Existence results for compressible radiation hydrodynamics equations with vacuum

In this paper, we consider the 3-D compressible isentropic radiation hydrodynamics (RHD) equations. The local existence of strong solutions with vacuum is firstly established when the initial data is arbitrarily large, contains vacuum and satisfy some initial layer compatibility condition. The initial mass density needs not be bounded away from zero, it may vanish in some open set or decay at infinity. We also prove that if the initial vacuum is not so irregular, then the compatibility condition of the initial data is necessary and sufficient to guarantee the existence of a unique strong solution. Finally, we prove a blow-up criterion for the local strong solution. The similar result also holds for the general barotropic flow with pressure law $p_m=p_m(\rho)\in C^1(\mathbb{\overline{R}}^+)$.


Introduction
The system of radiation hydrodynamic equations appears in high-temperature plasma physics [14] and in various astrophysical contexts [15]. The couplings between fluid field and radiation field involve momentum source and energy source depending on the specific radiation intensity driven by the so-called radiation transfer equation [19]. Suppose that the matter is in local thermodynamical equilibrium (LTE), the coupled system of Navier-Stokes-Boltzmann (RHD) equations for the mass density ρ(t, x), the velocity u(t, x) = (u (1) , u (2) , u (3) ) of the fluid and the specific radiation intensity I(v, Ω, t, x) in three-dimensional space reads as [19]                1 c I t + Ω · ∇I = A r , ρ t + div(ρu) = 0, where t ≥ 0 and x ∈ R 3 are the time and space variables, respectively. p m is the material pressure satisfying the equation of state: where A > 0 and γ > 1 are both constants, γ is the adiabatic exponent. T is the viscosity stress tensor given by T = µ(∇u + (∇u) ⊤ ) + λdivu I 3 , where I 3 is the 3 × 3 unit matrix, µ is the shear viscosity coefficient, λ + 2 3 µ is the bulk viscosity coefficient, µ and λ are both real constants satisfying which ensure the ellipticity of the Lamé operator defined by (1. 5) v and Ω are radiation variables. v ∈ R + is the frequency of photon, and Ω ∈ S 2 is the travel direction of photon. The radiation flux F r and the radiation pressure tensor P r are defined by where S 2 is the unit sphere in R 3 . The collision term on the right-hand side of the radiation transfer equation is where I = I(v, Ω, t, x), I ′ = I(v ′ , Ω ′ , t, x); S = S(v, Ω, t, x) ≥ 0 is the rate of energy emission due to spontaneous process; σ a = σ a (v, Ω, t, x, ρ) ≥ 0 denotes the absorption coefficient that may also depend on the mass density ρ; σ s is the "differential scattering coefficient" such that the probability of a photon being scattered from v ′ to v contained in dv, from Ω ′ to Ω contained in dΩ, and travelling a distance ds is given by σ s (v ′ → v, Ω ′ · Ω)dvdΩds, and . When there is no radiation effect, the local existence of strong solutions with vacuum has been solved by many authors, we refer the reader to [3][5] [6]. Huang-Li-Xin [12] obtained the well-posedness of classical solutions with large oscillations and vacuum for Cauchy problem [12] to the isentropic flow.
In general, the study of radiation hydrodynamics equations is challenging due to the high complexity and mathematical difficulty of the equations themselves. For the Euler-Boltzmann equations of the inviscid compressible radiation fluid, Jiang-Zhong [14] obtained the local existence of C 1 solutions for the Cauchy problem away from vacuum. Jiang-Wang [13] showed that some C 1 solutions will blow up in finite time, regardless of the size of the initial disturbance. Li-Zhu [16] established the local existence of Makino-Ukai-Kawashima type (see [18]) regular solutions with vacuum, and also proved that the regular solutions will blow up if the initial mass density vanishes in some local domain.
For the Navier-Stokes-Boltzmann equations of the viscous compressible radiation fluid, under some physical assumptions, Chen-Wang [2] studied the classical solutions of the Cauchy problem with the mass density away from vacuum. Ducomet and Nečasová [9] [10] obtained the global weak solutions and their large time behavior for the one-dimensional case. Li-Zhu [17] considered the formation of singularities on classical solutions in multidimensional space (d ≥ 2), when the initial mass density is compactly supported and the initial specific radiation intensity satisfies some directional condtions. Some special phenomenon has been observed, for example, it is known in contrast with the second law of thermodynamics, the associated entropy equation may contain a negative production term for RHD system, which has already been observed in Buet and Després [1]. Moreover, from Ducomet, Feireisl and Nečasová [8], in which they obtained the existence of global weak solution for some RHD model, we know that the velocity field u may develop uncontrolled time oscillations on the hypothetical vacuum zones.
The purpose of this paper is to provide a local theory of strong solutions (see Definition 2.1) to the RHD equations in the framework of Sobolev spaces. Via the radiation transfer equation (1.1) 1 and the definitions of F r and P r , system (1.1) can be rewritten as (1.6) where L is the Lamé operator defined by (1.5). We consider the Cauchy problem of (1.6) with the following initial data For (1.6)-(1.7), inspired by the argument used in [3][6], we introduce a similar initial layer compatibility condition (2.3), which will be used to compensate the loss of positive lower bound of the initial mass density when vacuum appears. The key point is to get a priori estimates independent of the lower bound of the initial mass density by this compatibility condition. Then the existence of the local strong solutions can be obtained by the approximation process from non-vacuum to vacuum. We also prove that if the initial vacuum is not so irregular, then the compatibility condition of the initial data is necessary and sufficient for the existence of a unique strong solution. Finally, we give a blow-up criterion for the local strong solution: if T < +∞ is the maximal existence time of the local strong solution (I, ρ, u), then where 3 < q ≤ 6 and ρ ≥ 0 are both constants. The similar results also hold for the barotropic flow with general pressure law p m = p m (ρ) ∈ C 1 (R + ).
Throughout this paper, we use the following simplified notations for standard homogenous and inhomogeneous Sobolev spaces: where 0 < T < ∞ and 1 ≤ p ≤ ∞ are both constants, X, X 1 , and X 2 are some Sobolev spaces. The following inequalities will be used in our paper: where 3 < q ≤ 6 and u X 1 ∩X 2 = u X 1 + u X 2 . A detailed study on homogeneous Sobolev spaces may be found in [11]. Now we make some assumptions on the physical coefficients σ a and σ s . First, let where λ 1 = 1 or 1 2 , and λ 2 = 1 or 2. Hereinafter we denote by C a generic positive constant depending only on the fixed constants µ, λ, γ, q, T and the norms of S. Second, let Remark 1.1. These assumptions are similar to those in [14] for the local existence of classical solutions to the Euler-Boltzmann equations with initial mass density away form vacuum and the assumptions in [16] for the local existence of regular solutions with vacuum. The evaluation of these radiation quantities is a difficult problem of quantum mechanics, and their general forms are usually not known. The expressions of σ a and σ s used for describing Compton Scattering process in [19] are given by where v 0 is the fixed frequency, D i (i = 1, 2) are positive constants and θ is the temperature.
The rest of this paper is organized as follows. In Section 2, we give our main results including the local existence of strong solutions with vacuum, the necessity and sufficiency of the initial layer compatibility condition and the corresponding blow-up criterion for the local strong solution that we obtained. In Section 3, we prove the existence and uniqueness of local strong solutions via establishing a priori estimates independent of the lower bound of ρ 0 . In Section 4, we show that the initial layer compatibility condition is necessary and sufficient for the existence of a unique local strong solution. Finally in Section 5, we prove the blow-up criterion that we claimed in Section 2.

Main results
We state our main results in this section. First, we give the definition of strong solutions to Cauchy problem (1.6)-(1.7). (1) (I, ρ, u) solves (1.6)-(1.7) in the following sense of distribution: (2) (I, ρ, u) satisfies the following regularities: As has been observed in 3-D compressible isentropic Navier-Stokes equations [6], in order to make sure that the Cauchy problem with initial mass density containing vacuum is well-posed, the lack of a positive lower bound of the initial mass density ρ 0 should be compensated by some initial layer compatibility condition on the initial data (ρ 0 , u 0 ). Now considering the 3-D compressible isentropic radiation hydrodynamic equations (1.6), if we , then the main result of this paper on the existence of the unique local strong solutions can be shown as Let the assumptions (1.8)-(1.10) hold, and assume that If the initial data (I 0 , ρ 0 , u 0 ) satisfy the regularities and the initial layer compatibility condition for some g 1 ∈ L 2 , then there exists a time T * > 0 and a unique strong solution (I, ρ, u) on R + × S 2 × [0, T * ] × R 3 to Cauchy problem (1.6)-(1.7).
Remark 2.1. For the case that the rate of energy emission S depends on the mass density ρ, that is, S = S(v, Ω, t, x, ρ), similar results can be obtained via the same argument as the case S = S(v, Ω, t, x), if we assume, for ρ i (t) (i = 1, 2) satisfying (1.9), that, Our second result can be regarded as an explanation for the compatibility between (2.2) and (2.3) when the initial vacuum is not so irregular. To be more precise, we denote by V the initial vacuum set, i.e, the interior of the zero-set of the initial density in R 3 , and define the Sobolev space D 1 0 (V ) as Let conditions in Theorem 2.1 hold. We assume that either the initial vacuum set V is empty or the elliptic system has only zero solution in D 1 0 (V ) ∩ D 2 (V ). Then there exists a unique local strong solution (I, ρ, u) satisfying if and only if the initial data satisfy the initial layer compatibility condition (2.3).

Remark 2.2.
From the regularities of the strong solution (I, ρ, u) in the Definition 2.1, we know that But since the strong solution (I, ρ, u) satisfies the Cauchy problem only in the sense of distribution, we only have I(v, Ω, t = 0, x) = I 0 , ρ(t = 0, x) = ρ 0 and ρu(t = 0, x) = ρ 0 u 0 . In the vacuum domain, the relation u(t = 0, x) = u 0 maybe not hold. Theorem 2.2 tells us that if the initial vacuum set V has a sufficiently simple geometry, for instance, it is a domain with Lipschitz boundary, we then have u(t = 0, x) = u 0 .
Finally, we give a blow-up criterion for strong solutions obtained in Theorem 2.1.

Theorem 2.3 (Blow-up criterion for the local strong solution).
Let conditions in Theorem 2.1 hold. If T < +∞ is the maximal existence time of the local strong solution (I, ρ, u) obtained in Theorem 2.1, then we have Assume that the initial data (I 0 , ρ 0 , u 0 ) satisfy the regularity conditions 8) and the compatibility condition for some g 2 ∈ L 2 , where p 0 m = p m (ρ 0 ), A 0 r is defined as before. Then the conclusions obtained in Theorems 2.1-2.3 also hold for (1.6)-(1.7).

The existence and uniqueness of local strong solutions
We prove Theorem 2.1 in this section, i.e., the existence and uniqueness of local strong solutions. For the rate of energy emission S, we always assume that In order to prove the local existence of strong solutions to the nonlinear problem, we need to consider the linearized system with the initial data (1.7), where w = w(t, x) ∈ R 3 is a known vector, the terms A r and A r are defined by is a known function. We assume that 3.1. A priori estimates to the linearized problem away from vacuum.
We immediately have the global existence of a unique strong solution (I, ρ, u) to (3.1) with (1.7) by the standard methods at least for the case that the initial mass density is away from vacuum.
Proof. First, the existence and regularity of the unique solution ρ to (3.1) 1 can be obtained essentially according to the same argument in [6] for Navier-Stokes equations, and ρ can be expressed by so we can easily get the positive lower bound of ρ. Second, (3.1) 2 can be rewritten into then we easily get the existence and regularity of a unique solution I to (3.5) such that and according to the classical imbedding theory for Sobolev spaces, it is easy to show that Finally, the momentum equations (3.1) 3 can be written into then the existence and regularity of the unique solution u to the corresponding linear parabolic problem can be obtained by standard methods as in [3] [5].
In order to pass to the limit as δ → 0, we need to establish a priori estimates independent of δ for the solution (I, ρ, u) to Cauchy problem (3.1) with (1.7) obtained in Lemma 3.1.
We fix a positive constant c 0 sufficiently large such that for some time T * ∈ (0, T ) and constants c i (i = 1, 2, 3, 4) such that The constants c i (i = 1, 2, 3, 4) and T * will be determined later and depend only on c 0 and the fixed constants ρ, q, A, µ, λ, γ, c and T . As defined in assumption (1.10), still denotes a strictly increasing continuous function depending only on fixed constants ρ, q, A, µ, λ, γ, c and T .
We first give the a priori estimates for density ρ. Hereinafter, we use C ≥ 1 to denote a generic positive constant depending only on fixed constants ρ, q, A, µ, λ, γ, c and T .  .7), there exists a time T 1 > 0 such that From the continuity equation and the standard energy estimates as shown in [6], Therefore, the desired estimate for ρ follows by observing that The estimate for ρ t is clear from ρ t = −div(ρw). Due to p m = Aρ γ (γ > 1), then the estimate for p m follows immediately from above. Now we give the a priori estimates for I.
Now we give the a priori estimates for u. Proof.
Step 1. The estimate of |u| D 1 . Multiplying (3.1) 3 by u t and integrating over R 3 , we have where According to Lemma 3.2, Hölder's inequality, Gagliardo-Nirenberg inequality and Young's inequality, we have for 0 < t ≤ T 2 . Now we estimate the radiation term E I , where We estimate J j term by term. From Lemmas 3.2-3.3, Gagliardo-Nirenberg inequality, Hölder's inequality, Young's inequality and (1.8)-(1.10), for 0 ≤ t ≤ T 2 , we have Su · ΩdxdΩdv + Cc 0 |∇u| 2 , Combining the above estimates for Λ i and J j , it turns out that Su · ΩdxdΩdv. (3.17) Then integrating (3.17) over (0, t), we have (3.18) According to Lemma 3.2, (3.18) and the standard elliptic regularity estimate, we have which means that From (3.18) and (3.20), we know that Step 2. The estimate of |u| D 2 . Differentiating (3.1) 3 with respect to t, we have multiplying (3.22) by u t and integrating the resulting equations over R 3 , we obtain First, we estimate the fluid terms 4 i=1 I i . According to Lemmas 3.2-3.3, Gagliardo-Nirenberg inequality, Hölder's inequality and Young's inequality, we easily have (3.24) Now we estimate the radiation term E II .
We estimate J j term by term. From Lemmas 3.2-3.3, Gagliardo-Nirenberg inequality, Hölder's inequality, Young's inequality and (1.8)-(1.10), for 0 ≤ t ≤ T 2 we have where we used the fact that And similarly Combining the above estimates for I i and J j , it turns out that (3.25) Integrating (3.25) over (τ, t) for τ ∈ (0, t), we easily get From the assumptions (1.8)-(1.10), Lemma 3.1, the regularity of S(v, Ω, t, x) and Minkowski inequality, we easily have According to the compatibility condition (2.3), it is easy to show that Therefore, by letting τ → 0 in (3.26) and (3.21), we have for 0 ≤ t ≤ T 2 . From Gronwall's inequality, we get (3.20) and (3.30) yields Finally, from the standard elliptic regularity estimate (see [3]) and Minkowski inequality, we conclude that Based on Lemmas 3.2-3.4, we obtain the following local (in time) a priori estimate independent of the lower bound δ of the initial mass density ρ 0 : then we deduce that I L 2 (R + ×S 2 ;C([0,T * ];H 1 ∩W 1,q (R 3 ))) ≤c 1 , (3.33)

The unique solvability of the linearized problem with vacuum.
First we give the following key lemma for the proof of our main result -Theorem 2.1.
Proof. We divide the proof into three steps.

Then from the compatibility condition (2.3) we have
where

ΩdΩdv.
It is easy to know from the assumptions (1.8)-(1.10), that for all small δ > 0, Therefore, corresponding to initial data (I 0 , ρ δ0 , u 0 ), there exists a unique strong solution (I δ , ρ δ , u δ ) satisfying the local estimate (3.33). Thus we can choose a subsequence of solutions (still denoted by (I δ , ρ δ , u δ )) converging to a limit (I, ρ, u) in weak or weak* sense. Furthermore, for any R > 0, thanks to the compact property [20], there exists a subsequence (still denoted by (I δ , ρ δ , u δ )) satisfying (3.35) where B R = {x ∈ R 3 : |x| < R}. By the lower semi-continuity of norms (see [11]), it follows from (3.35) that (I, ρ, u) also satisfies the estimate (3.33). For any ϕ ∈ C ∞ c (R + × S 2 × [0, T * ] × R 3 ), from (3.33), (3.35) and assumptions (1.8)-(1.10) we easily have Then it is easy to show that (I, ρ, u) is a weak solution in the sense of distribution and satisfies the following regularities: Step 2: Uniqueness. Let (I 1 , ρ 1 , u 1 ) and (I 2 , ρ 2 , u 2 ) be two solutions obtained in step 1 with the same initial data. Then by the same method as in [6] we can getρ 1 = ρ 2 and Here we omit the details. It is easy to show that I 1 − I 2 satisfies the following Cauchy problem: It follows immediately that I 1 = I 2 .

Proof of Theorem 2.1.
Our proof is based on the classical iteration scheme and the existence results for the linearized problem obtained in Section 3.2. Like in Section 3.2, we define constants c 0 and c 1 , c 2 , c 3 , c 4 and assume that Let I 0 ∈ L 2 (R + × S 2 ; C([0, T * ]; H 1 ∩ W 1,q (R 3 ))) be the solution to the linear parabolic problem Taking a small time T 1 ∈ (0, T * ), we then have We divided the proof of Theorem 2.1 into two cases: ρ > 0 and ρ = 0.
Proof. The proof of this case is divided into three steps.
Step 1. The existence of strong solutions. Let (I 1 , ρ 1 , u 1 ) be the strong solution to Cauchy problem (3.1) with (1.7) and (w, ψ) = (u 0 , I ′0 ). Then we construct approximate solutions (I k+1 , ρ k+1 , u k+1 ) inductively as follows. Assume that (I k , ρ k , u k ) was defined for k ≥ 1, let (I k+1 , ρ k+1 , u k+1 ) be the unique solution to the Cauchy problem (3.1) with (1.7) with (w, ψ)=(u k , I ′k ): with initial data According to the arguments in Sections 3.1-3.2, we know that the solution sequence (I k , ρ k , p k m , u k ) still satisfies the priori estimates (3.33). Now we show that (I k , ρ k , u k ) converges to a limit in a strong sense. Let where L 1 and L 2 are given by First, we estimate sequence ρ k+1 . Multiplying (3.39) 1 by ρ k+1 and integrating over R 3 , where we have used the facts σ a ≥ 0 and σ ′ s ≥ 0. σ k+1 , σ k+1,k and D k η (t) are defined by From the estimate (3.33), we also have t 0 D k η (s)ds ≤ C + C η t, for t ∈ [0, T 1 ]. Finally, multiplying (3.39) 3 by u k+1 and integrating over R 3 , we have For the fluid terms I 5 − I 8 , according to the Gagliardo-Nirenberg inequality, Minkowski inequality and Hölder's inequality, it is not hard to show that |ρ k+1 ||u k t ||u k+1 |dx, For the radiation related terms I 9 − I 14 , , . Then combining the above estimates, we have and we also have t 0 F k η (s) + F k 2 (s) + F k 3 (s) ds ≤ C + C η t, for t ∈ [0, T 1 ]. To estimate R 3 |ρ k+1 ||u k t ||u k+1 |dx, we need the following Lemma.
Lemma 3.6 (The lower bound of the mass density at far field).
There exists a sufficiently large R > 1 and a time T 2 ∈ (0, T 1 ) small enough such that where the constants R > 0 and T 2 is independent of k, and B C R = R 3 \ B R .
Proof. From ρ 0 − ρ ∈ H 1 ∩ W 1,q and the embedding W 1,q ֒→ C 0 , where C 0 is the set of all continuous functions on R 3 vanishing at infinity, we can choose a sufficiently large R > 1 such that From the proof of Lemma 3.1 we know that (3.45) The local estimate (3.33) leads to From the ODE problem (3.45), we get where C is a positive constant independent of k, and T 2 is a small positive time depending only on C and T 1 . That means, for some G k η such that t 0 G k η (s)ds ≤ C + C η t for 0 ≤ t ≤ T 2 . By using Gronwall's inequality, we have Since 0 < T 2 ≤ 1, we first choose η = η 0 small enough such that then we choose T 2 = T * small enough such that Therefore, the Cauchy sequence (I k , ρ k , u k ) converges to a limit (I, ρ, u) in the following strong sense: (3.50) Thanks to the local uniform estimate (3.33), the strong convergence in (3.50) and the lower semi-continuity of norms, we also have (I, ρ, u) still satisfies the a priori estimates (3.33). Then it is easy to show that (I, ρ, u) is a weak solution in the sense of distribution satisfying the a priori estimates (3.33).
Step 3. The time-continuity of the strong solution. It can be obtained by the same method used in the proof of Lemma 3.1.
Proof. Multiplying the first equation in (3.39) by sign(ρ k+1 )|ρ k+1 | 1 2 and integrating over for t ∈ [0, T 1 ], and from the local a priori estimate (3.33) we have According to (3.52), the key term can be estimated by We can also define the energy function by Then from (3.52)-(3.54) and Gronwall's inequality, we easily obtain  for some G k η such that t 0 G k η (s)ds ≤ C + C η t for 0 ≤ t ≤ T 1 . The rest of the proof are analogous to the proof for the case ρ > 0. We omit the details here.
Thus the proof of Theorem 2.1 is finished.

Necessity and sufficiency of the initial layer compatibility condition
We prove Theorem 2.2 in this section, that is, the initial layer compatibility condition is not only sufficient but also necessary if the initial vacuum set is not very irregular. Since the strong solution (I, ρ, u) only satisfies the Cauchy problem in the sense of distribution, we only have I(v, Ω, 0, x) = I 0 , ρ(0, x) = ρ 0 , ρu(0, x) = ρ 0 u 0 , x ∈ R 3 .
So, the key point of the proof is to make sure that the relation u(0, x) = u 0 holds in the vacuum domain. Now we give the proof of Theorem 2.2.
Proof. We prove the necessity and sufficiency, respectively.
Finally we remark that, for a special case that the mass density ρ(t, x) = 0 only holds in some single point or only decay in the far field, u(0, x) = u 0 obviously hold according to our proof of the sufficiency.