Yaojun Yang
College of Mathematics Science, Chongqing Normal University, Chongqing, China.
DOI: 10.4236/jamp.2024.122040   PDF    HTML   XML   43 Downloads   155 Views   Citations

Abstract

In this paper, we investigate a 1D pressureless Euler-alignment system with a non-local alignment term, describing a kind of self-organizing problem for flocking. As a result, by the transport equation theory and Lagrange coordinate transformation, the local well-posedness of the solutions for the 1D pressureless Euler-alignment in Besov spaces with 1≤p<∞ is established. Next, the ill-posedness of the solutions for this model in Besov spaces with 1≤p<∞ and is also deduced. Finally, the precise blow-up criteria of the solutions for this system is presented in Besov spaces with 1≤p<∞ .

Keywords

Euler-Alignment Equations, Local Well-Posedness, Blow-Up Criteria, Ill-Posedness

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1. Introduction

The Cucker-Smale Model. The flocking behaviors are widespread in biological systems, such as the swimming of fish, the movement of wildebeest groups and the migration of birds called as self-organization biological behaviors, which have attracted much attention in [1] [2] [3] [4] [5] . Understanding the population properties of interacting systems, and how individual components function, are important questions. In biology, studying the collective behaviors of animals can better understand the structure of ecosystems and provide guidance for ecosystem management and conservation. In medicine, such as cancer cells, there is a collective arrangement of patterns in the human body, and studying this collective pattern of destructive cells can more effectively affect them and promote the understanding of disease. The flocking behavior refers to the motion of a cluster of finite particles in which the velocity of each particle is consistent with the weighted average of the velocities of its neighbors. The discrete Vicsek model in time and two-dimensional spaces is usually used to describe the flocking behaviors in [6] : the velocity angle ${\theta }_j\left(t\right)$ of j-th particle satisfies

${\theta }_j\left(t+1\right)=\frac{1}{\left|{\mathcal{N}}_j\left(t\right)\right|}{\displaystyle \underset{i\in {\mathcal{N}}_j\left(t\right)}{\sum }}{\theta }_j\left(i\right)+\eta \Delta \theta \mathrm{.}$ (1)

Here, ${\mathcal{N}}_j\left(t\right)=\left\{i\mathrm{<mo>\; :\; </mo>}\left|x_i\left(t\right)-x_j\left(t\right)\right|\le r\right\}$ ( $r>0$ ), a random variable $\Delta \theta$ is uniformly distributed in $\left[-\mathrm{1,1}\right]$ , and a parameter $\eta >0$ is used to measure the intensity of the noise. Furthermore, a generalization of the Vicsek model was proposed by Cucker and Smale in [7] :

$\{\begin{array}{l}\frac{\text{d}{x}_i}{\text{d}t}=v_i,\\ \frac{\text{d}{v}_i}{\text{d}t}=\frac{1}{N}{\displaystyle \underset{j=1}{\overset{N}{\sum }}}\varphi \left(\left|x_i-x_j\right|\right)\left(v_j-v_i\right),\end{array}$ (2)

where ${\left\{x_i\mathrm{,}{v}_i\right\}}_{i\in \mathbb{N}}$ refers to the position and velocity of the agents i, and N is the total number of two groups. The nonnegative and decreased communication weight function $\varphi$ measures the strength of the interaction between two particles, since the distance of the particles increases, the interaction usually becomes smaller. The Cucker-Smale model can be used to analyze the flocking behaviors based on the decay properties of the kernel $\varphi \left(r\right)$ , that is to say, if $\varphi \left(r\right)$ decays weaker than $r^{-1}$ as $r\to +\infty$ , the velocities $v_i\left(t\right)$ of the agents converge to a limit velocity $\bar{v}\left(t\right)$ , and the relative positions $x_i\left(t\right)-x_j\left(t\right)$ also converge to a limit position ${\bar{x}}_{ij}$ as $r\to +\infty$ . This is what we would call the flocking behavior: all particles move with nearly identical velocities.

A Kinetic Cucker-Smale Model. When the number of particles is large, to make it easier to simulate the movement of each particle, kinetic models are often used to describe the behaviors of global flocking by the density function $f\left(t\mathrm{,}x\mathrm{,}v\right)$ with $x\in {ℝ}^d$ , $v\in {ℝ}^d$ . Furthermore, about the kinetic limit of the Cucker-Smale model (2), Ha and Tadmor established a nonlinear and non-local kinetic equation in [8] :

${\partial }_{t}f+v\cdot {\nabla }_{x}f+{\nabla }_v\cdot \left(Q\left[f\right]f\right)=\mathrm{0,}$ (3)

there $Q\left[f\right]$ is the velocity alignment force field given by

$Q\left[f\right]\left(t,x,v\right)={\displaystyle {\int }_{{ℝ}^{2d}}}\varphi \left(x-y\right)\left(w-v\right)f\left(t,y,w\right)\text{d}w\text{d}y.$ (4)

Here, (3)-(4) is a nonlinear and non-local kinetic version of the Cucker-Smale model. When $\varphi \left(r\right)$ decays weaker than $r^{-1}$ as $r\to +\infty$ , the solutions of Equations (3)-(4) show that the global flocking behaviors in [9] , where the size $S\left(t\right)$ of the support in x is uniformly bounded:

$S\left(t\right)=\sup \left\{\left|x-y\right|\mathrm{<mo>\; :\; </mo>}\left(x\mathrm{,}v\right)\mathrm{,}\left(y\mathrm{,}{v}^{\prime }\right)\in \text{supp}\left(f\left(t\mathrm{,}\cdot \mathrm{,}\cdot \mathrm{,}\right)\right)\right\}\mathrm{,}$

and the size $V\left(t\right)$ of the support in v remains decreasing:

$V\left(t\right)=\sup \left\{\left|v-v^{\prime }\right|\mathrm{<mo>\; :\; </mo>}\left(x\mathrm{,}v\right)\mathrm{,}\left(y\mathrm{,}{v}^{\prime }\right)\in \text{supp}\left(f\left(t\mathrm{,}\cdot \mathrm{,}\cdot \mathrm{,}\right)\right)\right\}\to 0\text{as}t\to +\infty \mathrm{.}$

A 1D Euler-alignment Model. In order to deduce the standard form of the hydrodynamic limit for nonlinear kinetic equations, one can consider the mono-kinetic ansatz of the form in (3)-(4)

$f\left(t\mathrm{,}x\mathrm{,}v\right)\simeq \rho \left(t\mathrm{,}x\right){\delta }_{v-u\left(t\mathrm{,}x\right)}\mathrm{.}$ (5)

In this case, the model becomes a local alignment model which is different from the global alignment model, the particles travel locally at a single speed in which the velocities of the particles are different in space. Under the one-dimensional pressureless condition, the flocking behaviors are known as a 1D pressureless Euler-alignment system with a non-local alignment term. Submitting (5) into (3)-(4), we arrive at the pressureless one-dimensional Euler-alignment system which consists of the mass conservation equation

${\partial }_t\rho +{\partial }_x\left(\rho u\right)=\mathrm{0,}$ (6)

and the momentum conservation equation

${\partial }_t\left(\rho u\right)+{\partial }_x\left(\rho u^2\right)={\displaystyle {\int }_{ℝ}}\varphi \left(x-y\right)\left(u\left(t\mathrm{,}y\right)-u\left(t\mathrm{,}x\right)\right)\rho \left(t\mathrm{,}y\right)\rho \left(t\mathrm{,}x\right)\text{d}y\mathrm{,}$ (7)

where the right side of (7) is the non-local alignment term, owing to the presence of the density $\rho$ , the density is higher, the alignment effect between the agents becomes stronger. The 1D pressureless Euler-alignment system with a non-local alignment term simulates the population movement from irregular movement to regular movement with constant relative distance and relative velocity in one-dimensional space.

Furthermore, by submitting (6) into (7), one can get the following 1D Euler-alignment system:

$\{\begin{array}{ll}{\partial }_{t}u+u{\partial }_{x}u=\varphi \ast \left(u\rho \right)-\left(\varphi \ast \rho \right)u,\hfill & t>0,x\in ℝ,\hfill \\ {\partial }_t\rho +{\partial }_x\left(\rho u\right)=0,\hfill & t>0,x\in ℝ,\hfill \\ u\left(0,x\right)=u^0\left(x\right),\rho \left(0,x\right)={\rho }^0\left(x\right),\hfill & t=0,x\in ℝ.\hfill \end{array}$ (8)

Many researchers showed that the characteristic of communication weight $\varphi$ plays an important role in the regularity of the solutions for system (8): when communication weight is symmetric and uniformly bounded, Carrillo showed that system (8) exists a critical threshold of initial data in [10] : if the initial data lies above the subcritical region in the sense that ${\partial }_x{u}_0\ge -\varphi \ast {\rho }_0$ for all $x\in ℝ$ , there has a global classical solution for 1D Euler-alignment system (8) if the initial data lies above the supercritical region in the sense that ${\partial }_x{u}_0<-\varphi \ast {\rho }_0$ for all $x\in ℝ$ , the solutions can blow up in a finite time for the 1D Euler-alignment system (8).

In [11] , let $G={\partial }_{x}u+\varphi \ast \rho$ , Tan rewrote model (8) as the following equivalent equations:

$\{\begin{array}{ll}{\partial }_{t}u+u{\partial }_{x}u=\varphi \ast \left(u\rho \right)-\left(\varphi \ast \rho \right)u,\hfill & t>0,x\in ℝ,\hfill \\ {\partial }_t\rho +u{\partial }_x\rho =-{\partial }_{x}u\rho ,\hfill & t>0,x\in ℝ,\hfill \\ {\partial }_{t}G+u{\partial }_{x}G=-{\partial }_{x}uG,\hfill & t>0,x\in ℝ,\hfill \\ {\partial }_{x}u=G-\varphi \ast \rho ,\hfill & t>0,x\in ℝ,\hfill \\ \rho \left(0,x\right)={\rho }^0\left(x\right),u\left(0,x\right)=u^0\left(x\right),\hfill & t=0,x\in ℝ,\hfill \end{array}$ (9)

where communication weight is integrable, so communication weight can be either regular or weakly singular (see [12] ). If communication weight is weakly singular, that is to say, it has an integrable singularity at the origin, Tan showed that system (8) exists a critical threshold of initial data: if the initial data lies

above the subcritical region in the sense that $\underset{x}{\inf }{G}_0\left(x\right)>0$ for all $x\in ℝ$ , then there exists a globally regular solution for 1D Euler-alignment system (8), if the initial data lies above the supercritical region in the sense that $\underset{x}{\inf }{G}_0\left(x\right)<0$ for all $x\in ℝ$ , then the solutions blow up in a finite time for the 1D Euler-alignment system (8) in [11] .

If communication weight $\varphi \in L^1\left(ℝ\right)$ , Tan established the local well-posedness in Sobelov spaces $H^{s+1}\times H^s$ with $s>\frac{1}{2}$ on the whole real line or the periodic domain in [11] . One natural question is: whether or not the system (8) is local well-posedness in $H^{s+1}\times H^s$ for $s=\frac{1}{2}$ . The Lagrange coordinate transformation does not change the dynamic nature of the system and can make the equation easier to solve. Indeed, if there exists a small enough time $T>0$ , based on that the characteristic $y\left(t\mathrm{,}\xi \right)$ is a homeomorphism in a small time interval $\left[\mathrm{0,}T\right]$ , we will obtain the uniqueness of the solutions. Note that $B_{\mathrm{2,1}}^s$ ↪ $B_{\mathrm{2,2}}^s\approx H^s$ , by compactness theory and coordinate transformation, we want to explore the local well-posedness of the Cauchy problem for the 1D Euler-alignment system

(8) in Besov spaces $B_{p\mathrm{,1}}^{1+\frac{1}{p}}\times B_{p\mathrm{,1}}^{\frac{1}{p}}$ with $1\le p<\infty$ (in the rest of this paper $\varphi \in L^1\left(ℝ\right)$ unless otherwise noted).

2. Main Results

Theorem 2.1. Suppose that $p\in \left[\mathrm{1,}\infty \right)$ and the initial data

$\left(u_0\mathrm{,}{\rho }_0\mathrm{,}{G}_0\right)\in {\left(B_{p\mathrm{,1}}^{\frac{1}{p}}\right)}^3$ . Then there exists a time $T>0$ such that Equation (9) has a unique solution $\left(u\mathrm{,}\rho \mathrm{,}G\right)$ in ${\left(E_T^p\right)}^3$ and $E_T^p=\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)\cap {\mathcal{C}}^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)$ . Moreover, the solutions depend continuously on the initial data.

Recall that $B_{p\mathrm{,}r}^{s+1}\times B_{p\mathrm{,}r}^s$ ↪ $B_{p\mathrm{,1}}^{1+\frac{1}{p}}\times B_{p\mathrm{,1}}^{\frac{1}{p}}$ ( $1\le p\mathrm{,}r\le \infty$ ) is locally compact (see

Proposition 1.3.5 in [13] ), using the same argument of the proof in Theorem 2.1, we can easily get the following local well-posedness of the solutions for Equation (8) in the Besov spaces $B_{p\mathrm{,}r}^{s+1}\times B_{p\mathrm{,}r}^s$ :

Assume that $1\le p$ , $r\le +\infty$ , $s>\max \left\{\frac{1}{p},\frac{1}{2}\right\}$ and the initial data

$\left(u_0\mathrm{,}{\rho }_0\right)\in B_{p\mathrm{,}r}^{s+1}\times B_{p\mathrm{,}r}^s$ . Then there exists a time $T>0$ such that the Cauchy problem (8) has a unique solution $\left(u\mathrm{,}\rho \right)\in B_{p\mathrm{,}r}^{s+1}\times B_{p\mathrm{,}r}^s$ , and the map $\left(u_0\mathrm{,}{\rho }_0\right)\to \left(u\mathrm{,}\rho \right)$ is continuous from a neighborhood of $\left(u_0\mathrm{,}{\rho }_0\right)$ in $B_{p\mathrm{,}r}^{s+1}\times B_{p\mathrm{,}r}^s$ into

$\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s^{\prime }+1}\right)\cap {\mathcal{C}}^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s^{\prime }}\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s^{\prime }}\right)\cap {\mathcal{C}}^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s^{\prime }-1}\right)\mathrm{,}$

$s^{\prime }<s$ when $r=+\infty$ whereas $s^{\prime }=s$ when $r<+\infty$ .

Now, another natural question is raised: whether or not the data-to-solution map of the system (8) continues in $B_{p\mathrm{,}r}^{s^{\prime }+1}\times B_{p\mathrm{,}r}^{s^{\prime }}$ for $s^{\prime }=s$ and $r=+\infty$ . In the following theorem, we deduce that this data-to-solution map is in ill-posedness in $B_{p\mathrm{,}\infty }^{s+1}\times B_{p\mathrm{,}\infty }^s$ .

Theorem 2.2. Suppose that $1\le p\le \infty$ and $s>\max \left\{\frac{1}{p},\frac{1}{2}\right\}$ , then the system

(9) is ill-posedness in Besov spaces ${\left(B_{p\mathrm{,}\infty }^s\right)}^3$ . More precisely, there exists $\left(u_0\mathrm{,}{\rho }_0\mathrm{,}{G}_0\right)\in {\left(B_{p\mathrm{,}\infty }^s\right)}^3$ and a positive constant $\delta$ such that the Cauchy problem

for system (9) has a unique solution $\left(u\mathrm{,}\rho \mathrm{,}G\right)\in L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{\left(B_{p\mathrm{,}\infty }^s\right)}^3\right)$ for some $T=T\left({\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^s}\mathrm{,}{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}\mathrm{,}{\Vert G_0\Vert }_{B_{p\mathrm{,}\infty }^s}\right)$ , while

$\underset{t\to 0}{\lim \inf }\left({\Vert u-u_0\Vert }_{B_{p,\infty }^s}+{\Vert \rho -{\rho }_0\Vert }_{B_{p,\infty }^s}+{\Vert G-G_0\Vert }_{B_{p,\infty }^s}\right)\ge C\delta .$

Theorem 1.3 in [11] shows that the solutions admit a finite time blow up for the 1D Euler-alignment system (8), in the sense of $\underset{x}{\inf }{G}_0\left(x\right)<0$ for all $x\in ℝ$ , if and only if $\underset{t\to T^{-}}{\lim }G\left(t,\cdot \right)=-\infty$ . Theorem 2.1 in [11] shows that the solutions of

the system (8) stay smooth up to time T, in the sense of $\left(\rho \mathrm{,}G\right)\in \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{H}^s\cap L_{+}^1\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{H}^s\right)$ , if and only if ${\displaystyle {\int }_0^T}\left({\Vert \rho \left(\cdot ,t\right)\Vert }_{L^{\infty }}+{\Vert G\left(\cdot ,t\right)\Vert }_{L^{\infty }}\right)\text{d}t<+\infty$ . Next, we show that the solutions of the system (8) stay smooth only depending on the slope of u but not involving the components of G and $\rho$ in the following theorem.

Theorem 2.3. Suppose that $\left(u\mathrm{,}\rho \mathrm{,}G\right)\in {\left(\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)\right)}^3$ is the solution of the Cauchy problem (9) with the initial data $\left(u_0\mathrm{,}{\rho }_0\mathrm{,}{G}_0\right)\in {\left(B_{p\mathrm{,1}}^{\frac{1}{p}}\right)}^3$ . Let T is the

maximal existence time of the solutions $\left(u\mathrm{,}\rho \mathrm{,}G\right)$ to Equation (9). Then the solutions blow up in finite time if and only if

$\underset{t\to T^{-}}{\lim }\underset{x\in ℝ}{\inf }{u}_x\left(t,\cdot \right)=-\infty .$

The paper is organized as follows. In Section 2, we introduce several important results on the Littlewood-Paley decomposition, the nonhomogeneous Besov spaces and their useful properties. In Section 3, we establish the local well-posedness result in Besov spaces of the solutions for Equation (9). Moreover, we prove the blow-up criteria of the solutions to the problem (9) in Section 4. Finally, the ill-posedness result of the solutions for Equation (9) is presented in Section 5. Note that we denote a general constant $C>0$ only depending on s and ${\Vert \varphi \Vert }_{L^1}$ , since all function spaces in the following sections are over $ℝ$ , for simplicity, we drop $ℝ$ in the notation of function spaces if there is no ambiguity.

3. Preliminaries

In this section, for the convenience of readers, we introduce some facts on the Littlewood-Paley theory, which is frequently used in the following arguments. Then we introduce some properties of nonhomogeneous Besov spaces which will play a key role in proving the local well-posedness and other properties of the system (9). One may refer to [13] [14] for more details.

Proposition 3.1. (See Proposition 2.10 in [13] ) Let $\mathbb{B}\doteq \left\{\xi \in {ℝ}^d,\left|\xi \right|\le \frac{4}{3}\right\}$ and $ℂ\doteq \left\{\xi \in {ℝ}^d,\frac{4}{3}\le \left|\xi \right|\le \frac{8}{3}\right\}$ . There exists two radial functions $\chi \in {\mathcal{C}}_c^{\infty }\left(\mathbb{B}\right)$ and $\phi \in {\mathcal{C}}_c^{\infty }\left(ℂ\right)$ such that

$\chi \left(\xi \right)+{\displaystyle \underset{q\ge 0}{\sum }}\phi \left(2^{-q}\xi \right)=\mathrm{1,}\text{for}\text{all}\xi \in {ℝ}^d\mathrm{,}$

$\left|q-q^{\prime }\right|\ge 2\Rightarrow \text{Supp}\phi \left(2^{-q}\cdot \right)\cap \text{Supp}\phi \left(2^{-q^{\prime }}\cdot \right)=\varnothing \mathrm{,}$

$q\ge 1\Rightarrow \text{Supp}\chi (\cdot )\cap \text{Supp}\phi \left(2^{-q^{\prime }}\cdot \right)=\varnothing \mathrm{,}$

$\frac{1}{3}\le \chi {\left(\xi \right)}^2+{\displaystyle \underset{q\ge 0}{\sum }}\phi {\left(2^{-q}\xi \right)}^2\le \mathrm{1,}\text{for}\text{all}\xi \in {ℝ}^d\mathrm{.}$

Moreover, let $h\doteq {\mathcal{F}}^{-1}\phi$ and $\tilde{h}\doteq {\mathcal{F}}^{-1}\chi$ . Then for all $f\in {\mathcal{S}}^{\prime }\left({ℝ}^d\right)$ , the dyadic operators ${\Delta }_q$ and $S_q$ can be defined as follows

${\Delta }_{q}f\doteq \phi \left(2^{-q}D\right)f=2^{qd}{\displaystyle {\int }_{{ℝ}^d}}h\left(2^{q}y\right)f\left(x-y\right)\text{d}y\mathrm{,}\text{for}q\ge \mathrm{0,}$

$S_{q}f\doteq \chi \left(2^{-q}D\right)f=2^{qd}{\displaystyle {\int }_{{ℝ}^d}}\tilde{h}\left(2^{q}y\right)f\left(x-y\right)\text{d}y,$

${\Delta }_{-1}f\doteq S_{0}f\text{and}{\Delta }_{q}f\doteq 0\text{for}q\le -\mathrm{2,}$

where $f={\displaystyle \underset{q\ge 0}{\sum }}{\Delta }_{q}f$ in ${\mathcal{S}}^{\prime }\left({ℝ}^d\right)$ , and the right-hand side is called the nonhomogeneous Littlewood-Paley decomposition of $f$ .

Definition 3.2. (See Definition 2.68 in [13] ) Let $s\in ℝ$ , $1\le p$ , $r\le \infty$ . The nonhomogenous Besov space $B_{p\mathrm{,}r}^s\left({ℝ}^d\right)$ is defined by

$B_{p,r}^s\doteq \left\{f\in {\mathcal{S}}^{\prime }\left({ℝ}^d\right);{\Vert f\Vert }_{B_{p,r}^s}<\infty \right\},$

where

${\Vert f\Vert }_{B_{p,r}^s}\doteq \{\begin{array}{ll}{\left({\displaystyle \underset{q\in \mathbb{Z}}{\sum }}{2}^{qsr}{\Vert {\Delta }_{q}f\Vert }_{L_p}^r\right)}^{\frac{1}{r}},\hfill & \text{for}r<\infty ,\hfill \\ \underset{q\in \mathbb{Z}}{\sup }{2}^{qs}{\Vert {\Delta }_{q}f\Vert }_{L_p},\hfill & \text{for}r=\infty .\hfill \end{array}$

If $s=\infty$ , $B_{p,r}^{\infty }\doteq {\displaystyle \underset{s\in ℝ}{\cap }}{B}_{p,r}^s$ .

Proposition 3.3. (See Corollary 2.86 in [13] ) For any positive real number s and any $\left(p\mathrm{,}r\right)$ in ${\left[\mathrm{1,}\infty \right]}^2$ , the space $L^{\infty }\left({ℝ}^d\right)\cap B_{p\mathrm{,}r}^s\left({ℝ}^d\right)$ is an algebra and a constant C exists such that

${\Vert uv\Vert }_{B_{p\mathrm{,}r}^s\left({ℝ}^d\right)}\le C\left({\Vert u\Vert }_{L^{\infty }\left({ℝ}^d\right)}{\Vert v\Vert }_{B_{p\mathrm{,}r}^s\left({ℝ}^d\right)}+{\Vert u\Vert }_{B_{p\mathrm{,}r}^s\left({ℝ}^d\right)}{\Vert v\Vert }_{L^{\infty }\left({ℝ}^d\right)}\right)\mathrm{.}$

If $s>\frac{d}{p}$ or $s=\frac{d}{p}$ , $r=1$ , then we have

${\Vert uv\Vert }_{B_{p,r}^s\left({ℝ}^d\right)}\le C{\Vert u\Vert }_{B_{p,r}^s\left({ℝ}^d\right)}{\Vert v\Vert }_{B_{p,r}^s\left({ℝ}^d\right)}.$

Proposition 3.4. Suppose that $s\in ℝ$ , $1\le p\mathrm{,}r\mathrm{,}{p}_i\mathrm{,}{r}_i\le \infty$ $\left(i=1,2\right)$ . We have

1) (See Proposition 1.3.5 in [14] ) Topological properties: $B_{p\mathrm{,}r}^s$ is a Banach space which is continuously embedded in ${\mathcal{S}}^{\prime }$ .

2) (See Proposition 1.3.5 in [14] ) Density: ${\mathcal{C}}_c^{\infty }$ is dense in $B_{p,r}^s$ $\iff$ $1\le p,r<\infty$ .

3) (See Proposition 1.3.5 in [14] ) Embedding: $B_{p_1\mathrm{,}{r}_1}^s$ ↪ $B_{p_2\mathrm{,}{r}_2}^{s-n\left(\frac{1}{p_1}-\frac{1}{p_2}\right)}$ , if $p_1\le p_2$ and $r_1\le r_2$ . $B_{p,r_2}^{s_2}$ ↪ $B_{p,r_1}^{s_1}$ is locally compact, if $s_1<s_2$ .

4) (See Proposition 1.4.3 in [14] ) Algebraic properties: for all $s>0$ , $B_{p\mathrm{,}r}^s\cap L^{\infty }$ is an algebra. Moreover, $B_{p\mathrm{,}r}^s$ is an algebra $\iff$ $B_{p,r}^s$ ↪ $L^{\infty }$ $\iff$ $s>\frac{n}{p}$ (or $s=\frac{n}{p}$ and $r=1$ ).

5) (See Proposition 1.3.5 in [14] ) Complex interpolation:

${\Vert u\Vert }_{B_{p\mathrm{,}r}^{\theta s_1+\left(1-\theta \right)s_2}}\le C{\Vert u\Vert }_{B_{p\mathrm{,}r}^{s_1}}^{\theta }{\Vert u\Vert }_{B_{p\mathrm{,}r}^{s_2}}^{1-\theta }\mathrm{,}$

for all $u\in B_{p\mathrm{,}r}^{s_1}\cap B_{p\mathrm{,}r}^{s_2}$ and $\theta \in \left[\mathrm{0,1}\right]$ .

6) (See Proposition 1.3.5 in [14] ) Fatou lemma: If ${\left(u_n\right)}_{n\in \mathbb{N}}$ is bounded in $B_{p\mathrm{,}r}^s$ and $u_n\to u$ in ${\mathcal{S}}^{\prime }$ , then $u\in B_{p\mathrm{,}r}^s$ and a subsequence ${\left(u_{n_k}\right)}_{k\in \mathbb{N}}$ exists such that

${\Vert u\Vert }_{B_{p,r}^s}\le \underset{k\to \infty }{\lim \inf }{\Vert u_{n_k}\Vert }_{B_{p,r}^s}.$

Lemma 3.5. (See Lemma 4.1 in [15] ) The transport equation is one of the fundamental partial differential equations and appears in many mathematical problems. By virtue of the uniqueness of the transport equation, one obtains the

estimates of the source term $f$ . Let $y_0\in B_{p\mathrm{,1}}^{\frac{1}{p}}$ with $1\le p<\infty$ and $f\in L^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)$ . Define $\bar{\mathbb{N}}=\mathbb{N}\cup \infty$ , for $n\in \bar{\mathbb{N}}$ , denote by $y_n\in \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)$ the solutions of

$\{\begin{array}{ll}{\partial }_t{y}_n+A_n\left(u\right){\partial }_x{y}_n=f,\hfill & x\in ℝ,\hfill \\ {y_n\left(t,x\right)|}_{t=0}=y_0\left(x\right).\hfill & \hfill \end{array}$

Assume for some $\alpha \left(t\right)\in L^1\left(\left[\mathrm{0,}T\right]\right)$ , $\underset{n\in \bar{\mathbb{N}}}{\sup }{\Vert A_n\left(u\right)\Vert }_{B_{p,1}^{1+\frac{1}{p}}}^{}\le \alpha \left(t\right)$ . If $A_n\left(u\right)$ converges in $A_{\infty }\left(u\right)$ in $L^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)$ , then the sequence ${\left(y_n\right)}_{n\in \mathbb{N}}$ converges in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)$ .

Lemma 3.6. (See Lemma 2.8 in [16] ) Suppose that $\left(p\mathrm{,}r\right)\in {\left[\mathrm{1,}+\infty \right]}^2$ and $s>-\min \left\{\frac{1}{p},1-\frac{1}{p}\right\}$ . Assume $f_0\in B_{p\mathrm{,}r}^s$ , $F\in L^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)$ and

$\{\begin{array}{ll}{\partial }_{x}v\in L^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^{s-1}\right)\mathrm{,}\hfill & \text{if}s>1+\frac{1}{p}\left(s=1+\frac{1}{p},r=1\right)\mathrm{,}\hfill \\ {\partial }_{x}v\in L^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\mathrm{,}\hfill & \text{if}s=1+\frac{1}{p},r>\mathrm{1,}\hfill \\ {\partial }_{x}v\in L^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}\infty }^{\frac{1}{p}}\cap L^{\infty }\right)\mathrm{,}\hfill & \text{if}s<1+\frac{1}{p}\mathrm{.}\hfill \end{array}$

If $f\in L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)\cap \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{\mathcal{S}}^{\prime }\right)$ solves

$\{\begin{array}{ll}{\partial }_{t}f+v\cdot {\partial }_{x}f=F,\hfill & t>0,x\in ℝ,\hfill \\ {f|}_{t=0}=f_0.\hfill & \hfill \end{array}$

1) Then there exists a constant C such that the following statements

${\Vert f\left(t\right)\Vert }_{B_{p\mathrm{,}r}^s}\le {\text{e}}^{CV\left(t\right)}\left({\Vert f_0\Vert }_{B_{p\mathrm{,}r}^s}+{\displaystyle {\int }_0^t}{\text{e}}^{-CV\left(\tau \right)}{\Vert F\left(\tau \right)\Vert }_{B_{p\mathrm{,}r}^s}\text{d}\tau \right)\mathrm{,}$

where

$V\left(t\right)=\{\begin{array}{ll}{\displaystyle {\int }_0^t}{\Vert {\partial }_{x}v\Vert }_{B_{p\mathrm{,}r}^{\frac{1}{p}}\cap L^{\infty }}^{}\text{d}\tau \mathrm{,}\hfill & \text{if}s<1+\frac{1}{p}\mathrm{,}\hfill \\ {\displaystyle {\int }_0^t}{\Vert {\partial }_{x}v\Vert }_{B_{p\mathrm{,}r}^{1+\frac{1}{p}}}^{}\text{d}\tau \mathrm{,}\hfill & \text{if}s=1+\frac{1}{p},r>\mathrm{1,}\hfill \\ {\displaystyle {\int }_0^t}{\Vert {\partial }_{x}v\Vert }_{B_{p\mathrm{,}r}^{s-1}}\text{d}\tau \mathrm{,}\hfill & \text{if}s>1+\frac{1}{p}\left(\text{or}s=1+\frac{1}{p}\mathrm{,}r=1\right).\hfill \end{array}$

2) If $f=v$ , then for all $s>0$ , (1) holds with $V\left(t\right)={\displaystyle {\int }_0^t}{\Vert {\partial }_{x}v\Vert }_{L^{\infty }}\text{d}\tau$ .

3) If $r<\infty$ , then $f\in \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,}r}^s\right)$ . If $r=\infty$ , then $f\in \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{s^{\prime }}\right)$ for all $s^{\prime }<s$ .

Lemma 3.7. (See Corollary 2.86 in [13] ) Assume that $1\le p$ , $r\le \infty$ , for $s_1\le \frac{1}{p}$ , $s_2>\frac{1}{p}$ , ( $s_2\ge \frac{1}{p}$ if $r=1$ ) and $s_1+s_2>0$ , the following estimates holds

${\Vert fg\Vert }_{B_{p\mathrm{,}r}^{s_1}}\le C{\Vert f\Vert }_{B_{p\mathrm{,}r}^{s_1}}{\Vert g\Vert }_{B_{p\mathrm{,}r}^{s_2}}\mathrm{,}$

where the constant C is independent of f and g.

Lemma 3.8. (See Theorem 2.100 in [13] ) Let $s>0$ , $1\le r\le \infty$ and $1\le p\le p_1\le \infty$ . Let v be a vector field over ${ℝ}^d$ . Define $R_j=\left[v\cdot \nabla \mathrm{,}{\Delta }_j\right]f$ . There exists a constant C such that

${\Vert {\left(2^{js}{\Vert R_j\Vert }_{L^p\left({ℝ}^d\right)}\right)}_j\Vert }_{l^r}\le C\left({\Vert \nabla v\Vert }_{L^{\infty }}{\Vert f\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert \nabla f\Vert }_{L^{p_2}}{\Vert \nabla v\Vert }_{B_{p_1\mathrm{,}r}^{s-1}}\right)\mathrm{,}$

where $\frac{1}{p_2}=\frac{1}{p}-\frac{1}{p_1}$ . Furthermore, if $s<1$ then

${\Vert {\left(2^{js}{\Vert R_j\Vert }_{L^p\left({ℝ}^d\right)}\right)}_j\Vert }_{l^r}\le C{\Vert \nabla v\Vert }_{L^{\infty }}{\Vert f\Vert }_{B_{p\mathrm{,}r}^s}\mathrm{.}$

4. Local Well-Posedness

From the relationship ${\partial }_{x}u=G-\varphi \ast \rho$ , we can claim that

${\Vert {\partial }_{x}u\Vert }_{L^{\infty }}\le {\Vert \varphi \Vert }_{L^1}{\Vert \rho \Vert }_{L^{\infty }}+{\Vert G\Vert }_{L^{\infty }}\mathrm{,}{\Vert {\partial }_{x}u\Vert }_{B_{p\mathrm{,}r}^s}\le {\Vert \varphi \Vert }_{L^1}{\Vert \rho \Vert }_{B_{p\mathrm{,}r}^s}+{\Vert G\Vert }_{B_{p\mathrm{,}r}^s}\mathrm{.}$ (1)

Firstly, Young inequality results in ${\Vert {\partial }_{x}u\Vert }_{L^{\infty }}\le {\Vert \varphi \ast \rho \Vert }_{L^{\infty }}+{\Vert G\Vert }_{L^{\infty }}\le {\Vert \varphi \Vert }_{L^1}{\Vert \rho \Vert }_{L^{\infty }}+{\Vert G\Vert }_{L^{\infty }}$ . On the other hand, according to the definition of Besov spaces, we can get

$\begin{array}{c}{\Delta }_j\left(\varphi \ast \rho \right)=2^j{\displaystyle {\int }_{ℝ}}h\left(2^{j}y\right)\left(\varphi \ast \rho \right)\left(x-y\right)\text{d}y\\ =2^j{\displaystyle {\int }_{ℝ}}{\displaystyle {\int }_{ℝ}}h\left(2^{j}y\right)\varphi \left(z\right)\rho \left(x-y-z\right)\text{d}z\text{d}y\\ ={\displaystyle {\int }_{ℝ}}\varphi \left(z\right){\displaystyle {\int }_{ℝ}}{2}^{j}h\left(2^{j}y\right)\rho \left(x-y-z\right)\text{d}z\text{d}y\\ =\varphi \ast {\Delta }_j\rho .\end{array}$

Applying ${\Delta }_j$ to ${\partial }_{x}u=G-\varphi \ast \rho$ and taking $L^p$ -norm to the above relationship yields

${\Vert {\Delta }_j{\partial }_{x}u\Vert }_{L^p}\le {\Vert {\Delta }_j\rho \ast \varphi \Vert }_{L^p}+{\Vert {\Delta }_{j}G\Vert }_{L^p}\le {\Vert \varphi \Vert }_{L^1}{\Vert {\Delta }_j\rho \Vert }_{L^p}+{\Vert {\Delta }_{j}G\Vert }_{L^p}\mathrm{.}$ (2)

By the definition of Besov spaces and Minkowski’s inequality, for $r<\infty$ , we can obtain

$\begin{array}{c}{\Vert {\partial }_{x}u\Vert }_{B_{p,r}^s}={\left({\displaystyle \underset{j\in \mathbb{Z}}{\sum }}{2}^{jsr}{\Vert {\Delta }_j{\partial }_{x}u\Vert }_{L_p}^r\right)}^{\frac{1}{r}}\\ \le {\left({\displaystyle \underset{j\in \mathbb{Z}}{\sum }}{\left({\Vert \varphi \Vert }_{L^1}{2}^{js}{\Vert {\Delta }_j\rho \Vert }_{L^p}+2^{js}{\Vert {\Delta }_{j}G\Vert }_{L^p}\right)}^r\right)}^{\frac{1}{r}}\\ \le {\Vert \varphi \Vert }_{L^1}{\left({\displaystyle \underset{j\in \mathbb{Z}}{\sum }}{2}^{jsr}{\Vert {\Delta }_j\rho \Vert }_{L^p}^r\right)}^{\frac{1}{r}}+{\left({\displaystyle \underset{j\in \mathbb{Z}}{\sum }}{2}^{jsr}{\Vert {\Delta }_{j}G\Vert }_{L^p}^r\right)}^{\frac{1}{r}}\\ ={\Vert \varphi \Vert }_{L^1}{\Vert \rho \Vert }_{B_{p,r}^s}+{\Vert G\Vert }_{B_{p,r}^s}.\end{array}$

The case $r=\infty$ can be easily treated as above, this completes the proof of the claim (3.1).

Moreover, by Bernstein-Type Lemmas (see Lemma 2.1 in [13] ), one can get

${\Vert u\Vert }_{B_{p\mathrm{,}r}^{s+1}}+{\Vert \rho \Vert }_{B_{p\mathrm{,}r}^s}\cong {\Vert u\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert {\partial }_{x}u\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert \rho \Vert }_{B_{p\mathrm{,}r}^s}\cong {\Vert u\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert G\Vert }_{B_{p\mathrm{,}r}^s}+{\Vert \rho \Vert }_{B_{p\mathrm{,}r}^s}\mathrm{.}$

Proof. In order to prove Theorem 2.1, we proceed as the following steps.

Step 1: Existence

Firstly, we aim to construct approximate solutions to smooth solutions of some linear equations. Let $z_n\left(t,\cdot \right)=\left({\rho }_n,G_n,u_n\right)$ , for $z_0=\left({\rho }_0,G_0,u_0\right)\triangleq \left(0,0,0\right)$ , we can define the induction sequence ${\left(z_n\right)}_{n\in \mathbb{N}}$ by solving the following linear transport equations

$\{\begin{array}{l}{\partial }_t{\rho }_{n+1}+u_n{\partial }_x{\rho }_{n+1}=-{\partial }_x{u}_n{\rho }_n,\\ {\partial }_t{G}_{n+1}+u_n{\partial }_x{G}_{n+1}=-{\partial }_x{u}_n{G}_n,\\ {\partial }_t{u}_{n+1}+u_n{\partial }_x{u}_{n+1}=\varphi \ast \left(u_n{\rho }_n\right)-\left(\varphi \ast {\rho }_n\right)u_n,\\ {\rho }_{n+1}\left(0\right)=S_{n+1}{\rho }^0,u_{n+1}\left(0\right)=S_{n+1}{u}^0.\end{array}$ (3)

By induction, we firstly assume that $z_n\in {\left(L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)\right)}^3$ for all $T>0$ . Owing to $s>\frac{1}{p}$ , it implies that $B_{p\mathrm{,}r}^s$ is an algebra. Combining Lemma 3.6 and (3), we deduce that there exists a global solution $z_{n+1}\in {\left(E_T^p\right)}^3$ and $E_T^p=\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)\cap {\mathcal{C}}^1\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)$ . Making use of Lemma 3.6, we can obtain the following inequality

$\begin{array}{c}{\Vert {\rho }_{n+1}\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\le \exp \left\{C{\displaystyle {\int }_0^t}{\Vert {\partial }_x{u}_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}\cap L^{\infty }}^{}\text{d}{t}^{\prime }\right\}({\Vert {\rho }_{n+1}\left(0\right)\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\\ +{\displaystyle {\int }_0^t}\exp \left\{-C{\displaystyle {\int }_0^{t^{\prime }}}{\Vert {\partial }_x{u}_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}\cap L^{\infty }}^{}\text{d}{t^{\prime }}^{\prime }\right\}{\Vert {\rho }_n{\partial }_x{u}_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\text{d}{t}^{\prime })\mathrm{,}\end{array}$ (4)

because $B_{p\mathrm{,1}}^{\frac{1}{p}}$ is an algebra and $B_{p\mathrm{,1}}^{\frac{1}{p}}$ ↪ $L^{\infty }$ , according to (1), we can obtain

${\Vert {\rho }_n{\partial }_x{u}_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\le {\Vert {\rho }_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}{\Vert {\partial }_x{u}_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\le C\left({\Vert {\rho }_n\Vert }_{B_{p,1}^{\frac{1}{p}}}^2+{\Vert G_n\Vert }_{B_{p,1}^{\frac{1}{p}}}^2\right),$

${\Vert {\partial }_x{u}_n\Vert }_{B_{p,1}^{\frac{1}{p}}\cap L^{\infty }}^{}\le {\Vert {\partial }_x{u}_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\le C\left({\Vert {\rho }_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert G_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\right),$

then submittting the above inequalities into (4), we can conclude

$\begin{array}{c}{\Vert {\rho }_{n+1}\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\le \exp \left\{C{\displaystyle {\int }_0^t}\left({\Vert {\rho }_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert G_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\right)\text{d}{t}^{\prime }\right\}({\Vert {\rho }_{n+1}\left(0\right)\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\\ +C{\displaystyle {\int }_0^t}\exp \left\{-C{\displaystyle {\int }_0^{t^{\prime }}}\left({\Vert {\rho }_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert G_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\right)\text{d}{t^{\prime }}^{\prime }\right\}\left({\Vert {\rho }_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^2+{\Vert G_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^2\right)\text{d}{t}^{\prime })\mathrm{.}\end{array}$ (5)

By a similar argument as above to the component G, we have

$\begin{array}{c}{\Vert G_{n+1}\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\le \exp \left\{C{\displaystyle {\int }_0^t}\left({\Vert {\rho }_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert G_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\right)\text{d}{t}^{\prime }\right\}({\Vert G_{n+1}\left(0\right)\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\\ +C{\displaystyle {\int }_0^t}\exp \left\{-C{\displaystyle {\int }_0^{t^{\prime }}}\left({\Vert {\rho }_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert G_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\right)\text{d}{t^{\prime }}^{\prime }\right\}\left({\Vert {\rho }_n\Vert }_{B_{p,1}^{\frac{1}{p}}}^2+{\Vert G_n\Vert }_{B_{p,1}^{\frac{1}{p}}}^2\right)\text{d}{t}^{\prime }),\end{array}$ (6)

and

$\begin{array}{c}{\Vert u_{n+1}\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\le \exp \left\{C{\displaystyle {\int }_0^t}{\Vert {\partial }_x{u}_n\Vert }_{B_{p,1}^{\frac{1}{p}}\cap L^{\infty }}^{}\text{d}{t}^{\prime }\right\}({\Vert u_{n+1}\left(0\right)\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\\ +{\displaystyle {\int }_0^t}\exp \left\{-C{\displaystyle {\int }_0^{t^{\prime }}}{\Vert {\partial }_x{u}_n\Vert }_{B_{p,1}^{\frac{1}{p}}\cap L^{\infty }}^{}\text{d}{t^{\prime }}^{\prime }\right\}{\Vert \varphi \ast \left(u_n{\rho }_n\right)-\left(\varphi \ast {\rho }_n\right)u_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\text{d}{t}^{\prime })\mathrm{,}\end{array}$ (7)

we can see that

${\Vert \varphi \ast \left(u_n{\rho }_n\right)-\left(\varphi \ast {\rho }_n\right)u_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\le C\left({\Vert u_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^2+{\Vert {\rho }_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^2\right)\mathrm{,}$

then submitting the inequality into (7), we can get

$\begin{array}{c}{\Vert u_{n+1}\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\le \exp \left\{C{\displaystyle {\int }_0^t}\left({\Vert {\rho }_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert G_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert u_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\right)\text{d}{t}^{\prime }\right\}({\Vert u_{n+1}\left(0\right)\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\\ +C{\displaystyle {\int }_0^{t^{\prime }}}\exp \left\{-C{\displaystyle {\int }_0^t}\left({\Vert {\rho }_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert G_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert u_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\right)\text{d}{t^{\prime }}^{\prime }\right\}({\Vert u_n\Vert }_{B_{p,1}^{\frac{1}{p}}}^2\\ +{\Vert {\rho }_n\Vert }_{B_{p,1}^{\frac{1}{p}}}^2)\text{d}{t}^{\prime }).\end{array}$ (8)

Let $Z_n\left(t\right)={\Vert {\rho }_n\left(t,\cdot \right)\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert G_n\left(t,\cdot \right)\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert u_n\left(t,\cdot \right)\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}$ , combining inequalities (5), (6), and (8), we can obtain

$\begin{array}{l}{Z}_{n+1}\left(t\right)\le \exp \left\{C{\displaystyle {\int }_0^t}{Z}_n\text{d}{t}^{\prime }\right\}{Z}^0+C{\displaystyle {\int }_0^t}{Z}_n^2\exp \left\{C{\displaystyle {\int }_{\tau }^{t^{\prime }}}{Z}_n\text{d}{t^{\prime }}^{\prime }\right\}\text{d}{t}^{\prime },\hfill \end{array}$ (9)

where $Z^0={\Vert {\rho }^0\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert G^0\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert u^0\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}$ . For fixed $T^{*}>0$ such that $2C{z}_0{T}^{*}<1$ and suppose that

$\forall t\in \left[\mathrm{0,}{T}^{\mathrm{<mo>\; *\; </mo>}}\right]\mathrm{,}{Z}_n\left(t\right)\le \frac{Z^0}{1-2C{Z}^{0}t}.$

In fact, we assume that the inequality for all $n\in \mathbb{N}$ is valid, then for $0\le \tau <t<T^{*}<\frac{1}{2C{Z}^0}$ , we can have

$\exp \left(C{\displaystyle {\int }_{\tau }^t}{Z}_n\text{d}{t}^{\prime }\right)\le \exp \left({\displaystyle {\int }_{\tau }^t}\frac{C{Z}^0}{1-2C{Z}^0{t}^{\prime }}\text{d}{t}^{\prime }\right)={\left(\frac{1-2C{Z}^0\tau }{1-2C{Z}^{0}t}\right)}^{\frac{1}{2}}\mathrm{,}$

applying the above inequalities into (9), we can get

$\begin{array}{c}{Z}_n\left(t\right)\le \frac{Z^0}{\sqrt{1-2C{Z}^{0}t}}+{\displaystyle {\int }_0^t}\frac{C{\left(Z^0\right)}^2{\left(1-2C{Z}^0\tau \right)}^{\frac{1}{2}}}{{\left(1-2C{Z}^0\tau \right)}^2{\left(1-2C{Z}^{0}t\right)}^{\frac{1}{2}}}\text{d}\tau \\ =\frac{1}{\sqrt{1-2C{Z}^{0}t}}\left(Z^0+{\displaystyle {\int }_0^t}\frac{C{\left(Z^0\right)}^2}{{\left(1-2C{Z}^0\tau \right)}^{\frac{3}{2}}}\text{d}\tau \right)\\ =\frac{1}{\sqrt{1-2C{Z}^{0}t}}\left(Z^0+Z^0{\frac{1}{\sqrt{1-2C{Z}^0\tau }}|}_0^t\right)\\ =\frac{Z^0}{1-2C{Z}^{0}t}.\end{array}$

The above derivation implies that ${\left(z_n\right)}_{n\in \mathbb{N}}$ is uniformly bounded in

${\left(L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)\right)}^3$ . Based on this, we can get that ${\left({\partial }_t{z}_n\right)}_{n\in \mathbb{N}}$ is uniformly bounded in ${\left(L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)\right)}^3$ . Therefore, ${\left(z_n\right)}_{n\in \mathbb{N}}$ is uniformly bounded in ${\left(\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)\cap {\mathcal{C}}^{\frac{1}{2}}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)\right)}^3$ .

In order to obtain a solution z of Equation (9), we make use of the compactness theory for the approximating sequence ${\left(z_n\right)}_{n\in \mathbb{N}}$ . We take a sequence ${\left({\phi }_j\right)}_{j\in \mathbb{N}}$ of smooth functions with values in $\left[\mathrm{0,1}\right]$ , supported in $B\left(\mathrm{0,}j+1\right)$ and equals to 1 on $B\left(\mathrm{0,}j\right)$ . It is easy to find that the map $z_n\mapsto {\phi }_j{z}_n$ is compact from

${\left(B_{p\mathrm{,1}}^{\frac{1}{p}}\right)}^3$ to ${\left(B_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)}^3$ by the virtue of Theorem 2.94 in [13] . Taking advantage of

Ascoli’s theorem, there exists some function $z_j$ such that the sequence ${\left({\phi }_j{z}_n\right)}_{j\in \mathbb{N}}$ converges to $z_j$ for any $j\in \mathbb{N}$ . At the same time, according to the Cantor diagonal process, there exists a subsequence of ${\left(z_j\right)}_{j\in \mathbb{N}}$ such that ${\phi }_j{z}_n$

converges to $z_j$ in ${\left(\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)\right)}^3$ for any $j\in \mathbb{N}$ . Owing to ${\phi }_j{\phi }_{j+1}={\phi }_j$ , we can get $z_j={\phi }_j{z}_{j+1}$ . Hence, there exists some function z such that the sequence ${\left(\phi z_n\right)}_{n\ge 1}$ tends to $\phi z$ in ${\left(\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)\right)}^3$ for any $\phi \in \mathcal{D}$ . Then on the basis of uniform boundeness of ${\left(z_j\right)}_{n\ge 1}$ and the Fatou property, we can obtain that z is bounded in ${\left(L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)\right)}^3$ . Taking advantage of the Fatou property yields that $\phi z_n$ tends to $\phi z$ in ${\left(\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-\epsilon }\right)\right)}^3$ for any $\epsilon >0$ small enough.

Furthermore, set any $\psi \in B_{p^{\prime }\mathrm{,}\infty }^{1-\frac{1}{p}}$ and combining with the duality for $\psi$ , as $n\to \infty$ , we can obtain

$\begin{array}{l}\langle {\partial }_t\left(\phi u_n\right)-{\partial }_t\left(\phi u\right)\mathrm{,}\psi \rangle -\langle \left(\phi u_n\right){\partial }_x\left(\phi u_n\right)-\left(\phi u\right){\partial }_x\left(\phi u\right)\mathrm{,}\psi \rangle \\ -\langle \varphi \ast \left(\phi u_n\phi {\rho }_n\right)-\left(\varphi \ast \phi {\rho }_n\right)\phi u_n-\varphi \ast \left(\phi u\phi \rho \right)-\left(\varphi \ast \phi \rho \right)\phi u\mathrm{,}\psi \rangle \to 0.\end{array}$

The main problem is the third term, for the sake of convenience, we treat only the term of $\langle \varphi \ast \left(\phi u_n\phi {\rho }_n\right)-\left(\varphi \ast \phi {\rho }_n\right)\phi u_n-\varphi \ast \left(\phi u\phi \rho \right)-\left(\varphi \ast \phi \rho \right)\phi u\mathrm{,}\psi \rangle$ . Therefore, we can have

$\begin{array}{l}\left|\langle \varphi \ast \left(\phi u_n\phi {\rho }_n\right)-\left(\varphi \ast \phi {\rho }_n\right)\phi u_n-\varphi \ast \left(\phi u\phi \rho \right)-\left(\varphi \ast \phi \rho \right)\phi u\mathrm{,}\psi \rangle \right|\\ \le \left|\langle \varphi \ast \left(\phi u_n\phi {\rho }_n\right)-\varphi \ast \left(\phi u\phi \rho \right)\mathrm{,}\psi \rangle \right|+\left|\langle \left(\varphi \ast \phi {\rho }_n\right)\phi u_n-\left(\varphi \ast \phi \rho \right)\phi u\mathrm{,}\psi \rangle \right|\mathrm{.}\end{array}$ (10)

Firstly, we can estimate the first term of the inequality (10)

$\begin{array}{l}\left|\langle \varphi \ast \left(\phi u_n\phi {\rho }_n\right)-\varphi \ast \left(\phi u\phi \rho \right)\mathrm{,}\psi \rangle \right|\\ \le {\Vert \varphi \ast \left(\phi u_n\phi {\rho }_n\right)-\varphi \ast \left(\phi u\phi \rho \right)\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}-1}}^{}{\Vert \psi \Vert }_{B_{p^{\prime }\mathrm{,}\infty }^{1-\frac{1}{p}}}^{}\\ \le C{\Vert \phi u_n\phi {\rho }_n-\phi u\phi \rho \Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}-\epsilon }}^{}{\Vert \psi \Vert }_{B_{p^{\prime }\mathrm{,}\infty }^{1-\frac{1}{p}}}^{}\\ \le C({\Vert \phi u_n\Vert }_{L^{\infty }}{\Vert \phi {\rho }_n-\phi \rho \Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}-\epsilon }}^{}+{\Vert \phi u_n\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}-\epsilon }}^{}{\Vert \phi {\rho }_n-\phi \rho \Vert }_{L^{\infty }}\\ +{\Vert \phi \rho \Vert }_{L^{\infty }}{\Vert \phi u_n-\phi u\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}-\epsilon }}^{}+{\Vert \phi \rho \Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}-\epsilon }}^{}{\Vert \phi u_n-\phi u\Vert }_{L^{\infty }}){\Vert \psi \Vert }_{B_{p^{\prime }\mathrm{,}\infty }^{1-\frac{1}{p}}}^{}\mathrm{.}\end{array}$ (11)

For the second term of the inequality (10), taking the same approach. We have Proven that $\phi z_n\to \phi z$ in ${\left(\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-\epsilon }\right)\right)}^3$ , and $\phi z_n$ is bounded in ${\left(L_T^{\infty }\left(B_{p\mathrm{,1}}^{\frac{1}{p}}\right)\right)}^3$ , then we can get that (11) tends to 0 uniformly on $\left[\mathrm{0,}T\right]$ as $n\to \infty$ . So (11) tends to 0 when $n\to \infty$ . Applying the similar argument, the second term of the inequality (10) also tends to 0 as $n\to \infty$ , then ${\partial }_t\left(\phi z\right)\in {\left(\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)\right)}^3$ . Hence, we deduce that z is the solution of the Equation (9), and belongs to ${\left(E_T^p\right)}^3$ .

Step 2: Uniqueness

In this step, taking advantage of the Lagrangian coordinate, we will prove the uniqueness of the smooth solution z. Introducing a new variable $\xi \in ℝ$ , and we define $y\left(t\mathrm{,}\xi \right)$ as

$\{\begin{array}{ll}{\partial }_{t}y\left(t,\xi \right)=u\left(t,y\left(t,\xi \right)\right),\hfill & t>0,x\in ℝ,\hfill \\ {y\left(t,\xi \right)|}_{t=0}=\xi .\hfill & \hfill \end{array}$

Next, Define the new variables $U\left(t\mathrm{,}\xi \right)=u\left(t\mathrm{,}y\left(t\mathrm{,}\xi \right)\right)$ and $V\left(t,\xi \right)=\rho \left(t,y\left(t,\xi \right)\right)$ , then we can obtain ${\partial }_{\xi }U\left(t\mathrm{,}\xi \right)={\partial }_{x}u\left(t\mathrm{,}y\left(t\mathrm{,}\xi \right)\right){\partial }_{\xi }y\left(t\mathrm{,}y\left(t\mathrm{,}\xi \right)\right)$ and ${\partial }_{\xi }V\left(t\mathrm{,}\xi \right)={\partial }_x\rho \left(t\mathrm{,}y\left(t\mathrm{,}\xi \right)\right){\partial }_{\xi }y\left(t\mathrm{,}y\left(t\mathrm{,}\xi \right)\right)$ . At the same time, the function $U\left(t\mathrm{,}\xi \right)$ is a solution of

${\partial }_{t}U\left(t\mathrm{,}\xi \right)={\partial }_{t}u\left(t,y\left(t\mathrm{,}\xi \right)\right)+{\partial }_{x}u\left(t,y\left(t\mathrm{,}\xi \right)\right){\partial }_{t}y\left(t\mathrm{,}\xi \right)\triangleq Q\left(t\mathrm{,}\xi \right),$ (12)

where

$Q\left(t\mathrm{,}\xi \right)=\varphi \ast \left(u\left(t\mathrm{,}y\left(t\mathrm{,}\xi \right)\right)\rho \left(t\mathrm{,}y\left(t\mathrm{,}\xi \right)\right)\right)-\left(\varphi \ast \rho \left(t\mathrm{,}y\left(t\mathrm{,}\xi \right)\right)\right)u\left(t\mathrm{,}y\left(t\mathrm{,}\xi \right)\right)\mathrm{.}$

Moreover, $V\left(t\mathrm{,}\xi \right)$ is a solution of

$\begin{array}{c}{\partial }_{t}V\left(t,\xi \right)={\partial }_t\rho \left(t,y\left(t,\xi \right)\right)+{\partial }_x\rho \left(t,y\left(t,\xi \right)\right){\partial }_{t}y\left(t,\xi \right)\\ =-\rho \left(t,y\left(t,\xi \right)\right){\partial }_{x}u\left(t,y\left(t,\xi \right)\right).\end{array}$ (13)

Taking the derivative of the Equation (12) with respect to variable $\xi$ , we can have

$\begin{array}{c}{\partial }_{t\xi }U\left(t,\xi \right)={\partial }_{\xi }Q\left(t,\xi \right)\\ =\varphi \ast \left({\partial }_{\xi }U\rho \right)+\varphi \ast \left(U{\partial }_{\xi }\rho \right)-\left(\varphi \ast \rho \right){\partial }_{\xi }U-\left(\varphi \ast {\partial }_{\xi }\rho \right)U.\end{array}$

Similarly, we can infer that

${\partial }_{t\xi }y\left(t,\xi \right)={\partial }_{\xi }u\left(t,y\left(t,\xi \right)\right)={\partial }_{\xi }U\left(t,\xi \right),$

and

$y\left(t,\xi \right)=\xi +{\displaystyle {\int }_0^t}U\left(t,\xi \right)\text{d}\tau ,$

and

${\partial }_{\xi }y\left(t\mathrm{,}\xi \right)=1+{\displaystyle {\int }_0^t}{\partial }_{\xi }U\left(t\mathrm{,}\xi \right)\text{d}\tau \mathrm{.}$

Since the fact that $\left(u\mathrm{,}\rho \right)$ is ${\mathcal{C}}_T\left(B_{p\mathrm{,1}}^{\frac{1}{p}}\right)\times {\mathcal{C}}_T\left(B_{p\mathrm{,1}}^{\frac{1}{p}}\right)$ and the embedding $B_{p\mathrm{,1}}^{\frac{1}{p}}$ ↪

$W^{\mathrm{1,}p}\cap W^{\mathrm{1,}\infty }$ , we can deduce that $\left(u\mathrm{,}\rho \right)$ is uniformly bounded in

${\mathcal{C}}_T\left(W^{\mathrm{1,}p}\cap W^{\mathrm{1,}\infty }\right)\times {\mathcal{C}}_T\left(W^{\mathrm{1,}p}\cap W^{\mathrm{1,}\infty }\right)$ easily. In addition, we can prove that ${\partial }_{\xi }y$ is uniformly bounded in $L_T^{\infty }\left(L^{\infty }\right)$ . Moreover, based on the above discussion, for

sufficiently small $T>0$ , we have $\frac{1}{2}\le {\partial }_{\xi }y\le C_{u_0}$ . By the continuous method and

the boundedness of $\left(u\mathrm{,}\rho \right)$ in ${\mathcal{C}}_T\left(W^{\mathrm{1,}p}\cap W^{\mathrm{1,}\infty }\right)\times {\mathcal{C}}_T\left(W^{\mathrm{1,}p}\cap W^{\mathrm{1,}\infty }\right)$ , let t is small enough, we can get

${\Vert U\left(t\mathrm{,}\xi \right)\Vert }_{L_p}^p={\displaystyle {\int }_R}{\left|u\left(t\mathrm{,}y\left(t\mathrm{,}\xi \right)\right)\right|}^p\frac{1}{y_{\xi }}\text{d}y\le {\Vert u\Vert }_{L^p}^p{\Vert \frac{1}{y_{\xi }}\Vert }_{L^{\infty }}\le 2{\Vert u\Vert }_{L^p}^p\le C\mathrm{,}$

${\Vert U{\left(t,\xi \right)}_{\xi }\Vert }_{L_p}^p\le {\Vert u_x\Vert }_{L^p}^p{\Vert y_{\xi }\Vert }_{L^{\infty }}^{p-1}\le C_{u_0}^{p-1}{\Vert u_x\Vert }_{L^p}^p\le C,$

${\Vert V\left(t,\xi \right)\Vert }_{L_p}^p={\displaystyle {\int }_R}{\left|\rho \left(t,y\left(t,\xi \right)\right)\right|}^p\frac{1}{y_{\xi }}\text{d}y\le {\Vert \rho \Vert }_{L^p}^p{\Vert \frac{1}{y_{\xi }}\Vert }_{L^{\infty }}\le 2{\Vert \rho \Vert }_{L^p}^p\le C,$

${\Vert V{\left(t,\xi \right)}_{\xi }\Vert }_{L_p}^p={\displaystyle {\int }_R}{\left|{\rho }_x\left(t,y\left(t,\xi \right)\right)\right|}^p{\left|y_{\xi }\right|}^{p-1}\text{d}\xi \le {\Vert {\rho }_x\Vert }_{L^p}^p{\Vert y_{\xi }\Vert }_{L^{\infty }}^{p-1}\le C.$

Through the above inequalities, for any $t\in \left[\mathrm{0,}T\right]$ , we have $U\left(t\mathrm{,}\xi \right)\in L_T^{\infty }\left(W^{\mathrm{1,}p}\cap W^{\mathrm{1,}\infty }\right)$ , $V\left(t\mathrm{,}\xi \right)\in L_T^{\infty }\left(W^{\mathrm{1,}p}\cap W^{\mathrm{1,}\infty }\right)$ ,

$y\left(t\mathrm{,}\xi \right)-\xi \in L_T^{\infty }\left(W^{\mathrm{1,}p}\cap W^{\mathrm{1,}\infty }\right)$ and $\frac{1}{2}\le {\partial }_{\xi }y\le C_{u_0}$ satisfying (12) and (13).

Set $z_1,z_2$ is two solutions in ${\left(E_T^p\right)}^3$ of Equation (9) with the same initial data, for $i=1,2$ , the function $U_j\left(t\mathrm{,}\xi \right)=u_j\left(t\mathrm{,}{y}_j\left(t\mathrm{,}\xi \right)\right)$ is a solution of

$U_{tj}\left(t,\xi \right)=u_{jt}\left(t,y_j\right)+u_{jx}\left(t,y_j\right)y_{jt}=\varphi \ast \left(u_j{\rho }_j\right)-\left(\varphi \ast {\rho }_j\right)u_j=Q_j.$

Obviously, we also have $U_j\left(t\mathrm{,}{y}_j\left(t\mathrm{,}\xi \right)\right)\mathrm{,}{y}_j\left(t\mathrm{,}\xi \right)-\xi \in L_T^{\infty }\left(W^{\mathrm{1,}p}\cap W^{\mathrm{1,}\infty }\right)$ , and $\frac{1}{2}\le {\partial }_{j\xi }y\le C_{u_0}$ for sufficiently small $T>0$ .

Next, we will establish the estimate ${\Vert U_1\left(t\mathrm{,}\xi \right)-U_2\left(t\mathrm{,}\xi \right)\Vert }_{W^{\mathrm{1,}p}\cap W^{\mathrm{1,}\infty }}$ , the key step is to estimate ${\Vert Q_1\left(t\mathrm{,}\xi \right)-Q_2\left(t\mathrm{,}\xi \right)\Vert }_{W^{\mathrm{1,}p}\cap W^{\mathrm{1,}\infty }}$ . For $Q_1\left(t\mathrm{,}\xi \right)$ and $Q_2\left(t\mathrm{,}\xi \right)$ , we can get

$Q_1\left(t\mathrm{,}\xi \right)-Q_2\left(t\mathrm{,}\xi \right)=\varphi \ast \left(U_1{V}_1-U_2{V}_2\right)-\left(\varphi \ast V_1\right)U_1+\left(\varphi \ast V_2\right)U_2\mathrm{.}$

By Young inequality and Hölder inequality, then we shall get an estimate of $L^p\cap L^{\infty }$ -the norm

$\begin{array}{l}{\Vert Q_1\left(t\mathrm{,}\xi \right)-Q_2\left(t\mathrm{,}\xi \right)\Vert }_{L^p\cap L^{\infty }}\\ \le {\Vert \varphi \ast \left(U_1{V}_1-U_2{V}_2\right)\Vert }_{L^p\cap L^{\infty }}+{\Vert \left(\varphi \ast V_1\right)U_1-\left(\varphi \ast V_2\right)U_2\Vert }_{L^p\cap L^{\infty }}\\ \le C\left({\Vert \left(U_1-U_2\right)V_1\Vert }_{L^p\cap L^{\infty }}+{\Vert \left(V_1-V_2\right)U_2\Vert }_{L^p\cap L^{\infty }}\right)\\ +{\Vert \left(\varphi \ast V_1\right)\left(U_1-U_2\right)\Vert }_{L^p\cap L^{\infty }}+{\Vert \left(\varphi \ast \left(V_1-V_2\right)\right)U_2\Vert }_{L^p\cap L^{\infty }}\end{array}$

$\begin{array}{l}\le C\left({\Vert U_1-U_2\Vert }_{L^p\cap L^{\infty }}{\Vert V_1\Vert }_{L^{\infty }}+{\Vert V_1-V_2\Vert }_{L^p\cap L^{\infty }}{\Vert U_2\Vert }_{L^{\infty }}\right)\\ +{\Vert U_1-U_2\Vert }_{L^p\cap L^{\infty }}{\Vert \varphi \ast V_1\Vert }_{L^{\infty }}+{\Vert \varphi \ast \left(V_1-V_2\right)\Vert }_{L^p\cap L^{\infty }}{\Vert U_2\Vert }_{L^{\infty }}\\ \le C\left({\Vert V_1-V_2\Vert }_{L^p\cap L^{\infty }}+{\Vert U_1-U_2\Vert }_{L^p\cap L^{\infty }}\right)\mathrm{.}\end{array}$

Similarly, we can have

$\begin{array}{c}{Q}_{1\xi }\left(t\mathrm{,}\xi \right)-Q_{2\xi }\left(t\mathrm{,}\xi \right)=\varphi \ast {\left(u_1{\rho }_1-u_2{\rho }_2\right)}_{\xi }-\left({\left(\varphi \ast {\rho }_1\right)}_{\xi }{u}_1-{\left(\varphi \ast {\rho }_2\right)}_{\xi }{u}_2\right)\\ -\left(\left(\varphi \ast {\rho }_1\right)u_{1\xi }-\left(\varphi \ast {\rho }_2\right)u_{2\xi }\right)\mathrm{,}\end{array}$

then we shall get an estimate about $L^p\cap L^{\infty }$ -norm

$\begin{array}{l}{\Vert Q_{1\xi }\left(t\mathrm{,}\xi \right)-Q_{2\xi }\left(t\mathrm{,}\xi \right)\Vert }_{L^p\cap L^{\infty }}\\ \le C{\Vert {\left(u_1{\rho }_1-u_2{\rho }_2\right)}_{\xi }\Vert }_{L^p\cap L^{\infty }}+{\Vert {\left(\varphi \ast {\rho }_1\right)}_{\xi }{u}_1-{\left(\varphi \ast {\rho }_2\right)}_{\xi }{u}_2\Vert }_{L^p\cap L^{\infty }}\\ +{\Vert \left(\varphi \ast {\rho }_1\right)u_{1\xi }-\left(\varphi \ast {\rho }_2\right)u_{2\xi }\Vert }_{L^p\cap L^{\infty }}\\ \le C\left({\Vert V_{1\xi }-V_{2\xi }\Vert }_{L^p\cap L^{\infty }}+{\Vert V_1-V_2\Vert }_{L^p\cap L^{\infty }}+{\Vert U_{1\xi }-U_{2\xi }\Vert }_{L^p\cap L^{\infty }}+{\Vert U_1-U_2\Vert }_{L^p\cap L^{\infty }}\right)\\ \le C\left({\Vert V_1-V_2\Vert }_{L^p\cap L^{\infty }}+{\Vert U_1-U_2\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}\right)\mathrm{.}\end{array}$

Similarly, applying this routine process to prove, we can also get

${\Vert {\partial }_{x_1}{u}_1{\rho }_1-{\partial }_{x_2}{u}_2{\rho }_2\Vert }_{L^p\cap L^{\infty }}\le C\left({\Vert V_1-V_2\Vert }_{L^p\cap L^{\infty }}+{\Vert U_1-U_2\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}\right)\mathrm{.}$

Moreover,

$\begin{array}{l}{\Vert U_1-U_2\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}+{\Vert y_1-y_2\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}+{\Vert V_1-V_2\Vert }_{L^p\cap L^{\infty }}\\ \le {\Vert U_1\left(0\right)-U_2\left(0\right)\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}+{\Vert y_1\left(0\right)-y_2\left(0\right)\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}\\ +{\Vert V_1\left(0\right)-V_2\left(0\right)\Vert }_{L^p\cap L^{\infty }}+C{\displaystyle {\int }_0^T}{\Vert Q_1-Q_2\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}\\ +{\Vert y_1-y_2\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}+{\Vert {\partial }_{x_1}{u}_1{\rho }_1-{\partial }_{x_2}{u}_2{\rho }_2\Vert }_{L^p\cap L^{\infty }}\text{d}t\end{array}$

$\begin{array}{l}\le {\Vert U_1\left(0\right)-U_2\left(0\right)\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}+{\Vert y_1\left(0\right)-y_2\left(0\right)\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}\\ +{\Vert V_1\left(0\right)-V_2\left(0\right)\Vert }_{L^p\cap L^{\infty }}+C{\displaystyle {\int }_0^T}\parallel U_1-U_2{\parallel }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}\\ +{\Vert y_1-y_2\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}+{\Vert V_1-V_2\Vert }_{L^p\cap L^{\infty }}\text{d}t\mathrm{.}\end{array}$

Using the Growall inequality yields

$\begin{array}{l}{\Vert U_1-U_2\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}+{\Vert y_1-y_2\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}+{\Vert V_1-V_2\Vert }_{L^p\cap L^{\infty }}\\ \le {\text{e}}^{CT}\left({\Vert U_1\left(0\right)-U_2\left(0\right)\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}+{\Vert y_1\left(0\right)-y_2\left(0\right)\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}+{\Vert V_1\left(0\right)-V_2\left(0\right)\Vert }_{L^p\cap L^{\infty }}\right)\\ \le {\text{e}}^{CT}\left({\Vert U_1\left(0\right)-U_2\left(0\right)\Vert }_{W^{\mathrm{1,}\infty }\cap W^{\mathrm{1,}p}}+{\Vert V_1\left(0\right)-V_2\left(0\right)\Vert }_{L^p\cap L^{\infty }}\right)\\ \le {\text{e}}^{CT}\left({\Vert u_1\left(0\right)-u_2\left(0\right)\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}+{\Vert {\rho }_1\left(0\right)-{\rho }_2\left(0\right)\Vert }_{B_{p\mathrm{,1}}^{\frac{1}{p}}}^{}\right)\mathrm{,}\end{array}$

where owing to $y_1\left(0\right)=y_2\left(0\right)=\xi$ , it follows that

$\begin{array}{c}{\Vert u_1-u_2\Vert }_{L^{p-1}}\le C{\Vert u_1\circ y_1-u_2\circ y_1\Vert }_{L^{p-1}}\\ \le C{\Vert u_1\circ y_1-u_2\circ y_2+u_2\circ y_2-u_2\circ y_1\Vert }_{L^{p-1}}\\ \le C{\Vert U_1-U_2\Vert }_{L^{p-1}}+C{\Vert u_{2x}\Vert }_{L^{\infty }}{\Vert y_1-y_2\Vert }_{L^{p-1}}\\ \le C{\Vert u_1\left(0\right)-u_2\left(0\right)\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\mathrm{.}\end{array}$

Using similar estimate, we can obtain ${\Vert {\rho }_1-{\rho }_2\Vert }_{L^{p-1}}\le C{\Vert {\rho }_1\left(0\right)-{\rho }_2\left(0\right)\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\mathrm{.}$ Based on the embedding $L^{p-1}$ ↪ $B_{p-\mathrm{1,}\infty }^0$ , we can deduce

$\begin{array}{l}{\Vert u_1-u_2\Vert }_{B_{p-\mathrm{1,}\infty }^0}\le C{\Vert u_1-u_2\Vert }_{L^{p-1}}\le C{\Vert u_1\left(0\right)-u_2\left(0\right)\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\mathrm{,}\\ {\Vert {\rho }_1-{\rho }_2\Vert }_{B_{p-\mathrm{1,}\infty }^0}\le C{\Vert {\rho }_1-{\rho }_2\Vert }_{L^{p-1}}\le C{\Vert {\rho }_1\left(0\right)-{\rho }_2\left(0\right)\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\mathrm{.}\end{array}$

Therefore, if $u_1\left(0\right)=u_2\left(0\right)$ and ${\rho }_1\left(0\right)={\rho }_2\left(0\right)$ , the uniqueness of the solutions is deduced. By a similar argument as above to the component G, we can conclude that z is the unique solution of the Equation (9).

Step 3: The continuous dependence

Let $\left(u_n\mathrm{,}{\rho }_n\right)$ , $\left(u_{\infty }\mathrm{,}{\rho }_{\infty }\right)$ is the solution with the initial data $\left(u_{0n}\mathrm{,}{\rho }_{0n}\right)$ , $\left(u_{0\infty }\mathrm{,}{\rho }_{0\infty }\right)$ , and the initial data $\left(u_{0n}\mathrm{,}{\rho }_{0n}\right)$ ↪ $\left(u_{0\infty }\mathrm{,}{\rho }_{0\infty }\right)$ in $B_{p\mathrm{,1}}^{\frac{1}{p}}\times B_{p\mathrm{,1}}^{\frac{1}{p}}$ . Combining Step 1 and Step 2, we can obtain that $\left(u_n\mathrm{,}{\rho }_n\right)$ , $\left(u_{\infty }\mathrm{,}{\rho }_{\infty }\right)$ are uniformly bounded in $L_T^{\infty }\left(B_{p\mathrm{,1}}^{\frac{1}{p}}\right)\times L_T^{\infty }\left(B_{p\mathrm{,1}}^{\frac{1}{p}}\right)$ , and

$\begin{array}{l}{\Vert u_n-u_{\infty }\Vert }_{B_{p-\mathrm{1,}\infty }^0}\le C{\Vert u_{0n}-u_{0\infty }\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\mathrm{,}\\ {\Vert {\rho }_n-{\rho }_{\infty }\Vert }_{B_{p-\mathrm{1,}\infty }^0}\le C{\Vert {\rho }_{0n}-{\rho }_{0\infty }\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\mathrm{.}\end{array}$ (14)

This means that $\left(u_n\mathrm{,}{\rho }_n\right)$ tends to $\left(u_{\infty }\mathrm{,}{\rho }_{\infty }\right)$ in

$\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{,}{B}_{p-\mathrm{1,}\infty }^0\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{,}{B}_{p-\mathrm{1,}\infty }^0\right)$ . Under the theorem of interpolation, for any

$\epsilon >0$ , we can see that $\left(u_n\mathrm{,}{\rho }_n\right)\to \left(u_{\infty }\mathrm{,}{\rho }_{\infty }\right)$ in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{,}{B}_{p\mathrm{,1}}^{\frac{1}{p}-\epsilon }\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{,}{B}_{p\mathrm{,1}}^{\frac{1}{p}-\epsilon }\right)$ . If $\epsilon =1$ , we will have $\left(u_n\mathrm{,}{\rho }_n\right)\to \left(u_{\infty }\mathrm{,}{\rho }_{\infty }\right)$ in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{,}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{,}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)$ . Combining (14) and the above relationship, we just need to prove that $\left({\partial }_x{u}_n\mathrm{,}{\partial }_x{\rho }_n\right)\to \left({\partial }_x{u}_{\infty }\mathrm{,}{\partial }_x{\rho }_{\infty }\right)$ in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{,}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{,}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)$ . Set $v_{1n}={\partial }_x{u}_n$ ,

$v_{2n}={\partial }_x{\rho }_n$ , we split $v_{1n}=z_{1n}+w_{1n}$ and $v_{2n}=z_{2n}+w_{2n}$ with $\left(z_{1n}\mathrm{,}{w}_{1n}\right)$ , $\left(z_{2n}\mathrm{,}{w}_{2n}\right)$ satisfying the following equations

$\{\begin{array}{l}{\partial }_t{z}_{1n}+u_n{\partial }_x{z}_{1n}={\partial }_x{Q}_{\infty }-{\left({\partial }_x{u}_{\infty }\right)}^2,\\ {\partial }_t{z}_{2n}+u_n{\partial }_x{z}_{2n}=-{\partial }_x\left({\rho }_{\infty }{\partial }_x{u}_{\infty }\right)-{\partial }_x{u}_{\infty }{\partial }_x{\rho }_{\infty },\\ {z_{1n}|}_{t=0}={\partial }_x{u}_{0\infty },{z_{2n}|}_{t=0}={\partial }_x{\rho }_{0\infty },\end{array}$

and

$\{\begin{array}{l}{\partial }_t{w}_{1n}+u_n{\partial }_x{w}_{1n}={\partial }_x{Q}_n-{\partial }_x{Q}_{\infty }-{\left({\partial }_x{u}_n\right)}^2+{\left({\partial }_x{u}_{\infty }\right)}^2,\\ {\partial }_t{w}_{2n}+u_n{\partial }_x{w}_{2n}=-{\partial }_x\left({\rho }_n{\partial }_x{u}_n-{\rho }_{\infty }{\partial }_x{u}_{\infty }\right)-{\partial }_x{u}_n{\partial }_x{\rho }_n+{\partial }_x{u}_{\infty }{\partial }_x{\rho }_{\infty },\\ {w_{1n}|}_{t=0}={\partial }_x{u}_{0n}-{\partial }_x{u}_{0\infty },{z_{2n}|}_{t=0}={\partial }_x{\rho }_{0n}-{\partial }_x{\rho }_{0\infty }.\end{array}$

Taking advantage of the fact that $\left(u_n\mathrm{,}{\rho }_n\right)$ , $\left(u_{\infty }\mathrm{,}{\rho }_{\infty }\right)$ are uniformly bounded in $L_T^{\infty }\left(B_{p\mathrm{,1}}^{\frac{1}{p}}\right)\times L_T^{\infty }\left(B_{p\mathrm{,1}}^{\frac{1}{p}}\right)$ , we can have

$\begin{array}{l}{\Vert {\partial }_x{Q}_n-{\partial }_x{Q}_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}\\ \le {\Vert \varphi \ast \left(u_n{\rho }_n-u_{\infty }{\rho }_{\infty }\right)-\left(\varphi \ast {\rho }_n\right)u_n+\left(\varphi \ast {\rho }_{\infty }\right)u_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\\ \le C\left({\Vert {\rho }_n-{\rho }_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert u_n-u_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\right)\\ \le C({\Vert u_n-u_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert {\partial }_x{u}_n-{\partial }_x{u}_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert {\rho }_n-{\rho }_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}\\ +{\Vert {\partial }_x{\rho }_n-{\partial }_x{\rho }_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{})\mathrm{,}\end{array}$

and

$\begin{array}{l}{\Vert {\partial }_x\left({\rho }_n{\partial }_x{u}_n-{\rho }_{\infty }{\partial }_x{u}_{\infty }\right)\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}\\ \le C\left({\Vert {\rho }_n-{\rho }_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert {\partial }_x{u}_n-{\partial }_x{u}_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\right)\\ \le C({\Vert {\rho }_n-{\rho }_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert {\partial }_x{\rho }_n-{\partial }_x{\rho }_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert u_n-u_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}\\ +{\Vert {\partial }_x{u}_n-{\partial }_x{u}_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}).\end{array}$

Here, owing to $G={\partial }_{x}u+\varphi \ast \rho$ and $G\in C\left(\left[\mathrm{0,}T\right]\mathrm{,}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)$ , we can get ${\Vert {\partial }_{x}u\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\le C$ . In addition, applying Lemma 3.7, let $s_1=\frac{1}{p}-1$ and $s_2=\frac{1}{p}$ , we can get

$\begin{array}{l}{\Vert {\left({\partial }_x{u}_n\right)}^2-{\left({\partial }_x{u}_{\infty }\right)}^2\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}\\ \le C{\Vert {\partial }_x{u}_n-{\partial }_x{u}_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}{\Vert {\partial }_x{u}_n+{\partial }_x{u}_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\\ \le C{\Vert {\partial }_x{u}_n-{\partial }_x{u}_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}.\end{array}$

Then for all $n\in \mathbb{N}$ , using the above inequalities, we can have

$\begin{array}{l}{\partial }_t{\Vert w_{1n}\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\partial }_t{\Vert w_{2n}\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}\\ \le C\left({\Vert {\rho }_n-{\rho }_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert {\partial }_x{\rho }_n-{\partial }_x{\rho }_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert u_n-u_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert {\partial }_x{u}_n-{\partial }_x{u}_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}\right)\\ \le C({\Vert {\rho }_n-{\rho }_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert u_n-u_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert w_{1n}-w_{1\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}\\ +{\Vert z_{1n}-z_{1\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert w_{2n}-w_{2\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert z_{2n}-z_{2\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{})\\ \le C({\Vert {\rho }_n-{\rho }_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert u_n-u_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert w_{1n}\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert z_{1n}-z_{1\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}\\ +{\Vert w_{2n}\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert z_{2n}-z_{2\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}),\end{array}$

from which it follows

$\begin{array}{l}{\Vert w_{1n}\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert w_{2n}\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}\\ \le C{\text{e}}^{Ct}({\Vert v_{1n}\left(0\right)-v_{1\infty }\left(0\right)\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert v_{2n}\left(0\right)-v_{2\infty }\left(0\right)\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}\\ +{\displaystyle {\int }_0^t}{\text{e}}^{-C{t}^{\prime }}({\Vert {\rho }_n-{\rho }_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert u_n-u_{\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert w_{1n}\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}\\ +{\Vert z_{1n}-z_{1\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert w_{2n}\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{}+{\Vert z_{2n}-z_{2\infty }\Vert }_{B_{p,1}^{\frac{1}{p}-1}}^{})\text{d}{t}^{\prime })\mathrm{.}\end{array}$

Since we prove that $v_{1n}\left(0\right)\to v_{1\infty }\left(0\right)$ and $v_{2n}\left(0\right)\to v_{2\infty }\left(0\right)$ in $B_{p\mathrm{,1}}^{\frac{1}{p}-1}$ , $\left({\rho }_n\mathrm{,}{u}_n\right)\to \left({\rho }_{\infty }\mathrm{,}{u}_{\infty }\right)$ in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)$ , and according to Lemma 3.5, we can deduce that $z_{1n}\left(t\right)\to z_{1\infty }\left(t\right)$ in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)$ , $z_{2n}\left(t\right)\to z_{2\infty }\left(t\right)$ in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)$ . Furthermore, we obtain that $w_{1n}\to 0$ in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)$ and $w_{2n}\to 0$ in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)$ . Then according to Lemma 3.6 and $w_{1\infty }=w_{2\infty }=0$ , we can get that $w_{1n}\to w_{1\infty }$ in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)$ , and $w_{2n}\to w_{2\infty }$ in $\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)$ .

Finally, we can deduce

$\begin{array}{l}{\Vert v_{1n}-v_{1\infty }\Vert }_{L_T^{\infty }\left(B_{p,1}^{\frac{1}{p}-1}\right)}^{}+{\Vert v_{2n}-v_{2\infty }\Vert }_{L_T^{\infty }\left(B_{p,1}^{\frac{1}{p}-1}\right)}^{}\\ \le {\Vert z_{1n}-z_{1\infty }\Vert }_{L_T^{\infty }\left(B_{p,1}^{\frac{1}{p}-1}\right)}^{}+{\Vert w_{1n}-w_{1\infty }\Vert }_{L_T^{\infty }\left(B_{p,1}^{\frac{1}{p}-1}\right)}^{}+{\Vert z_{2n}-z_{2\infty }\Vert }_{L_T^{\infty }\left(B_{p,1}^{\frac{1}{p}-1}\right)}^{}+{\Vert w_{2n}-w_{2\infty }\Vert }_{L_T^{\infty }\left(B_{p,1}^{\frac{1}{p}-1}\right)}^{}\\ \le {\Vert z_{1n}-z_{1\infty }\Vert }_{L_T^{\infty }\left(B_{p,1}^{\frac{1}{p}-1}\right)}^{}+{\Vert w_{1n}\Vert }_{L_T^{\infty }\left(B_{p,1}^{\frac{1}{p}-1}\right)}^{}+{\Vert z_{2n}-z_{2\infty }\Vert }_{L_T^{\infty }\left(B_{p,1}^{\frac{1}{p}-1}\right)}^{}+{\Vert w_{2n}\Vert }_{L_T^{\infty }\left(B_{p,1}^{\frac{1}{p}-1}\right)}^{},\end{array}$

which indicates that $\left({\partial }_x{u}_n\mathrm{,}{\partial }_x{\rho }_n\right)\to \left({\partial }_x{u}_{\infty }\mathrm{,}{\partial }_x{\rho }_{\infty }\right)$ in

$\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{,}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)\times \mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{,}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)$ . By a similar argument as above to the component G, we can have ${\partial }_x{z}_n\to {\partial }_x{z}_{\infty }$ in ${\left(\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{,}{B}_{p\mathrm{,1}}^{\frac{1}{p}-1}\right)\right)}^3$ . Combining Step 1 to Step 3, we complete the proof of Theorem 2.1. □

5. Ill-Posedness

In this section, we shall explore that the solutions of the Equation (9) are

ill-posedness in Besov spaces ${\left(B_{p\mathrm{,}r}^s\right)}^3$ with the index $s>\max \left\{\frac{1}{2},\frac{1}{p}\right\}$ and

$r=\infty$ . To localize the frequency region, we have to introduce smooth radial cut-off functions. Set ${\hat{\psi }}_i\in {\mathcal{C}}_0^{\infty }$ , $i=1,2,3$ is an non-negative, even and real-valued function on $ℝ$ satisfying

${\hat{\psi }}_i\left(\xi \right)=(\begin{array}{cc}\mathrm{1,}& \text{if}\left|\xi \right|\le \frac{1}{4}\mathrm{,}\\ \mathrm{0,}& \text{if}\left|\xi \right|\ge \frac{1}{2}\mathrm{.}\end{array}$

Define the initial data

$\{\begin{array}{l}{u}_0\left(x\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{2}^{-ns}{\psi }_1\left(x\right)\cos \left({\lambda }_1{2}^{n}x\right),\\ {\rho }_0\left(x\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{2}^{-ns}{\psi }_2\left(x\right)\cos \left({\lambda }_2{2}^{n}x\right),\\ G_0\left(x\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{2}^{-ns}{\psi }_3\left(x\right)\cos \left({\lambda }_3{2}^{n}x\right),\end{array}$ (15)

there ${\lambda }_i\in \left[\frac{67}{48}\mathrm{,}\frac{69}{48}\right]$ and $i=1,2,3$ . Then we can have

${\Delta }_j\left({\psi }_i\left(x\right)\cos \left({\lambda }_i{2}^{n}x\right)\right)=(\begin{array}{ll}{\psi }_i\left(x\right)\cos \left({\lambda }_i{2}^{n}x\right)\mathrm{,}\hfill & \text{if}j=n\mathrm{,}\hfill \\ \mathrm{0,}\hfill & \text{if}j\ne n\mathrm{.}\hfill \end{array}$

So we can get

$\begin{array}{c}{\Vert u_0\Vert }_{B_{p,\infty }^s}={\Vert {\left(2^{js}{\Vert {\Delta }_j{u}_0\Vert }_{L^p}\right)}_{j\in \mathbb{Z}}\Vert }_{l^{\infty }}\\ ={\Vert {\left(2^{js}{\Vert {\Delta }_j{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{2}^{-ns}{\psi }_1\left(x\right)\cos \left({\lambda }_1{2}^{n}x\right)\Vert }_{L^p}\right)}_{j\in \mathbb{Z}}\Vert }_{l^{\infty }}\\ ={\Vert {\left(2^{js}{\Vert 2^{-js}{\psi }_1\left(x\right)\cos \left({\lambda }_1{2}^{j}x\right)\Vert }_{L^p}\right)}_{j\in \mathbb{Z}}\Vert }_{l^{\infty }}\\ \le {\Vert {\left({\Vert {\psi }_1\left(x\right)\cos \left({\lambda }_1{2}^{j}x\right)\Vert }_{L^p}\right)}_{j\in \mathbb{Z}}\Vert }_{l^{\infty }}\\ \le C.\end{array}$

At the same time, we can obtain

${\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}\le C\mathrm{,}{\Vert G_0\Vert }_{B_{p\mathrm{,}\infty }^s}\le C\mathrm{.}$

On the basis of definition of Besov spaces, we can make an estimate

${\Vert \varphi \ast u\Vert }_{B_{p,\infty }^s}=\underset{q\in \mathbb{Z}}{\sup }{2}^{qs}{\Vert {\Delta }_q\varphi \ast u\Vert }_{L^p}\le \underset{q\in \mathbb{Z}}{\sup }{2}^{qs}{\Vert \varphi \Vert }_{L^1}{\Vert {\Delta }_{q}u\Vert }_{L^p}\le C{\Vert u\Vert }_{B_{p,\infty }^s},$

where C is some positive constant.

Lemma 5.1. For the above constructed initial data $\left(u_0\mathrm{,}{\rho }_0\mathrm{,}{G}_0\right)$ , if some n large enough, we can obtain

$\begin{array}{l}{\Vert u_0{\Delta }_n{\partial }_x{u}_0\Vert }_{L^p}\ge C{2}^{-n\left(s-1\right)}\mathrm{,}\\ {\Vert u_0{\Delta }_n{\partial }_x{\rho }_0\Vert }_{L^p}\ge C{2}^{-n\left(s-1\right)}\mathrm{,}\\ {\Vert u_0{\Delta }_n{\partial }_x{G}_0\Vert }_{L^p}\ge C{2}^{-n\left(s-1\right)}\mathrm{.}\end{array}$ (16)

Proof. According to the definition of (15), we can get

$\begin{array}{c}{u}_0{\Delta }_n{\partial }_x{u}_0=u_0{2}^{-ns}{\partial }_x\left({\psi }_1\left(x\right)\cos \left({\lambda }_1{2}^{n}x\right)\right)\\ =u_0{2}^{-ns}\left[{\partial }_x{\psi }_1\left(x\right)\cos \left({\lambda }_1{2}^{n}x\right)-{\lambda }_1{2}^n{\psi }_1\left(x\right)\sin \left({\lambda }_1{2}^{n}x\right)\right]\\ =2^{-ns}{u}_0{\psi }_{1x}\cos \left({\lambda }_1{2}^{n}x\right)-{\lambda }_1{2}^{-n\left(s-1\right)}{u}_0{\psi }_1\left(x\right)\sin \left({\lambda }_1{2}^{n}x\right).\end{array}$

Because $u_0\left(x\right)$ is a real-valued continuous function on $ℝ$ , there exists ${\lambda }_1>0$ such that for any $x\in B_{{\lambda }_1}(\; 0\; )$

$\left|u_0\left(x\right)\right|\ge \frac{1}{2}{u}_0\left(0\right)=\frac{1}{2}{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{2}^{-ns}{\psi }_1\left(0\right)=\frac{2^{s-1}{\psi }_1\left(0\right)}{2^s-1},$ (17)

by applying (17), we can obtain

$\begin{array}{c}{\Vert u_0{\Delta }_n{\partial }_x{u}_0\Vert }_{L^p}={\Vert 2^{-ns}{u}_0{\psi }_{1x}\cos \left({\lambda }_1{2}^{n}x\right)-{\lambda }_1{2}^{-n\left(s-1\right)}{u}_0{\psi }_1\left(x\right)\sin \left({\lambda }_1{2}^{n}x\right)\Vert }_{L^p}\\ \ge C{\Vert 2^{-n\left(s-1\right)}{u}_0{\psi }_1\left(x\right)\sin \left({\lambda }_1{2}^{n}x\right)\Vert }_{L^p}-{\Vert 2^{-ns}{u}_0{\psi }_{1x}\cos \left({\lambda }_1{2}^{n}x\right)\Vert }_{L^p}\\ \ge \left(C{2}^n-C_1\right)2^{-ns}\mathrm{,}\end{array}$

taking a large enough $n$ such that $C_1\le C{2}^{n-1}$ , then we obtain (16). □

Lemma 5.2. For the above constructed initial data $\left(u_0\mathrm{,}{\rho }_0\mathrm{,}{G}_0\right)\in {\left(B_{p\mathrm{,}r}^s\right)}^3$ , there exists $T\left({\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^s}\mathrm{,}{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}\mathrm{,}{\Vert G_0\Vert }_{B_{p\mathrm{,}\infty }^s}\right)$ for $0\le t\le T$ , we can obtain

${\Vert u\left(t\right)-u_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\le Ct\mathrm{,}{\Vert \rho \left(t\right)-{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\le Ct\mathrm{,}{\Vert G\left(t\right)-G_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\le Ct\mathrm{.}$ (18)

Proof. Owing to $\left(u_0\mathrm{,}{\rho }_0\mathrm{,}{G}_0\right)\in {\left(B_{p\mathrm{,}\infty }^s\right)}^3$ , the system (9) has a unique solution $\left(u\mathrm{,}\rho \mathrm{,}G\right)\in L^{\infty }\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{\left(B_{p\mathrm{,}\infty }^s\right)}^3\right)$ , and

$\begin{array}{l}\underset{0\le t\le T}{\sup }\left({\Vert u\left(t\right)\Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert \rho \left(t\right)\Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert G\left(t\right)\Vert }_{B_{p\mathrm{,}\infty }^s}\right)\\ \le C\left({\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert G_0\Vert }_{B_{p\mathrm{,}\infty }^s}\right)\mathrm{.}\end{array}$ (19)

Applying the differential mean value theorem, Lemma 3.7 and (19) $t\in \left[\mathrm{0,}T\right]$ , we can obtain

$\begin{array}{c}{\Vert u-u_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\le {\displaystyle {\int }_0^t}{\Vert {\partial }_{\tau }u\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\text{d}\tau \\ \le {\displaystyle {\int }_0^t}{\Vert u{\partial }_{x}u\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}+{\Vert \varphi \ast \left(u\rho \right)-\left(\varphi \ast \rho \right)u\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\text{d}\tau \\ \le C{\displaystyle {\int }_0^t}{\Vert u\Vert }_{B_{p\mathrm{,}\infty }^s}{\Vert u_x\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}+{\Vert u\rho \Vert }_{B_{p\mathrm{,}\infty }^{s-1}}+{\Vert \varphi \ast \rho \Vert }_{B_{p\mathrm{,}\infty }^s}{\Vert u\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\text{d}\tau \\ \le C{\displaystyle {\int }_0^t}{\Vert u\Vert }_{B_{p\mathrm{,}\infty }^s}^2+{\Vert u\Vert }_{B_{p\mathrm{,}\infty }^s}{\Vert \rho \Vert }_{B_{p\mathrm{,}\infty }^{s-1}}+{\Vert \rho \Vert }_{B_{p\mathrm{,}\infty }^s}{\Vert u\Vert }_{B_{p\mathrm{,}\infty }^s}\text{d}\tau \\ \le Ct\mathrm{,}\end{array}$

and

$\begin{array}{c}{\Vert \rho -{\rho }_0\Vert }_{B_{p,\infty }^{s-1}}\le {\displaystyle {\int }_0^t}{\Vert {\partial }_{\tau }\rho \Vert }_{B_{p,\infty }^{s-1}}\text{d}\tau \le {\displaystyle {\int }_0^t}{\Vert -{\partial }_{x}u\rho \Vert }_{B_{p,\infty }^{s-1}}+{\Vert -u{\partial }_x\rho \Vert }_{B_{p,\infty }^{s-1}}\text{d}\tau \\ \le {\displaystyle {\int }_0^t}{\Vert u\Vert }_{B_{p,\infty }^s}{\Vert {\partial }_x\rho \Vert }_{B_{p,\infty }^{s-1}}+{\Vert {\partial }_{x}u\Vert }_{B_{p,\infty }^{s-1}}{\Vert \rho \Vert }_{B_{p,\infty }^s}\text{d}\tau \\ \le C{\displaystyle {\int }_0^t}{\Vert u\Vert }_{B_{p,\infty }^s}{\Vert \rho \Vert }_{B_{p,\infty }^s}+{\Vert u\Vert }_{B_{p,\infty }^s}{\Vert \rho \Vert }_{B_{p,\infty }^s}\text{d}\tau \\ \le Ct.\end{array}$

By a similar argument as above to the component G, ${\Vert G\left(t\right)-G_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\le Ct$ . Therefore, we complete the proof of Lemma 5.2. □

Lemma 5.3. According to the assumption of Theorem 2.2 for the above-constructed initial data $\left(u_0\mathrm{,}{\rho }_0\mathrm{,}{G}_0\right)$ , $0\le t\le T$ , there $T\left({\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^s}\mathrm{,}{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}\mathrm{,}{\Vert G_0\Vert }_{B_{p\mathrm{,}\infty }^s}\right)$ , we can obtain

$\begin{array}{l}{\Vert u\left(t\right)-u_0-t{w}_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\le C\left(t^2+t\right)\mathrm{,}\\ {\Vert \rho \left(t\right)-{\rho }_0-t{v}_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\le C\left(t^2+t\right)\mathrm{,}\\ {\Vert G\left(t\right)-G_0-t{m}_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\le C\left(t^2+t\right)\mathrm{,}\end{array}$ (20)

there $w_0=-u_0{\partial }_x{u}_0+\varphi \ast \left(u_0{\rho }_0\right)-\left(\varphi \ast {\rho }_0\right)u_0$ , $v_0=-u_0{\partial }_x{\rho }_0-{\partial }_x{u}_0{\rho }_0$ , and $m_0=-u_0{\partial }_x{G}_0-{\partial }_x{u}_0{G}_0$ .

Proof. Define

$\tilde{u}=u\left(t\right)-u_0-t{w}_0,\tilde{\rho }=\rho \left(t\right)-{\rho }_0-t{v}_0,\tilde{G}=G\left(t\right)-G_0-t{m}_0.$

Applying the differential mean value theorem, $t\in \left[\mathrm{0,}T\right]$ , we can obtain

$\begin{array}{c}{\Vert \tilde{\rho }\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\le {\displaystyle {\int }_0^t}{\Vert {\partial }_{\tau }\rho -v_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\text{d}\tau \\ \le {\displaystyle {\int }_0^t}{\Vert u{\partial }_x\rho -u_0{\partial }_x{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\text{d}\tau +{\displaystyle {\int }_0^t}{\Vert \rho {\partial }_{x}u-{\rho }_0{\partial }_x{u}_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\text{d}\tau ,\end{array}$ (21)

and

$\begin{array}{c}{\Vert \tilde{u}\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\le {\displaystyle {\int }_0^t}{\Vert {\partial }_{\tau }u-w_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\text{d}\tau \\ \le {\displaystyle {\int }_0^t}{\Vert u{\partial }_{x}u-u_0{\partial }_x{u}_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\text{d}\tau +{\displaystyle {\int }_0^t}{\Vert \varphi \ast \left(u\rho \right)-\varphi \ast \left(u_0{\rho }_0\right)\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\\ +{\Vert \left(\varphi \ast \rho \right)u-\left(\varphi \ast {\rho }_0\right)u_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\text{d}\tau .\end{array}$ (22)

Using proposition 3.4, Lemma 3.7, and (19), we can deduce

$\begin{array}{l}{\Vert u{\partial }_x\rho -u_0{\partial }_x{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\\ \le {\Vert \left(u-u_0\right){\partial }_x\rho +u_0{\partial }_x\left(\rho -{\rho }_0\right)\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\\ \le {\Vert u-u_0\Vert }_{B_{p\mathrm{,}\infty }^s}{\Vert {\partial }_x\rho \Vert }_{B_{p\mathrm{,}\infty }^{s-1}}+{\Vert {\partial }_x\left(\rho -{\rho }_0\right)\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}{\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^s}\\ \le {\Vert u-u_0\Vert }_{B_{p\mathrm{,}\infty }^s}{\Vert \rho \Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert \rho -{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}{\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^s}\\ \le C\mathrm{.}\end{array}$ (23)

Taking advantage of proposition 3.4, Lemma 3.7, Lemma 5.2, and (19), we can obtain

$\begin{array}{l}{\Vert \rho {\partial }_{x}u-{\rho }_0{\partial }_x{u}_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\\ \le {\Vert \left(\rho -{\rho }_0\right){\partial }_{x}u+{\rho }_0{\partial }_x\left(u-u_0\right)\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\\ \le {\Vert \rho -{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}{\Vert {\partial }_{x}u\Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert {\partial }_x\left(u-u_0\right)\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}\\ \le C{\Vert \rho -{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}+{\Vert u-u_0\Vert }_{B_{p\mathrm{,}\infty }^s}{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}\\ \le C\left(\tau +1\right)\mathrm{,}\end{array}$ (24)

and

$\begin{array}{l}{\Vert u{\partial }_{x}u-u_0{\partial }_x{u}_0\Vert }_{B_{p,\infty }^{s-2}}\\ \le {\Vert \left(u-u_0\right){\partial }_{x}u+u_0{\partial }_x\left(u-u_0\right)\Vert }_{B_{p,\infty }^{s-1}}\\ \le {\Vert u-u_0\Vert }_{B_{p,\infty }^{s-1}}{\Vert {\partial }_{x}u\Vert }_{B_{p,\infty }^s}+{\Vert u-u_0\Vert }_{B_{p,\infty }^s}{\Vert u_0\Vert }_{B_{p,\infty }^s}\\ \le C\left(\tau +1\right).\end{array}$ (25)

Applying proposition 3.4, Lemma 3.7, and (19), we can get

$\begin{array}{l}{\Vert \varphi \ast \left(u\rho -u_0{\rho }_0\right)\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\\ \le C{\Vert u\rho -u_0{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\\ \le C{\Vert \left(u-u_0\right)\rho +\left(\rho -{\rho }_0\right)u_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\\ \le C{\Vert u-u_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}{\Vert \rho \Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert \rho -{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}{\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^s}\\ \le C\tau \mathrm{,}\end{array}$ (26)

and

$\begin{array}{l}{\Vert \left(\varphi \ast \rho \right)u-\left(\varphi \ast {\rho }_0\right)u_0\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\\ \le {\Vert \left(\varphi \ast \rho \right)u-\left(\varphi \ast \rho \right)u_0+\left(\varphi \ast \rho \right)u_0-\left(\varphi \ast {\rho }_0\right)u_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\\ \le {\Vert u-u_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}{\Vert \varphi \ast \rho \Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert \varphi \ast \left(\rho -{\rho }_0\right)\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}{\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^s}\\ \le C\tau \mathrm{.}\end{array}$ (27)

Taking (23)-(24) into (21) and (25)-(27) into (22), we can deduce

$\begin{array}{l}{\Vert \tilde{\rho }\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\le C{\displaystyle {\int }_0^t}\left(\tau +1\right)\text{d}\tau \le C\left(t^2+t\right)\mathrm{,}\\ {\Vert \tilde{u}\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\le C{\displaystyle {\int }_0^t}\left(\tau +1\right)\text{d}\tau \le C\left(t^2+t\right)\mathrm{,}\end{array}$

similarly,

${\Vert \tilde{G}\Vert }_{B_{p\mathrm{,}\infty }^{s-2}}\le C{\displaystyle {\int }_0^t}\left(\tau +1\right)\text{d}\tau \le C\left(t^2+t\right)\mathrm{.}$

Thus, we complete the proof of Lemma 5.3. Next, we prove Theorem 8. □

Proof. On the basis of the definition of the Besov norm, we can obtain

$\begin{array}{l}{\Vert \rho -{\rho }_0\Vert }_{B_{p,\infty }^s}\\ \ge 2^{ns}{\Vert {\Delta }_n\left(\rho -{\rho }_0\right)\Vert }_{L^p}=2^{ns}{\Vert {\Delta }_n\left(\tilde{\rho }+t{v}_0\right)\Vert }_{L^p}\\ \ge t{2}^{ns}{\Vert {\Delta }_n{v}_0\Vert }_{L^p}-2^{ns}{\Vert {\Delta }_n\tilde{\rho }\Vert }_{L^p}\\ \ge t{2}^{ns}{\Vert {\Delta }_n\left(u_0{\partial }_x{\rho }_0\right)\Vert }_{L^p}-Ct{2}^{ns}{\Vert {\Delta }_n\left({\partial }_x{u}_0{\rho }_0\right)\Vert }_{L^p}-2^{ns}{\Vert {\Delta }_n\tilde{\rho }\Vert }_{L^p}\\ \ge t{2}^{ns}{\Vert {\Delta }_n\left(u_0{\partial }_x{\rho }_0\right)\Vert }_{L^p}-Ct{2}^n{\Vert {\partial }_x{\rho }_0{u}_0\Vert }_{B_{p,\infty }^{s-1}}-2^{2n}{\Vert \tilde{\rho }\Vert }_{B_{p,\infty }^{s-2}}.\end{array}$ (28)

Making the following analysis

${\Vert {\partial }_x{\rho }_0{u}_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}\le {\Vert {\partial }_x{\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^{s-1}}{\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^s}\le {\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^s}{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}\le C\mathrm{,}$

and

$\begin{array}{c}{\Delta }_n\left(u_0{\partial }_x{\rho }_0\right)={\Delta }_n\left(u_0{\partial }_x{\rho }_0\right)-u_0{\Delta }_n\left({\partial }_x{\rho }_0\right)+u_0{\Delta }_n\left({\partial }_x{\rho }_0\right)\\ =\left[{\Delta }_n,u_0\right]{\partial }_x{\rho }_0+u_0{\Delta }_n\left({\partial }_x{\rho }_0\right).\end{array}$

Applying Lemma 3.8, we can obtain

${\Vert 2^{ns}{\Vert \left[{\Delta }_n\mathrm{,}{u}_0\right]{\partial }_x{\rho }_0\Vert }_{L^p}\Vert }_{l^{\infty }}\le C\left({\Vert {\partial }_x{u}_0\Vert }_{L^{\infty }}{\Vert {\rho }_0\Vert }_{B_{p\mathrm{,}\infty }^s}+{\Vert {\partial }_x{\rho }_0\Vert }_{L^{\infty }}{\Vert u_0\Vert }_{B_{p\mathrm{,}\infty }^s}\right)\le C\mathrm{.}$

Then using above inequalities into (28), we can get

$\begin{array}{c}{\Vert \rho -{\rho }_0\Vert }_{B_{p,\infty }^s}\ge t{2}^{ns}{\Vert u_0{\Delta }_n\left({\partial }_x{\rho }_0\right)\Vert }_{L^p}-C_{1}t{2}^n-2^{2n}{\Vert \tilde{\rho }\Vert }_{B_{p,\infty }^{s-2}}\\ \ge C_{2}t{2}^{ns}{2}^{-n\left(s-1\right)}-C_{1}t{2}^n-C_3\left(t^2+t\right)2^{2n}\\ =C_{2}t{2}^n-C_{1}t{2}^n-C_3\left(t^2+t\right)2^{2n},\\ \ge C_{2}t{2}^n-C_{1}t{2}^n-C_3{t}^2{2}^{2n}.\end{array}$

Here, we take that $C_3$ is a enough large number. If taking that $C_1$ and $C_2$ satisfy $C_2>2C_1$ , then we can have

$\begin{array}{l}{\Vert \rho -{\rho }_0\Vert }_{B_{p,\infty }^s}\ge \frac{C_{2}t{2}^n}{2}-C_3{t}^2{2}^{2n}.\hfill \end{array}$

When n is a fixed number, let $t\to 0$ , choosing $t{2}^n\approx \delta <\frac{C_2}{4C_3}$ yields

$\begin{array}{l}{\Vert \rho -{\rho }_0\Vert }_{B_{p,\infty }^s}\ge \frac{C_2\delta }{2}-C_3{\delta }^2=\frac{C_2\delta }{4},\hfill \end{array}$

similarly, we can deduce

$\begin{array}{l}{\Vert u-u_0\Vert }_{B_{p,\infty }^s}\ge \frac{C_4\delta }{4},\\ {\Vert G-G_0\Vert }_{B_{p,\infty }^s}\ge \frac{C_5\delta }{4}.\end{array}$

This completes the proof of Theorem 2.2. □

6. Blow-Up Criteria

In this section, we will present blow-up criteria for Equation (9). We first need to establish support from the following Lemma.

Lemma 6.1. Suppose that $\left(u\mathrm{,}\rho \mathrm{,}G\right)\in {\left(\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)\right)}^3$ is the solution of the Cauchy problem (9) with the initial data $\left({\rho }_0\mathrm{,}{u}_0\mathrm{,}{G}_0\right)\in {\left(B_{p\mathrm{,1}}^{\frac{1}{p}}\right)}^3$ . Then the component $\rho$ satisfies the following expression

$\rho \left(t\mathrm{,}x\right)={\rho }_0\left({\zeta }^{-1}\left(t\mathrm{,}x\right)\right){\text{e}}^{-{\displaystyle {\int }_0^t{u}_x\left(s\mathrm{,}x\right)\text{d}s}}\mathrm{,}t\in \left[\mathrm{0,}T\right)\mathrm{.}$

Proof. Considering the following initial value problem

$\{\begin{array}{ll}{\zeta }_t=u\left(t,\zeta \left(t,x\right)\right),\hfill & t\in \left[0,T\right),x\in ℝ,\hfill \\ \zeta \left(0,x\right)=x,\hfill & t=0,x\in ℝ.\hfill \end{array}$

Set $T>0$ is the maximal existence time. Owing to ${\zeta }_{tx}=u_{\zeta }\left(t\mathrm{,}\zeta \left(t,x\right)\right){\zeta }_x\left(t,x\right)$ and ${\zeta }_x\left(\mathrm{0,}x\right)=1$ , this implies

${\zeta }_x\left(t\mathrm{,}x\right)=\exp \left({\displaystyle {\int }_0^t}{u}_{\zeta }\left(s\mathrm{,}\zeta \left(s\mathrm{,}x\right)\right)\text{d}s\right)\mathrm{.}$

According to the second formula for (9), we can get

$\begin{array}{l}\frac{\text{d}}{\text{d}t}\left(\rho \left(t,\zeta \left(t,x\right)\right){\zeta }_x\right)\\ ={\rho }_t\left(t,\zeta \left(t,x\right)\right){\zeta }_x+{\rho }_{\zeta }\left(t,\zeta \left(t,x\right)\right){\zeta }_t{\zeta }_x+\rho \left(t,\zeta \left(t,x\right)\right){\zeta }_{xt}\\ ={\zeta }_x\left({\rho }_t\left(t,\zeta \left(t,x\right)\right)+{\rho }_{\zeta }\left(t,\zeta \left(t,x\right)\right)u\left(t,\zeta \left(t,x\right)\right)+\rho \left(t,\zeta \left(t,x\right)\right)u_{\zeta }\left(t,\zeta \left(t,x\right)\right)\right)\\ =0,\end{array}$

thus, $\rho \left(t\mathrm{,}\zeta \left(t\mathrm{,}x\right)\right){\zeta }_x$ is independent on t, we can obtain

$\rho \left(t\mathrm{,}\zeta \left(t\mathrm{,}x\right)\right){\zeta }_x={\rho }_0\left(x\right)\mathrm{.}$

Consequently, we prove the Lemma 6.1. □

Theorem 6.2. Set the initial value $\left(u_0\mathrm{,}{\rho }_0\mathrm{,}{G}_0\right)\in {\left(\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)\right)}^3$ and $\varphi \in L^1$ .

Suppose that the maximal existence time of the Cauchy problem (9) solutions is $T>0$ . If $M>0$ satisfies that

${\Vert u_x\Vert }_{L^{\infty }}\le M\mathrm{,}t\in \left[\mathrm{0,}T\right)\mathrm{,}$

then the solution $z\left(x\mathrm{,}t\right)$ of the Cauchy problem (9) does not blow up in $\left[\mathrm{0,}T\right)$ .

Proof. According to Lemma 6.1, we can establish the following estimate

$\begin{array}{c}{\Vert \rho \left(t,x\right)\Vert }_{L^{\infty }}={\Vert {\rho }_0\left({\zeta }^{-1}\left(t,x\right)\right){\text{e}}^{-{\displaystyle {\int }_0^t{u}_x\left(s,x\right)\text{d}s}}\Vert }_{L^{\infty }}\\ \le {\Vert {\rho }_0\left({\zeta }^{-1}\left(t,x\right)\right)\Vert }_{L^{\infty }}{\text{e}}^{{\displaystyle {\int }_0^t}{\Vert u_x\left(s,x\right)\Vert }_{L^{\infty }}\text{d}s}\\ \le {\Vert {\rho }_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}{\text{e}}^{{\displaystyle {\int }_0^t}{\Vert u_x\left(s,x\right)\Vert }_{L^{\infty }}\text{d}s},\end{array}$

this is because the embedding $B_{p\mathrm{,1}}^{\frac{1}{p}}$ ↪ $W^{\mathrm{1,}p}\cap W^{\mathrm{1,}\infty }$ . If $M>0$ satisfies that ${\Vert u_x\Vert }_{L^{\infty }}\le {\Vert u_x\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\le M$ , $t\in \left[\mathrm{0,}T\right)$ , then we can have

${\Vert \rho \left(t\mathrm{,}x\right)\Vert }_{L^{\infty }}\le {\Vert {\rho }_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}{\text{e}}^{MT}\mathrm{,}$ (29)

similarly,

${\Vert G\left(t\mathrm{,}x\right)\Vert }_{L^{\infty }}\le {\Vert G_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}{\text{e}}^{MT}\mathrm{.}$ (30)

According to the result of local well-posedness, applying ${\Delta }_j$ to (9) yields

$\{\begin{array}{l}{\Delta }_j{u}_t+u{\Delta }_j{u}_x={\Delta }_j\left[\varphi \ast \left(u\rho \right)-\left(\varphi \ast \rho \right)u\right]+\left[u{\partial }_x,{\Delta }_j\right]u,\hfill \\ {\Delta }_j{\rho }_t+u{\Delta }_j{\rho }_x=-{\Delta }_j\left(u_x\rho \right)+\left[u{\partial }_x,{\Delta }_j\right]\rho ,\hfill \\ {\Delta }_j{G}_t+u{\Delta }_j{G}_x=-{\Delta }_j\left(u_{x}G\right)+\left[u{\partial }_x,{\Delta }_j\right]G,\hfill \\ {\Delta }_j\rho \left(x,0\right)={\Delta }_j{\rho }_0\left(x\right),{\Delta }_{j}u\left(x,0\right)={\Delta }_j{u}_0\left(x\right).\hfill \end{array}$

Multiplying both sides of the above equation by $sgn\left({\Delta }_{j}u\right){\left|{\Delta }_{j}u\right|}^{p-1}$ , $sgn\left({\Delta }_j\rho \right){\left|{\Delta }_j\rho \right|}^{p-1}$ , and $sgn\left({\Delta }_{j}G\right){\left|{\Delta }_{j}G\right|}^{p-1}$ and integrating over $ℝ$ respectively, we can obtain

$\begin{array}{c}{\Vert {\Delta }_{j}u\Vert }_{L^p}\le {\Vert {\Delta }_j{u}_0\Vert }_{L^p}+{\displaystyle {\int }_0^t}({\Vert {\Delta }_j\left[\varphi \ast \left(u\rho \right)-\left(\varphi \ast \rho \right)u\right]\Vert }_{L^p}\\ +{\Vert \left[u{\partial }_x\mathrm{,}{\Delta }_j\right]u\Vert }_{L^p}+{\Vert u_x\Vert }_{L^{\infty }}{\Vert {\Delta }_{j}u\Vert }_{L^p})\text{d}{t}^{\prime }\mathrm{,}\end{array}$

and

${\Vert {\Delta }_j\rho \Vert }_{L^p}\le {\Vert {\Delta }_j{\rho }_0\Vert }_{L^p}+{\displaystyle {\int }_0^t}\left({\Vert {\Delta }_j\left(u_x\rho \right)\Vert }_{L^p}+{\Vert \left[u{\partial }_x\mathrm{,}{\Delta }_j\right]\rho \Vert }_{L^p}+{\Vert u_x\Vert }_{L^{\infty }}{\Vert {\Delta }_j\rho \Vert }_{L^p}\right)\text{d}{t}^{\prime }\mathrm{,}$

multiplying both sides of the above inequality by $2^{\frac{j}{p}}$ and taking $l^1$ -norm respectively, we can get

$\begin{array}{c}{\Vert u\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\le {\Vert u_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\displaystyle {\int }_0^t}({\Vert \varphi \ast \left(u\rho \right)-\left(\varphi \ast \rho \right)u\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\\ +{\Vert 2^{\frac{j}{p}}{\Vert \left[u{\partial }_x\mathrm{,}{\Delta }_j\right]u\Vert }_{L^p}\Vert }_{l^1}+{\Vert u_x\Vert }_{L^{\infty }}{\Vert u\Vert }_{B_{p,1}^{\frac{1}{p}}}^{})\text{d}{t}^{\prime },\end{array}$

and

${\Vert \rho \Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\le {\Vert {\rho }_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\displaystyle {\int }_0^t}\left({\Vert u_x\rho \Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert 2^{\frac{j}{p}}{\Vert \left[u{\partial }_x\mathrm{,}{\Delta }_j\right]\rho \Vert }_{L^p}\Vert }_{l^1}+{\Vert u_x\Vert }_{L^{\infty }}{\Vert \rho \Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\right)\text{d}{t}^{\prime }\mathrm{.}$

On the basis of Lemma 3.8, we can deduce

${\Vert 2^{\frac{j}{p}}{\Vert \left[u{\partial }_x\mathrm{,}{\Delta }_j\right]u\Vert }_{L^p}\Vert }_{l^1}\le C{\Vert u_x\Vert }_{L^{\infty }}{\Vert u\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\mathrm{,}$

and

${\Vert 2^{\frac{j}{p}}{\Vert \left[u{\partial }_x\mathrm{,}{\Delta }_j\right]\rho \Vert }_{L^p}\Vert }_{l^1}\le C{\Vert u_x\Vert }_{L^{\infty }}{\Vert \rho \Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\mathrm{,}$

by the estimate (1) and Lemma 3.4, we can obtain

${\Vert u\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\le {\Vert u_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\displaystyle {\int }_0^t}C\left({\Vert u\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}{\Vert \rho \Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert u_x\Vert }_{L^{\infty }}{\Vert u\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\right)\text{d}{t}^{\prime }\mathrm{,}$

and

${\Vert \rho \Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\le {\Vert {\rho }_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\displaystyle {\int }_0^t}C\left({\Vert u_x\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}{\Vert \rho \Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert u_x\Vert }_{L^{\infty }}{\Vert \rho \Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\right)\text{d}{t}^{\prime }.$

Similarly, by the same deduction, we can deduce

${\Vert G\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\le {\Vert G_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\displaystyle {\int }_0^t}C\left({\Vert u_x\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}{\Vert G\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert u_x\Vert }_{L^{\infty }}{\Vert G\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\right)\text{d}{t}^{\prime }\mathrm{,}$

from the above inequalities, we can get

$\begin{array}{c}{\Vert u\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert \rho \Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert G\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\le \left({\Vert u_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert {\rho }_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert G_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\right)\\ +C{\displaystyle {\int }_0^t}{\Vert u_x\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\left({\Vert u\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert \rho \Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert G\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\right)\text{d}{t}^{\prime }\mathrm{.}\end{array}$

Applying Gronwall’s inequality, we can obtain

$\begin{array}{l}{\Vert u\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert \rho \Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert G\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\\ \le \left({\Vert u_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert {\rho }_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert G_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\right)\exp \left(C{\displaystyle {\int }_0^t}{\Vert u_x\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\text{d}{t}^{\prime }\right)\\ \le \left({\Vert u_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert {\rho }_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}+{\Vert G_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}\right)\exp \left(CMt\right)\mathrm{,}\end{array}$

so we complete the proof. □

Then we shall establish the proof of Theorem 2.3.

Proof. Set $\left(u\mathrm{,}\rho \mathrm{,}G\right)\in {\left(\mathcal{C}\left(\left[\mathrm{0,}T\right]\mathrm{<mo>\; ;\; </mo>}{B}_{p\mathrm{,1}}^{\frac{1}{p}}\right)\right)}^3$ is the solution of the Cauchy problem (9) with the initial data $\left(u_0\mathrm{,}{\rho }_0\mathrm{,}{G}_0\right)\in {\left(B_{p\mathrm{,1}}^{\frac{1}{p}}\right)}^3$ , and the maximal existence time of

the Cauchy problem (9) solutions is $T>0$ . We suppose that the solution of Equation (9) blows up in finite time T, and there exists a constant $M>0$ such that

$u_x\left(t\mathrm{,}x\right)\ge -M\mathrm{,}\forall \left(t\mathrm{,}x\right)\in \left[\mathrm{0,}T\right)\times ℝ\mathrm{.}$ (31)

Applying to (29) and (30), then we can get

$\begin{array}{c}{\Vert u_x\left(t,x\right)\Vert }_{L^{\infty }}\le {\Vert \varphi \Vert }_{L^1}{\Vert \rho \Vert }_{L^{\infty }}+{\Vert G\Vert }_{L^{\infty }}\\ \le C{\Vert \rho \Vert }_{L^{\infty }}+{\Vert G\Vert }_{L^{\infty }}\\ \le C{\Vert {\rho }_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}{\text{e}}^{MT}+{\Vert G_0\Vert }_{B_{p,1}^{\frac{1}{p}}}^{}{\text{e}}^{MT}.\end{array}$ (32)

By (31) and (32), we can obtain a contradiction with Theorem 6.2. On the basis of Theorem 6.2 and Sobolev embedding theorem, if

$\underset{t\to T^{-}}{\lim }\underset{x\in ℝ}{\inf }{u}_x\left(t,x\right)=-\infty ,$

then the solution will blow up in a finite time.

This completes the proof of the Theorem 2.3. □