm{d}s_{1} \\&\quad =\int _{\mathcal {H}}\! \int _{\mathbb {R}} (P_{2} \chi _{2}^{\tau })(\tilde{\rho }, \tilde{u}, s_{1})\, \big (\chi (s_1)\, P_{3} \sigma _{3}^{\tau }\!-\! P_{3} \chi _{3}^{\tau } \,\sigma (s_1)\big )(\rho , u, s_{1})\,\psi (s_{1}) \,\textrm{d}s_{1} \textrm{d} \nu _{(t,r)}(\tilde{\rho }, \tilde{u}). \end{aligned} \end{aligned}$$

It follows from Lemma 4.14 that $P_{j} \chi _{j}^{\tau }, j=2,3$, are measures in $s_{1}$ such that

$$\begin{aligned} \Vert P_{j} \chi _{j}^{\tau }(\tilde{\rho }, \tilde{u}, \cdot )\Vert _{\mathfrak {M}, \alpha } \le C_{\alpha }\tilde{\rho } \qquad \text { for large } \tilde{\rho }, \end{aligned}$$

where $\Vert \mu \Vert _{\mathfrak {M}, \alpha } =\sup \left\{ |\langle \mu , f\rangle |\,:\, f\in C_{0}^{ \alpha }(\mathbb {R})\right.$ and $\left. \Vert f\Vert _{C^{ \alpha }(\mathbb {R})} \le 1\right\}$ with $\alpha \in (0,1)$. Then we use Lemma 8.2 and Lebesgue’s dominated convergence theorem to pass the limit inside the Young measure to obtain

$$\begin{aligned}&\int _{\mathbb {R}} \overline{P_{2} \chi _{2}^{\tau }}\,\big (\chi (s_1)\, P_{3} \sigma _{3}^{\tau } -P_{3} \chi _{3}^{\tau }\, \sigma (s_1)\big ) \,\psi (s_{1}) \,\textrm{d}s_{1}\\&\quad \longrightarrow \int _{\mathcal {H}}\int _{\mathbb {R}}(P_{1} \chi )(\tilde{\rho }, \tilde{u}, s_{1}) X(\rho , u, s_{1}) \psi (s_{1})\,\textrm{d}s_{1}\textrm{d}\nu _{(t,r)}(\tilde{\rho }, \tilde{u}) \end{aligned}$$

pointwise in $(\rho ,u)$ as $\tau \rightarrow 0$. Now we are going to prove

$$\begin{aligned} \Big \vert \int _{\mathbb {R}}\overline{P_2\chi _2^{\tau }}\,\big (\chi (s_1)\,P_3\sigma _3^{\tau } -P_3\chi _3^{\tau }\,\sigma (s_1)\big )\,\psi (s_1)\,\textrm{d}s_1\Big \vert \le C\big (\rho ^{\beta (\gamma _2+1)}+1\big ) \end{aligned}$$

(8.17)

for some constants $C>0$ and $\beta \in (0,1]$, which are both independent of $\tau$. Once (8.17) is proved, it follows from Lebesgue’s dominated convergence theorem that

$$\begin{aligned} \begin{aligned}&\lim \limits _{\tau \rightarrow 0}\int _{\mathbb {R}}\overline{P_2\chi _2^{\tau }}(s_1)\,\overline{\chi (s_1)\,P_3\sigma _3^{\tau } -P_3\chi _3^{\tau }\,\sigma (s_1))}\,\psi (s_1)\,\textrm{d}s_1\\&\quad =\lim \limits _{\tau \rightarrow 0}\int _{\mathcal {H}}\int _{\mathbb {R}}\overline{P_2\chi _2^{\tau }}(s_1)\,\big (\chi (s_1)\,P_3\sigma _3^{\tau } -P_3\chi _{3}^{\tau }\,\sigma (s_1)\big )(\rho ,u,s_1)\,\psi (s_1)\,\textrm{d}s_1\textrm{d}\nu _{(t,r)}(\rho ,u)\\&\quad =\int _{\mathcal {H}}\int _{\mathcal {H}}\int _{\mathbb {R}}(P_{1} \chi )(\tilde{\rho }, \tilde{u}, s_{1}) X(\rho , u, s_{1}) \psi (s_{1})\,\textrm{d}s_{1}\textrm{d}\nu _{(t,r)}(\tilde{\rho },\tilde{u})\,\textrm{d}\nu _{(t,r)}(\rho ,u). \end{aligned} \end{aligned}$$

Since $X(\rho ,u,s_{1})$ is independent of the choice of the mollifying functions $\phi _{2}^{\tau }$ and $\phi _{3}^{\tau }$ from Lemma 8.7, we may interchange the roles of $s_{2}$ and $s_{3}$ to conclude the proof of (8.16).

To see the validity of (8.17), we begin by observing that, for $j=2,3$, $\overline{P_j\chi _{j}^{\tau }}(s_1)$ and $\psi (s_1)$ are independent of $(\rho ,u)$. Then it suffices to estimate the function:

$$\begin{aligned} \chi (s_1)\,P_j\sigma _j^{\tau }-P_j\chi _{j}^{\tau }\,\sigma (s_1)=\chi (s_1)\,P_{j}(\sigma _{j}^{\tau }-u\chi _{j}^{\tau })-(\sigma (s_1)-u\chi (s_1))\,P_j\chi _{j}^{\tau }. \end{aligned}$$

(8.18)

It follows from Lemmas 4.2–4.3 and 4.14 (also see [11, Proof of Lemma 4.2]) that

$$\begin{aligned} \text{ RHS } \text{ of } (8.18) = E^{1,\tau }+E^{2,\tau }+E^{3,\tau }+E^{4,\tau }, \end{aligned}$$

with

$$\begin{aligned} E^{1,\tau }&=\sum \limits _{\pm }\big (A_{1,\pm }(\rho )b_{1}(\rho )G_{\lambda _1}(s_1)+A_{1,\pm }(\rho )b_2(\rho )G_{\lambda _1+1}(s_1)\big )\\&\qquad \quad \times \big ((s_j-s_1)\delta (s_j-u\pm k(\rho ))\big )*\phi _{j}^{\tau }\\&\quad +\sum \limits _{\pm }\big (A_{3,\pm }(\rho )b_1(\rho )G_{\lambda _1}(s_1)+A_{3,\pm }(\rho )b_2(\rho )G_{\lambda _1+1}(s_1)\big )\\&\quad \qquad \times \big ((s_j-s_1)PV(s_j-u\pm k(\rho ))\big )*\phi _{j}^{\tau }\\&:=E_{1}^{1,\tau }+E_{2}^{1,\tau },\\ E^{2,\tau }=&-\sum _{\pm }A_{1,\pm }(\rho )g_2(s_1)\big (\delta (s_{j}-u\pm k(\rho ))*\phi _{j}^{\tau }\big )\\&-\sum _{\pm }A_{3,\pm }(\rho )g_2(s_1)\big (PV(s_{j}-u\pm k(\rho ))*\phi _{j}^{\tau }\big )\\ :=&E_{1}^{2,\tau }+E_{2}^{2,\tau },\\ E^{3, \tau }&= \sum _{\pm }\big (B_{1,\pm }(\rho )a_1(\rho )-A_{1,\pm }(\rho )b_{1}(\rho )\big )G_{\lambda _1}(s_1)\big ((s_j-u)\delta (s_j-u\pm k(\rho ))\big )*\phi _{j}^{\tau }\\&\quad +\sum _{\pm }\big (B_{1,\pm }(\rho )a_2(\rho )-A_{1,\pm }(\rho )b_2(\rho )\big )G_{\lambda _1+1}(s_1)\big ((s_j-u)\delta (s_j-u\pm k(\rho ))\big )*\phi _{j}^{\tau }\\&\quad +\sum _{\pm }B_{1,\pm }(\rho )g_1(s_1)\big ((s_j-u)\delta (s_j-u\pm k(\rho ))\big )*\phi _{j}^{\tau } \\&\quad +\sum _{\pm } \big (B_{3,\pm }(\rho )a_1(\rho )-A_{3,\pm }(\rho )b_{1}(\rho )\big )G_{\lambda _1}(s_1)\big ((s_j-u)PV(s_j-u\pm k(\rho ))\big )*\phi _{j}^{\tau }\\&\quad +\sum _{\pm } \big (B_{3,\pm }(\rho )a_2(\rho )-A_{3,\pm }(\rho )b_{2}(\rho )\big )G_{\lambda _1+1}(s_1)\big ((s_j-u)PV(s_j-u\pm k(\rho ))\big )*\phi _{j}^{\tau }\\&\quad +\sum _{\pm } B_{3,\pm }(\rho )g_1(s_1)\big ((s_j-u)PV(s_j-u\pm k(\rho ))\big )*\phi _{j}^{\tau }\\&:=E_{1}^{3,\tau }+E_{2}^{3,\tau }+E_{3}^{3,\tau }+E_{4}^{3,\tau }+E_{5}^{3,\tau }+E_{6}^{3,\tau }, \end{aligned}$$

and $E^{4,\tau }$ is the remainder term which consists of the mollification of continuous functions, where we have used the notation: $G_{\lambda _1}(s_1)=[k(\rho )^2-(u-s_1)^2]^{\lambda _1}$, and $g_{i}(s_1)=g_{i}(\rho ,u-s_1)$ for $i=1,2$.

We first demonstrate the uniform bound on the term involving the delta measures. By direct calculation, we have

$$\begin{aligned}&\delta (s_{j}-u+k(\rho ))*\phi _{j}^{\tau }=\frac{1}{\tau }\phi _{j}(\frac{s_{1}-u+k(\rho )}{\tau }),\nonumber \\&\big ((s_{j}-s_{1})\delta (s_{j}-u+k(\rho ))\big )*\phi _{j}^{\tau }=-\frac{s_1-u+k(\rho )}{\tau }\phi _{j}(\frac{s_{1}-u+k(\rho )}{\tau }),\nonumber \\&\big ((s_{j}-u)\delta (s_{j}-u+k(\rho ))\big )*\phi _{j}^{\tau }\nonumber \\&\qquad =\big ((s_{j}-s_{1})\delta (s_{j}-u+k(\rho ))\big )*\phi _{j}^{\tau } +\big ((s_{1}-u)\delta (s_{j}-u+k(\rho ))\big )*\phi _{j}^{\tau }\nonumber \\&\qquad =(s_1-u)\tau ^{-1}\phi _{j}(\frac{s_{1}-u+k(\rho )}{\tau }) -\frac{s_1-u+k(\rho )}{\tau }\phi _{j}(\frac{s_{1}-u+k(\rho )}{\tau }). \end{aligned}$$

(8.19)

Noting that

$$\begin{aligned} \begin{aligned}&G_{\lambda _1+1}(s_1)=G_{\lambda _1}(s_1)\,(k(\rho )-u+s_1)\,(k(\rho )+u-s_1),\\&|g_{i}(s_1)|\le \Vert \partial _{u}g_{i}(\rho ,\cdot )\Vert _{L^{\infty }(\mathbb {R})}|s_{1}-u\pm k(\rho )|\qquad \text{ for } \, i=1,2, \end{aligned} \end{aligned}$$

(8.20)

using (8.19)–(8.20) and Lemmas 4.2–4.3 and 4.11–4.14, we obtain

$$\begin{aligned} E_{1}^{3,\tau }=0,\qquad |E_{1}^{1,\tau }|+|E_{1}^{2,\tau }|+|E_{2}^{3,\tau }|+|E_{3}^{3,\tau }|\le C_{\phi }\big (1+\rho ^{\frac{3}{2}+\frac{\theta _2}{2}}\big ). \end{aligned}$$

(8.21)

For the term involving with the principal value distribution, a direct calculation shows that

$$\begin{aligned} |PV*\phi _{j}^{\tau }(x)|&=\Big \vert \int _{0}^{\infty }\frac{\phi _{j}^{\tau }(x-y)-\phi _{j}^{\tau }(x+y)}{y}\,\textrm{d}y\Big \vert \\&=\frac{1}{\tau }\Big \vert \int _{0}^{\infty }\frac{1}{y}\big (\phi _{j}(\frac{x+y}{\tau }) -\phi _{j}(\frac{x-y}{\tau })\big )\,\textrm{d}y\Big \vert . \end{aligned}$$

If $|x|\le 2\tau$, we have

$$\begin{aligned} |PV*\phi _j^{\tau }(x)|&\le \frac{1}{\tau }\int _{0}^{4\tau }\frac{1}{|y|}\big \vert \phi _j(\frac{x-y}{\tau }) -\phi _j(\frac{y+x}{\tau })\big \vert \,\textrm{d}y \le C\frac{1}{\tau }\Vert \phi _{j}'\Vert _{L^{\infty }}=\frac{C_{\phi }}{|x|}. \end{aligned}$$

(8.22)

On the other hand, if $|x|\ge 2\tau$, we can obtain

$$\begin{aligned} |PV*\phi _{j}^{\tau }(x)| \le \frac{2}{\tau }\int _{|x|-\tau }^{|x|+\tau }\frac{\Vert \phi \Vert _{L^{\infty }}}{|x|-\tau }\,\textrm{d}y\le \frac{C_{\phi }}{|x|}. \end{aligned}$$

(8.23)

Notice that

$$\begin{aligned}{} & {} \big ((s_{j}-s_1)PV(s_{j}-u-k(\rho ))\big )*\phi _{j}^{\tau }\nonumber \\{} & {} \quad =\big ((s_{j}-u+k(\rho ))PV(s_{j}-u+k(\rho ))\big )*\phi _{j}^{\tau }\nonumber \\{} & {} \qquad \quad +\big ((u-k(\rho )-s_1)PV(s_{j}-u+k(\rho ))\big )*\phi _{j}^{\tau },\nonumber \\{} & {} \big ((s_{j}-u)PV(s_{j}-u-k(\rho ))\big )*\phi _{j}^{\tau }\nonumber \\{} & {} \quad =\big ((s_{j}-u+k(\rho ))PV(s_{j}-u+k(\rho ))\big )*\phi _{j}^{\tau } -k(\rho )PV(s_{j}-u+k(\rho ))*\phi _{j}^{\tau },\nonumber \\ \end{aligned}$$

(8.24)

which, with (8.20), (8.22)–(8.24), and Lemmas 4.2–4.3 and 4.11–4.14, yields

$$\begin{aligned} E_{4}^{3,\tau }=0,\qquad |E_{2}^{1,\tau }|+|E_{2}^{2,\tau }|+|E_{5}^{3,\tau }|+|E_{6}^{3,\tau }|\le C_{\phi }\big (1+\rho ^{\frac{3}{2}+\frac{\theta _2}{2}}\big ). \end{aligned}$$

(8.25)

Combining (8.21) with (8.25) yields that there exists $\beta _1=\frac{3+\theta _2}{2(\gamma _2+1)}\in (0,1)$ such that

$$\begin{aligned} |E^{1,\tau }+E^{2,\tau }+E^{3,\tau }|\le C_{\phi }(1+\rho ^{\frac{3}{2}+\frac{\theta _2}{2}})\le C_{\phi }(1+\rho ^{\beta _1(\gamma _2+1)}). \end{aligned}$$

(8.26)

For $E^{4,\tau }$ consisting of the mollification of continuous functions, direct calculations show that

$$\begin{aligned} |E^{4,\tau }|\le C_{\phi }\, \big (1+\rho ^{2+\theta _2} |\ln \rho |\big )\le C_{\phi }\,\big (1+\rho ^{\beta _2(\gamma _2+1)}\big ), \end{aligned}$$

(8.27)

with $\beta _2=\frac{4+3\theta _2}{2(\gamma _2+1)}\in (0,1)$. Combining (8.27) with (8.26), we conclude the proof of (8.17). $\square$

9 Existence of Global Finite-Energy Solutions of CEPEs

In this section, we complete the proof of Theorem 2.3. Since the proof is similar to [10], we sketch the proof for the self-containedness of this paper. We divide the proof into four steps.

1. Since $(\rho ^{\varepsilon },m^{\varepsilon })(t,r)$ obtained in Theorem 2.1 satisfies all the assumptions of Theorem 2.2, then it follows from Theorem 2.2 that there exists a vector function $(\rho ,m)(t,r)$ such that, up to a subsequence as $\varepsilon \rightarrow 0$,

$$\begin{aligned}&(\rho ^{\varepsilon },m^{\varepsilon })\longrightarrow (\rho ,m) \quad { a.e.}\, (t,r)\in \mathbb {R}_{+}^2, \end{aligned}$$

(9.1)

$$\begin{aligned}&(\rho ^{\varepsilon }, m^{\varepsilon }) \longrightarrow (\rho , m) \quad \text {in}\, L_{\textrm{loc}}^{p_{1}}(\mathbb {R}_{+}^{2}) \times L_{\textrm{loc}}^{p_{2}}(\mathbb {R}_{+}^{2}) \end{aligned}$$

(9.2)

for $p_{1} \in [1, \gamma _2+1)$ and $p_{2} \in [1, \frac{3(\gamma _2+1)}{\gamma _2+3})$, where $L_{\textrm{loc}}^{p_{j}}(\mathbb {R}_{+}^{2})$ represents $L^{p_{j}}([0, T] \times K)$ for any $T>0$ and $K \Subset (0, \infty ), j=1,2$.

Noting (9.1) and $\rho ^{\varepsilon }\ge 0$ a.e. from Lemma 6.1, it is direct to show that $\rho (t,r)\ge 0$ a.e. on $\mathbb {R}_{+}^2$. Moreover, it follows from (2.22) that $\sqrt{\rho ^{\varepsilon }}u^{\varepsilon }r=\frac{m^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}}r$ is uniformly bounded in $L^{\infty }(0,T;L^2(\mathbb {R}))$. Then using Fatou’s lemma yields

$$\begin{aligned} \int _{0}^{T}\int _{0}^{\infty }\frac{|m(t,r)|^2}{\rho (t,r)}\,r^{2}\textrm{d}r\textrm{d}t\le \liminf _{\varepsilon \rightarrow 0} \int _{0}^{T}\int _{0}^{\infty }\frac{|m^{\varepsilon }(t,r)|^2}{\rho ^{\varepsilon }(t,r)}\,r^{2}\textrm{d}r\textrm{d}t<\infty . \end{aligned}$$

Thus, $m(t,r)=0$ a.e. on $\{(t,r)\,:\,\rho (t,r)=0\}$, and we can define the limit velocity u(t, r) as

$$\begin{aligned} \begin{aligned} u(t,r)&=\frac{m(t,r)}{\rho (t,r)}\qquad { a.e.}\,\text { on }\{(t,r)\,:\,\rho (t,r)\ne 0\},\\ u(t,r)&=0 \qquad { a.e.}\,\text { on }\{(t,r)\,:\,\rho (t,r)=0\text { or }r=0\}. \end{aligned} \end{aligned}$$

Therefore, $m(t,r)=\rho (t,r)u(t,r)$ a.e. on $\mathbb {R}_{+}^2$. Also, we can define $\big (\frac{m}{\sqrt{\rho }}\big )(t,r):=\sqrt{\rho (t,r)}u(t,r)$, which is zero a.e. on $\{(t,r):\,\rho (t,r)=0\}$. Moreover, using (2.24) and Fatou’s lemma, we obtain

$$\begin{aligned} \int _{0}^{T}\int _{d}^{D}\rho |u|^3\,\textrm{d}r\textrm{d}t \le \liminf _{\varepsilon \rightarrow 0}\int _{0}^{T}\int _{d}^{D}\rho ^{\varepsilon }|u^{\varepsilon }|^3\,\textrm{d}r\textrm{d}t\le C(d,D,M,E_0,T)<\infty \end{aligned}$$

for any $[d,D]\Subset (0,\infty )$.

By similar calculations as in [10, Sect. 5], we obtain that, as $\varepsilon \rightarrow 0$,

$$\begin{aligned} \frac{m^{\varepsilon }}{\sqrt{\rho ^{\varepsilon }}} \equiv \sqrt{\rho ^{\varepsilon }} u^{\varepsilon } \longrightarrow \frac{m}{\sqrt{\rho }} \equiv \sqrt{\rho } u \qquad \text {strongly in }L^{2}([0, T] \times [d, D]; r^{n-1} \,\textrm{d} r \textrm{d}t) \end{aligned}$$

(9.3)

for any $T>0$ and $[d, D] \Subset (0, \infty )$.

From (9.2)–(9.3), we also obtain the convergence of the mechanical energy as $\varepsilon \rightarrow 0$:

$$\begin{aligned} \eta ^{*}(\rho ^{\varepsilon }, m^{\varepsilon }) \longrightarrow \eta ^{*}(\rho , m) \qquad \text { in } L_{\textrm{loc}}^{1}(\mathbb {R}_{+}^{2}). \end{aligned}$$

(9.4)

Using (9.2), (9.4), and Fatou’s lemma, and taking limit $\varepsilon \rightarrow 0$ in (2.21)–(2.22), we have

$$\begin{aligned} \int _{t_{1}}^{t_{2}} \int _{0}^{\infty } \big (\eta ^{*}(\rho , m)(t, r)+\rho ^{\gamma _2}(t,r)+\rho (t, r)\big )\, r^{2}\textrm{d} r \textrm{d} t \le C(M, E_{0})(t_{2}-t_{1}), \end{aligned}$$

(9.5)

which implies

$$\begin{aligned} \sup _{0\le t\le T}\int _{0}^{\infty } \big (\eta ^{*}(\rho , m)(t, r)+\rho ^{\gamma _2}(t,r)+\rho (t, r)\big )\,r^{2}\textrm{d}r \le C(M, E_{0}). \end{aligned}$$

(9.6)

This indicates that $\rho (t,r)\in L^{\infty }([0,T];L^{\gamma _2}(\mathbb {R};r^{2}\textrm{d}r))$, which implies that $\rho (t,\textbf{x})$ is a function in $L^{\infty }([0,T];L^{\gamma _2}(\mathbb {R}^3))$ with $\gamma _2>1$ (rather than a measure in $(t,\textbf{x})$). Therefore, no delta measure (i.e., concentration) is formed in the density in the time interval [0, T], especially at the origin: $r=0$.

2. For the convergence of the gravitational potential functions $\Phi ^{\varepsilon }(t,r)$, by similar calculation in [10, Sect. 5], we obtain that, as $\varepsilon \rightarrow 0$ (up to a subsequence),

$$\begin{aligned} \Phi _{r}^{\varepsilon }(t, r) r^{2} = \int _{0}^{r} \rho ^{\varepsilon }(t, y)\,y^{2}\textrm{d}y \longrightarrow \int _{0}^{r} \rho (t, y)\,y^{2}\textrm{d}y \qquad { a.e.}\, (t,r)\in \mathbb {R}_{+}^2. \end{aligned}$$

(9.7)

Thus, using (6.3), (9.1), (9.7), Fatou’s lemma, and similar arguments as in (9.5)–(9.6), we have

$$\begin{aligned} \int _{0}^{\infty }\Big (\int _{0}^{r} \rho (t, y)\,y^{2}\textrm{d}y\Big ) \rho (t, r)\,r \textrm{d} r \le C(M, E_{0}) \qquad \text { for }\, { a.e.\,\, t} \ge 0. \end{aligned}$$

On the other hand, it follows from (6.4) that there exists a function $\Phi (t, \textbf{x})=\Phi (t, r)$ such that, as $\varepsilon \rightarrow 0$ (up to a subsequence),

$$\begin{aligned} \begin{aligned}&\Phi ^{\varepsilon } \rightharpoonup \Phi \qquad \text { weak-star in}\, L^{\infty }(0, T ; H_{{\text {loc}}}^{1}(\mathbb {R}^{3}))\, \text {and weakly in}\, L^{2}(0, T ; H_{\textrm{loc}}^{1}(\mathbb {R}^3),\\&\Vert \Phi (t)\Vert _{L^6(\mathbb {R}^{3})}+\Vert \nabla \Phi (t)\Vert _{L^{2} \left( \mathbb {R}^{3}\right) } \le C(M, E_{0}) \qquad { a.e.}\, t \ge 0. \end{aligned} \end{aligned}$$

Thus, by (9.7) and the uniqueness of limit, we obtain that $\Phi _{r}(t, r) r^{2}=\int _{0}^{r} \rho (t, z) z^{2} \,\textrm{d} z$ a.e. $(t, r) \in \mathbb {R}_{+}^{2}$. By similar arguments in [10, Sect. 5], we also have the strong convergence of the potential functions:

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \int _{0}^{T}\int _{0}^{\infty }\big |(\Phi _{r}^{\varepsilon }-\Phi _{r})(t, r)\big |^{2} r^{2} \,\textrm{d} r \textrm{d} t=0 \qquad \text { for }\, \gamma _2>\frac{6}{5}. \end{aligned}$$

(9.8)

3. Now we define

$$\begin{aligned} (\rho ,\mathcal {M},\Phi )(t,\textbf{x} ):=(\rho (t,r),m(t,r)\frac{\textbf{x}}{r}, \Phi (t,r)). \end{aligned}$$

Then it follows from (2.20), (9.8), and Fatou’s lemma that

$$\begin{aligned}&\int _{t_{1}}^{t_{2}} \int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}}{\sqrt{\rho }}\Big |^{2} +\rho e(\rho )-\frac{1}{2}|\nabla \Phi |^2\Big )(t, \textbf{x}) \,\textrm{d} \textbf{x} \textrm{d} t\\&\quad \le (t_{2}-t_{1}) \int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}_{0}}{\sqrt{\rho _{0}}}\Big |^{2} +\rho _{0} e(\rho _{0})-\frac{1}{2}|\nabla \Phi _{0}|^2\Big )(\textbf{x}) \,\textrm{d}\textbf{x}, \end{aligned}$$

which implies that, for a.e. $t \ge 0$,

$$\begin{aligned}&\int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}}{\sqrt{\rho }}\Big |^{2} +\rho e(\rho )-\frac{1}{2}|\nabla \Phi |^2\Big )(t, \textbf{x}) \,\textrm{d}\textbf{x}\nonumber \\&\quad \le \int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}_{0}}{\sqrt{\rho _{0}}}\Big |^{2}+\rho _{0} e(\rho _{0}) -\frac{1}{2}|\nabla \Phi _{0}|^2\Big )(\textbf{x})\,\textrm{d}\textbf{x}. \end{aligned}$$

(9.9)

On the other hand, using (2.22), (9.6), and (9.8), we obtain

$$\begin{aligned} \int _{\mathbb {R}^{3}}\Big (\frac{1}{2}\Big |\frac{\mathcal {M}}{\sqrt{\rho }}\Big |^{2}+\rho e(\rho ) +\frac{1}{2}|\nabla \Phi |^2\Big )(t, \textbf{x}) \,\textrm{d}\textbf{x}\le C(M,E_0). \end{aligned}$$

(9.10)

Combining (9.9) with (9.10), we complete the proof of (2.28).

4. Using (6.7), (6.9)–(6.10), and similar arguments as in [17, Sect. 5], we conclude the proof of (2.29)–(2.31) which, along with Steps 1–3, shows that $(\rho , \mathcal {M}, \Phi )(t, \textbf{x})$ is indeed a global weak solution of problem (1.1) and (1.13)–(1.14) in sense of Definition 2.2. This completes the proof. $\square$

Appendix A. Some Inequalities

1.1 A.1. A sharp Sobolev inequality

In this subsection, we recall a sharp Sobolev inequality, which is used in Sect. 5.1. The proof can be found in [45, Sect. 8.3].

Lemma A.1

(Sobolev inequality). Let $n\ge 3$ and $\nabla f\in L^2(\mathbb {R}^n)$ with $\lim _{|\textbf{x}|\rightarrow \infty }f(\textbf{x})=0$. Then

$$\begin{aligned} \Vert f\Vert _{L^{\frac{2n}{n-2}}}^2\le A_{n}\Vert \nabla f\Vert _{L^2}^2, \end{aligned}$$

where $A_{n}=\frac{4}{n(n-2)}\omega _{n+1}^{-\frac{2}{n}}$ is the best constant and $\omega _{k}=\frac{2\pi ^{\frac{k}{2}}}{\Gamma (\frac{k}{2})}$ is the surface area of the unit sphere in $\mathbb {R}^k$.

1.2 A.2. Some variants of the Grönwall inequality

In this subsection, we introduce some variants of the Grönwall inequality, which plays an essential role in identifying the singularities of the entropy kernel and entropy flux kernel; see also [64].

Lemma A.2

(A variant of Grönwall inequality [59, Theorem 1.2.4]). Let x(t), y(t), z(t), and w(t) be non-negative continuous functions on $J=[t_0,t_1]$ with $t_0\ge 0$. If

$$\begin{aligned} x(t)\le y(t)+z(t)\int _{t_0}^tw(s) x(s)\,\textrm{d} s\qquad \text {for }\, t\in J, \end{aligned}$$

then

$$\begin{aligned} x(t)\le y(t)+z(t)\int _{t_0}^t w(s)y(s)\exp \Big (\int _{s}^tw(r)z(r)\,\textrm{d} r\Big )\,\textrm{d} s\qquad \text {for }\, t\in J. \end{aligned}$$

Lemma A.3

Let $\theta \ge 0$, and let d(s) be defined in (3.9). Assume that $x(t)\ge 0$ is measurable and locally integrable, and satisfies

$$\begin{aligned} x(t)\le Ct^{\theta }+\frac{1}{t}\int _{0}^td(s)x(s)\,\textrm{d}s\qquad \text {for }\, t\ge \rho ^{*} \end{aligned}$$

(A1)

for some constant $C>0$. Then there exists a possibly larger constant $\tilde{C}>0$ independent of t such that, for $t\ge \rho ^{*}$,

$$\begin{aligned} x(t)\le {\left\{ \begin{array}{ll} \tilde{C}t^{\theta _2} \quad &{}\text {if } \,0\le \theta <\theta _2,\\ \tilde{C}t^{\theta _2}\ln t\,\,\,&{}\text {if } \,\theta =\theta _2,\\ \tilde{C}t^{\theta } \quad &{}\text {if }\, \theta >\theta _2. \end{array}\right. } \end{aligned}$$

Proof

Since x(t) is positive and locally integrable, then, using Lemma 3.2, there exists a constant $C>0$ that may depend on $\rho ^{*}$, but independent of t, such that

$$\begin{aligned} \frac{1}{t}\int _{0}^{\rho ^{*}}d(s)x(s)\,\textrm{d} s\le Ct^{-1}\qquad \text {for }\, t\ge \rho ^{*}. \end{aligned}$$

This, with (A1), yields that $x(t)\le Ct^{\theta }+\frac{1}{t}\int _{\rho ^{*}}^{t}d(s)x(s)\,\textrm{d} s$ for $t\ge \rho ^{*}$. Applying Lemma A.2, we obtain

$$\begin{aligned} x(t)\le Ct^{\theta }+\frac{1}{t}\int _{\rho ^{*}}^{t}Cs^{\theta }d(s)\textrm{exp}\Big (\int _{s}^{t}\frac{d(r)}{r} \,\textrm{d} r\Big )\,\textrm{d} s\qquad \text {for }\, t\ge \rho ^{*}. \end{aligned}$$

(A2)

It is clear that

$$\begin{aligned} \Big \vert \int _{s}^{t}\frac{d(r)}{r}\,\textrm{d} r\Big \vert \le \int _{s}^{t}\frac{1+\theta _2}{r}\,\textrm{d} r+\int _{s}^{t}\frac{|d(r)-(1+\theta _2)|}{r}\,\textrm{d} r. \end{aligned}$$

(A3)

It follows from Lemma 3.2 that $\frac{|d(r)-(1+\theta _2)|}{r}\le Cr^{-\epsilon -1}$ for $r\ge \rho ^{*}$ which, with (A3), yields

$$\begin{aligned} \textrm{exp}\Big (\int _{s}^{t}\frac{d(r)}{r}\,\textrm{d} r\Big )\le C\Big (\frac{t}{s}\Big )^{1+\theta _2} \qquad \text {for }\, t\ge s\ge \rho ^{*}. \end{aligned}$$

Combining (A2) with $|d(s)|\le 3$, we obtain that, for $t\ge \rho ^{*}$,

$$\begin{aligned} x(t)\le Ct^{\theta }+\frac{1}{t}\int _{\rho ^{*}}^{t}Cs^{\theta }\Big (\frac{t}{s}\Big )^{1+\theta _2}\,\textrm{d} s \le Ct^{\theta }+Ct^{\theta _2}\int _{\rho ^{*}}^{\theta }s^{-1-\theta _2+\theta }\,\textrm{d} s. \end{aligned}$$

(A4)

Case 1. If $0\le \theta <\theta _2$, it follows from (A4) that

$$\begin{aligned} x(t)\le Ct^{\theta }+Ct^{\theta _2}\Big (\int _{\rho ^{*}}^{\infty }s^{-1-\theta _2+\theta }\,\textrm{d} s\Big ) \le \tilde{C}t^{\theta _2}\qquad \text {for }\, t\ge \rho ^{*}. \end{aligned}$$

Case 2. If $\theta =\theta _2$, it follows from (A4) that

$$\begin{aligned} x(t)\le Ct^{\theta _2}+Ct^{\theta _2}\Big (\int _{\rho ^{*}}^{t}s^{-1}\,\textrm{d} s\Big )\le \tilde{C}t^{\theta _2}\ln t \qquad \text{ for } t\ge \rho ^{*}. \end{aligned}$$

Case 3. If $\theta >\theta _2$, then it follows from (A4) that

$$\begin{aligned} x(t)\le Ct^{\theta }+Ct^{\theta _2}\Big (\int _{\rho ^{*}}^{t}s^{-1-\theta _{2}+\theta }\,\textrm{d} s\Big ) \le \tilde{C}t^{\theta }\qquad \text {for }\, t\ge \rho ^{*}. \end{aligned}$$

This completes the proof. $\square$

Corollary A.4

If x(t) satisfies

$$\begin{aligned} x(t)\le Ct^{\theta }\ln t+\frac{1}{t}\int _{0}^td(s)x(s)\,\textrm{d} s\qquad \text {for }\, t\ge \rho ^{*}, \end{aligned}$$

with $\theta >\theta _2$, then $\,x(t)\le Ct^{\theta } \ln t\,$ for $t\ge \rho ^{*}$.