On regular solutions of the 3-D compressible isentropic Euler-Boltzmann equations with vacuum

In this paper, we discuss the Cauchy Problem for the compressible isentropic Euler-Boltzmann equations with vacuum in radiation hydrodynamics. Firstly, we establish the local existence of regular solutions by the fundamental methods in the theory of quasi-linear symmetric hyperbolic systems under some physical assumptions. Then we give the non-global existence of regular solutions caused by the effect of vacuum for $1<\gamma\leq 3$. Finally, we extend our result to the initial-boundary value problem under some suitable boundary conditions. These blow-up results tell us that the radiation cannot prevent the formation of singularities caused by the appearance of vacuum.


Introduction
This paper is concerned with the local existence of regular solutions (see Definition 2.1) and the formation of singularities to the Cauchy problem for the isentropic Euler-Boltzmann equations with vacuum arising from the radiation hydrodynamics.
This system appears in various astrophysical contexts [7] and in high-temperature plasma physics [15]. The couplings of fluid field and radiation field involve momentum source and energy source depending on the specific radiation intensity driven by the so called radiation transfer integro-differential equation [15]. Suppose that the matter is in local thermodynamical equilibrium, the coupled system of Euler-Boltzmann equations for the mass density ρ(t, x), the fluid velocity u(t, x) = (u 1 , u 2 , u 3 ), and the specific radiation intensity I(v, Ω, t, x) in three-dimensional space reads as [15]                1 c ∂ t I + Ω · ∇I = A r , ∂ t ρ + ∇ · (ρu) = 0, ∂ t ρu + 1 c 2 F r + ∇ · (ρu ⊗ u + P r ) + ∇p m = 0, where (t, x) ∈ R + ∩ R 3 , v ∈ R + is the frequency of photons and Ω ∈ S 2 is the travel direction of photons, here S 2 stands for the unit sphere in R 3 ; p m is the material pressure satisfying the equation of state where γ is the adiabatic exponent.
3) is the collision term involving emission, absorption and scattering of energy, where I = I(v, Ω, t, x), I ′ = I(v ′ , Ω ′ , t, x). S = S(v, t, x, ρ) is the rate of energy emission due to spontaneous process; σ a = σ a (v, t, x, ρ) stands for the absorption coefficient that may also depend on the mass density ρ; σ s = σ s (v ′ → v, Ω ′ · Ω, ρ) is the "differential scattering coefficient" (see [8] or [15]) such that the probability of a photon being scattered from v ′ to v contained in dv, and from Ω ′ to Ω contained in dΩ, and traveling a distance ds is given by σ s (v ′ → v, Ω ′ · Ω)dvdΩds.
For the isentropic flow, the impact of radiation on the dynamical properties of the fluid is described by the following two quantities (1.4) which are called the radiation flux and the radiation pressure tensor, respectively. The radiation field affects the dynamical properties of the fluid significantly, which makes it difficult to get the estimates of some physical quantities. For example, the material momentum ρudx of the fluid is not conserved because of the impact coming from the radiation flux F r and the radiation pressure tensor P r .
For pure compressible hydrodynamics equations without radiation, there have been many results on the local existence of regular solutions and the formation of singularities caused by the appearance of vacuum. The study on the appearance of vacuum in fluid dynamics can be traced back at least to the collected work of von Neumann [14]. He made some remarks on the general hydrodynamical discussion about motions in one dimension following Riemann's theory. Makino-Ukai-Kawashima [13] discussed the Cauchy problem for the compressible Euler equations with both initial density and velocity compactly supported. They established the local existence of the regular solutions and showed that the life span is finite for any non-trivial solution. Liu-Yang [10] first showed that the regular solution of three-dimensional compressible Euler equations with damping will not be global if the initial density has compact support. Xu-Yang [18] established the local existence of smooth solutions to Euler equations with damping under the assumption of physical vacuum boundary condition.
Recently, similar problems for compressible radiation hydrodynamics equations started drawing attention of people. For Euler-Boltzmann equations, when the initial density is away from vacuum, Jiang-Zhong [5] obtained the local existence of C 1 solutions for the Cauchy problem. Jiang-Wang [4] showed that some C 1 solutions will blow up in finite time regardless of the size of the initial perturbation. For Navier-Stokes-Boltzmann equations, in addition to the local existence of strong solutions with vacuum we obtained in [9], we also established the non-global existence of classical solutions to the Cauchy problem with compactly supported initial density by introducing a new functional which is a linear combination of some mechanical quantities and some radiation quantities in [8], we even studied the case that the viscosity coefficients depend on the mass density ρ.

Ducomet-Nečasová [2] [3] studied the global weak solutions and their large time behavior
for one-dimensional case.
In this paper, we are interested in the isentropic Euler-Boltzmann equations with the occurrence of vacuum. We first prove the local existence of regular solutions to the Cauchy problem, then we studied the formation of singularities caused by the vacuum. Our paper is greatly inspired by the arguments in Makino-Ukai-Kawashima [13] , Jiang-Zhong [5] and Xin-Yan [19]. An important technique for symmetrization is adopted from [13], which will be used to prove the local existence of the regular solutions with nonnegative initial density. Due to the complexity of this physical model, we have to make some structure assumptions to the corresponding physical quantities such as σ a , σ s and S, etc. Then we showed the formation of singularities for Cauchy problem when the initial data contain vacuum in some local domain, which is similar to the assumptions in [19]. Additionally, via the analysis of the finite influence domain, we also get some blow-up results for classical solutions to some initial-boundary value problems. These blow-up results imply that the radiation effect is not strong enough to prevent the formation of singularities caused by the appearance of vacuum. Similar results have been proved for damped Euler equations [10] and Euler-Possion equations [12], which are very different from the results obtained in [16] [17] for the case without vacuum. Some discussion on the relation between this kind of singularities and the formation of shock can be seen in [10].
We organize this paper as follows. In section 2, we first reformulate the Cauchy problem for system (1.1) into a simpler form for the case σ s = 0, and then we establish the local existence of regular solutions. In section 3, we show that the regular solution obtained in Section 2 will develop singularities in finite time provided that the initial data contain vacuum in some local domain for γ > 1, and we extend our blow-up result for Cauchy problem to some initial-boundary value problems. Finally, in Section 4, we show the local existence of regular solutions for the case σ s = 0 under some assumptions to S, σ a and σ s .

Reformulation.
We only consider the case σ s = 0 in this section. For the case σ s = 0, some corresponding results will be shown in Section 4. We reformulate the Cauchy problem of the compressible Euler-Boltzmann equations (1.1) to a quasi-linear symmetric hyperbolic system, so that we can get the local existence of regular solutions (see Definition 2.1).
From the assumptions of "induced process" and local thermal equilibrium (LTE, see [8] [15]), S and σ a can be written as where B(v) ∈ L 2 (R + ) is a function of v, h is the Planck constant, and where K a ∈ C ∞ for (v, t, x, ρ) and lim ρ→0 K a (v, t, x, ρ) = 0. More comments on S(v, t, x, ρ) and σ a (v, t, x, ρ) can be seen in [5] as well as in [15]. So, when σ s = 0, the photon transport equation in (1.1) can be written as Then the compressible isentropic Euler-Boltzmann equations (1.1) can be reduced to We consider the Cauchy problem with initial data We first introduce the definition of regular solutions to Cauchy problem (2.4)-(2.5).
(1) We point out that this definition for regular solutions is almost the same as that of Makio-Ukai-Kawashima [13], in which the local existence of regular solutions is studied for Euler equations with initial data arbitrarily large and inf ρ 0 = 0. Similar result has been obtained for damped Euler equations in [10]. Moreover, √ γρ γ−1 2 is a very important physical quantity called local sound speed in gas dynamics.
(2) When ρ > 0, if (ρ, u) has the regularity showed in (i)-(ii), then it naturally holds Passing to the limit as ρ → 0, we have So condition (iii) is reasonable at points (t, x)(t > 0) satisfying ρ(t, x) = 0 due to the continuity of ρ and properties of K a .
(3) We emphasize that condition (iii) is very important to make the velocity u well defined at vacuum points and to ensure the uniqueness of regular solutions. Without condition (iii), it is very difficult to get enough information about velocity even when considering specific cases such as point vacuum or continuous vacuum of one piece.
We note that A j (U )(j = 0, 1, 2, 3) are C ∞ for U and G(I, U ) are C ∞ for I and U .
Moreover, A j (U ) (j = 1, 2, 3) are all symmetric, and A 0 (U ) is bounded and positively definite. In fact, we have In order to get the local existence of regular solutions to the Cauchy problem (2.4)-(2.5), it sufficies to prove the local existence of classcal solutions to the reformulated Cauchy problem (2.9)-(2.10).
We first introduce some notations. Denote by W s,p (R n ) and H s (R n ) the ordinary Sobolev spaces, and The following well-known estimates for the derivatives of product are useful in the energy estimates for local existence.
and when s ≥ 1, where C s is a constant depending on s.
Remark 2.2. The proof of Lemma 2.1 can be found in [1]. From Sobolev imbedding theorem we know that, if s > n 2 , then we have where f ∈ L ∞ H s . Then letting r = 2 and p = ∞, q = 2 in Lemma 2.1, we obtain To obtain the local existence of classical solutions to (2.9)-(2.10), we need some assumptions on the coefficient K a . If there exists a positive constant M such that w i s ≤ M (i = 1, 2), and we denote K i a = K a (v, t, x, w i ) and K i a = K a (v, t, x, w i ) ≥ 0, then we assume that Theorem 2.1. Let s ≥ 3 be an integer. If the initial data satisfy then there exists T > 0 such that Cauchy problem (2.9)-(2.10) admits a unique classical Remark 2.3. The assumptions in Theorem 2.1 for isentropic flows can be satisfied when the absorption coefficient is given by, for example (see [5] or [15]), where θ is the temperature, v 0 is the fixed frequency, D i (i = 1, 2) are positive constants.
For isentropic polytropic gas, we know that p m = Rρθ = ρ γ , where R is a positive constant.
Then in the case (2.15), we have (2.17) When 1 < γ ≤ 3, it is easy to verify that the assumptions in Theorem 2.1 are satisfied if Now we start proving Theorem 2.1.
Proof. The proof is based on standard energy estimates as well as Banach contraction where ǫ 0 is to be chosen later. We construct approximate solutions to (2.9)-(2.10) through the following iteration scheme. We take inductively as the solution of the following linearized problem: It follows immediately that with T k being the largest time of existence for (2.18) where the estimates are valid for any given constant C > 0. Hereinafter, C stands for a generic positive constant.
Of course, in order to get the compactness, we have to guarantee that there exists a T > 0 such that T k ≥ T for each k. So the following lemma which gives the uniform estimates of high order norms is very important. There exist constants C 1 > 0 and T * > 0 such that the solutions (I (k) , U (k) ) (k = 0, 1, 2, ...) Proof. By induction, it is sufficient to prove that (2.22) holds for (I (k+1) , U (k+1) ) under the assumption that (2.22) holds for (I (k) , U (k) ). We divide the proof into three steps.
It is obvious that Taking T = T 1 to be small enough, we have Step 2. The estimate of source terms D α G(I (k) , U (k) ) , ∀|α| ≤ s.
Due to Minkowski's inequality, Holder's inequality and (2.13), for |α| ≤ s, we have which implies that Then from (2.28) and assumptions (2.14), we see that Step 3. In order to estimate (2.22), define M (k+1) = U (k+1) − U (0) 0 , and it is easy to get where With the aid of the steps 1 and 2, it is easy to follow the standard procedure as in [1] and obtain that there exists a time T 2 such that Let T * = min{T 1 , T 2 } . Then the conclusions in Lemma 2.2 are obtained.
Now we continue to prove Theorem 2.1.

Lemma 2.3 tells us that
(2.48) In addition, from Lemma 2.2 we know that sequence {U (k) (t, ·)} ⊂⊂ Φ for any fixed t and Then from Sobolev interpolation inequalities, we have for any 0 < s ′ < s. So from (2.49) and (2.50), we get

Similarly, using Sobolev interpolation inequalities and Holder's inequality, we get
According to (2.48), we conclude that for any 0 < s ′ < s.
Therefore, if we choose s ′ > 5 2 , then from Sobolev embedding theorem, there exists (I, U ) such that Furthermore, from the second equation in (2.18), we have Similarly to the proof of Lemma 2.3, we can prove that (I, U ) = ( I, U ).
The proof of Theorem 2.1 is finished.
Remark 2.4. by the standard method in Majda [1], the classical solution obtained in the above theorem also satisfies Back to the Cauchy problem (2.4)-(2.5), we will give the local existence and uniqueness of regular solutions based on the above results for classical solutions to Cauchy problem (2.9)-(2.10).

Local existence and uniqueness of regular solutions to (2.4)-(2.5).
In this section, we will give the local existence and uniqueness of regular solutions to the original Cauchy problem (2.4)-(2.5) based on the results obtained in Section 2.2.
Theorem 2.2. Let s ≥ 3 be an integer. If the initial data satisfy then there exists a time T > 0 such that Cauchy problem (2.4)-(2.5) admits a unique regular solution (I, ρ, u).
which is exactly the continuity equation in (2.4).
we get the momentum equations in (2.4): That is to say, (I, ρ, u) satisfies the Euler-Boltzmann equations classically. Then from the continuity equation, it is easy to get that ρ can be expressed by where X ∈ C [0, T ] × [0, T ] × R 3 is the solution of the initial value problem d dt X(t, s, x) = u(t, X(t, s, x)), 0 ≤ t ≤ T, X(s, s, x) = x, 0 ≤ s ≤ T, x ∈ R 3 , (2.57) In conclusion, Cauchy problem (2.4)-(2.5) has a unique regular solution (I, ρ, u).

Finite time Blow-up of regular solutions
In this section, we consider the formation of singularities to regular solutions obtained in Section 2.3. We first assume that the initial data (2.5) satisfy the following local vacuum state condition: Let A 0 and B 0 be two bounded open sets in R 3 , B 0 is connected and then we say that the initial data (I 0 , ρ 0 , u 0 ) contain local vacuum state. In order to observe the evolution of A 0 and B 0 , we need the following definition. Let x(t; x 0 ) be the particle path starting from x 0 at t = 0, i.e., Then we denote by A(t), B(t), (B − A)(t) the images of A 0 , B 0 , and B 0 − A 0 , respectively, under the flow map of (3.2), i.e., It is easy to know that (B − A)(t) is the vacuum domain.
Then we have Proof. Firstly, because B(v) is independent of x and t, the first equation of system (2.4) can be rewritten as 1 We denote by y(t; y 0 ) the photon path starting from y 0 at t = 0, i.e., d ∂t y(t; y 0 ) = cΩ, y(0; y 0 ) = y 0 .
Then from the initial conditions, it is easy to have x · Ω ≤ 0, then along the photon path Due to (3.6), it yields Secondly, on the domain (B − A)(t), ρ = K a (v, t, x, ρ) ≡ 0. Due to the momentum equations in (2.8) and the definition of regular solutions, we have That is to say, u is invariant along the particle path. Thus, according to the local vacuum state condition, we have Using the continuity of u(t, x), we get Now we give the main result of this section, which shows the formation of singularities caused by the appearance of vacuum in some local domain. We first introduce the mass and second moment over B(t): ρ(t, x)|x| 2 dx (second moment). From the momentum equations and integration by parts, we get It follows from (3.10)-(3.11) that From Holder's inequality, we give (3.14) Then (3.12) yields So, using Taylor's expansion, we have Combining (3.16)-(3.17), we have Solving this inequality, we get In other words, the life span T must be finite. The corresponding results for Euler equations, damped Euler equations, and Euler-Possion equations can be found in [10], [11], [12], [16], [17], etc. Moreover, the result obtained in Theorem 3.1 also improved the conclusion in [10] [12] in the sense that we removed the crucial assumption that the initial mass density is compactly supported.
The similar blow-up estimate can be extended to the initial-boundary problem in a smooth and bounded domain Ξ ⊆ R 3 under some suitable boundary condition. ∀T > 0, we assume that a solution (I, ρ, u) of the initial-boundary problem is regular if We divide our proof into three steps.
Step 1. We claim that there exists a positive lower bound ǫ > 0 such that T * ≥ ǫ. In fact, without loss of generality, we assume that T > 1. Then from the definition of particle path x(t; x 0 ), we have If we let T 5 ∈ (0, 1] be small enough such that So, we get that T * ≥ ǫ = T 5 > 0.
Step 2. We claim that a finite T * does not exist. In fact, if there exists a finite T * such that ǫ ≤ T * < T , then due to the definition of T * , we have Via the same analysis as in Lemma 3.1, we have which contradicts with the definition of T * .
Step 3. Now we show that the life span T of regular solutions is finite, i.e., T < +∞.

From
Step 2, we know that Using again the same analysis in Lemma 3.1, we have Therefore, we can handle the initial-boundary value problem in the same way as the Cauchy problem. In other words, we can introduce the same functionals m(t) and M (t) to prove the finiteness of T accordingly. We omit the details here.

4.
Local Existence for the case σ s = 0 In this section, we will give the corresponding local existence of regular solutions for the case σ s = 0, which is similar to the result obtained in Section 2, and we use the same notation as in Section 2. When σ s = 0, if we still consider the assumptions of 'induced process' and local thermal equilibrium as in Section 2 for the case σ s = 0, then the Euler-Boltzmann equations (1.1) become very complicated, for example, the radiation transfer equation in (1.1) reads as which is rather complicated and hard to deal with. Therefore, for simplicity, we start from the original Euler-Boltzmann equations (1.1). For this, we need some assumptions. Let where λ 1 = 1 or 1 2 , and λ 2 = 1 or 2. Let  and denote U = U (t, x) = (w(t, x), u(t, x)) ⊤ . Then system (1.1) of the isentropic Euler-Boltzmann equations can be reduced to the following system: where A 0 (U ) and A j (U ) are defined in Section 2, and Similarly to Theorem 2.1, in order to get the local existence of the original Cauchy problem (1.1) and (2.5), we need the following key theorem.
We can follow the same procedure as the proof of Theorem 2.1 to prove Theorem 4.1.
For k = 0, 1, 2, ..., we define U (k+1) (t, x) and I (k+1) (v, Ω, t, x) inductively as the solution of the following linearized problem: , x, w (k) ). (4.6) It follows immediately that where T k is the largest time of existence for (4.5) such that the estimates are valid for any given constant C > 0. Next we give two key lemmas as in Section 2, which imply the compactness of the above-constructed approximate solutions. There exist constants C 2 > 0 and T * > 0 such that the solution (I (k) , U (k) ) to (4.5) and for k = 0, 1, 2, ....
We also have and where With the aid of the steps 1 and 2, we can easily follow the standard procedure as in [1] to show that there is a time T 2 satisfying where η 1 < 1 2 and k |β k | < +∞. To bound I (k+1) − I (k) , we use equation (2.18)