On regular solutions to compressible radiation hydrodynamic equations with far ﬁ eld vacuum

: The Cauchy problem for three - dimensional ( 3D ) isentropic compressible radiation hydrodynamic equations is considered. When both shear and bulk viscosity coe ﬃ cients depend on the mass density ρ in a power law ρ δ ( with δ 0 1 < < ) , based on some elaborate analysis of this system ’ s intrinsic singular structures, we establish the local - in - time well - posedness of regular solution with arbitrarily large initial data and far ﬁ eld vacuum in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions. Note that due to the appearance of the vacuum, the momentum equations are degenerate both in the time evolution and viscous stress tensor, which, along with the strong coupling between the ﬂ uid and the radiation ﬁ eld, make the study on corresponding well - posedness challenging. For proving the existence, we ﬁ rst introduce an enlarged reformulated structure by considering some new variables, which can transfer the degeneracies of the radiation hydrodynamic equations to the possible singularities of some special source terms, and then carry out some singularly weighted energy estimates carefully designed for this reformulated system.


Introduction
It is well known that the radiation effects become remarkable in some regime when the temperature is high. Radiation sometimes contributes largely to energy density, momentum density, and pressure, for instance, in astrophysics and inertial confinement fusion. Radiation transfer is usually the most effective mechanism that affects the energy exchange in fluids, so it is necessary to take effects of the radiation field into consideration in the classical hydrodynamic framework. The equations of radiation hydrodynamics result from the balances of particles, momentum, and energy. From a microscopic point of view, the radiation field is composed of photons. We first introduce some basic concepts necessary for describing the radiation field and its interaction with matter. At any time t, we need d 2 variables to specify the state of a photon in phase space, namely, d position variables and d velocity (or momentum) variables. Usually, we can denote by x the d position variables, and replace the d momentum variables equivalently with frequency ν and the travel direction Ω of the photon. Via these variables, we then define the phase-space distribution function f f t x ν , , , Ω ( ) = for photons such that where n is the number of photons; n d is the number of photons (at time t) at space point x in a volume element x d , with local frequency ν in a frequency interval ν d , and traveling in a direction Ω in the cubic angel element dΩ. In the radiation transport, we usually use the specific radiation intensity I I t x ν , , , Ω ( ) = to replace the distribution function f . The specific radiation intensity is defined as follows: I t x ν chνf t x ν , , , Ω , , , Ω , where c is the vacuum speed of light and h is Planck constant. The physical interpretation of I is contained in the relationship where E d 1 is the amount of the radiation energy in ν d centered at ν, travelling in a direction Ω confined to a solid angle element dΩ, which crosses, in a time element t d , an area dΣ oriented such that Θ is the angel, which the direction Ω makes with the normal to dΣ ( n cosΘ Ω = ⋅ , and here n is the outward unit normal vector of dΣ).
Regarding the three basic interactions between photons and matter, namely, absorption, scattering, and emission, we have transport equation in the general form (see [23]): where A r is a collision term given by denotes the unit sphere in 3 . σ t x ν ρ , , , Ω, 0 e ( )≥ is the rate of energy emission due to spontaneous process, and σ t x ν ρ , , , Ω, 0 a ( )≥ denotes the absorption coefficient that may depend on the mass density ρ.
Similarly to absorption, a photon can undergo scattering interactions with matter, and the scattering interaction serves to change the photon's characteristics ν′ and Ω′ to a new set of characteristics ν and Ω. To quantitatively describe the scattering event, one requires a probabilistic statement concerning this change, which leads to the definition of the "differential scattering coefficient" σ ν ν σ ν ν ρ , Ω Ω , Ω Ω, s s ( ) ( ) ′ → ′⋅ ≡ ′ → ′⋅ that may depend on ρ such that the probability of a photon being scattered from ν′ to ν contained in ν d , from Ω′ to Ω contained in dΩ, and traveling a distance s d is given by σ ν ν ρ ν s , Ω Ω, d dΩd s ( ) ′ → ′⋅ . Therefore, the time rates of outscattering and inscattering within a unit volume element are expressed as follows: Concerning the effect of radiation on the dynamic properties of the fluid is very significant, we introduce the following two quantities to describe this effect: which are called the radiation flux and the radiation pressure tensor, respectively. Now we take radiation effect into consideration for viscous (barotropic) fluids to have the following isentropic radiation hydrodynamics equations in 3D space: where A r is defined by (1.2). The unknown functions ρ u u u u , , , ( ) ( ) ( ) ( ) = ⊤ represent the density and the velocity, respectively. P m is the material pressure with the following equation of state for the polytropic fluid: where A is a constant and γ is the adiabatic exponent. denotes the viscous stress tensor with the form where D u u u 1 2 ( ) ( ( ) ) = ∇ + ∇ ⊤ is the deformation tensor and 3 is the 3 3 × identity matrix, for some constant δ 0 ≥ , μ ρ ( ) is the shear viscosity coefficient, λ ρ μ ρ In the current paper, we are concerned with the local-in-time well-posedness of regular solution to the Cauchy problem (1.9) with the following initial data and far field behavior: Throughout this paper, we will adopt the following simplified notations, and most of them are for the homogeneous and inhomogeneous Sobolev spaces:  (i.e., δ 0 = in (1.7)), when ρ x inf 0 x 0 ( ) > , the local well-posedness of classical solutions to the Cauchy problem follows from the standard symmetric hyperbolic-parabolic structure satisfying the well-known Kawashima's condition, cf. [13,22,27], which has been extended to be a global one by Matsumura-Nishida [21] for initial data close to a nonvacuum equilibrium in some Sobolev space H s 3 ( ) s 5 2 ( ) > . However, these approaches do not work when ρ x inf 0 x 0 ( ) = , which occurs when some physical requirements are imposed, such as finite total initial mass and energy in the whole space. One of the main issues in the presence of vacuum is the degeneracy of the time evolution operator, which makes it hard to understand the behavior of the velocity field near the vacuum. Via imposing some initial compatibility conditions, Cho et al. [4] established the local well-posedness of strong solutions with vacuum, which, recently, has been shown to be a global one with small energy by Huang et al. [11]. We also refer to Lions [19], and Feireisl et al. [8] to readers and the references therein for the existence theory of global weak solutions with finite energy.
For degenerate viscous flow (i.e., δ 0 > in (1.7)) without considering radiation, when ρ x inf 0 x 0 ( ) = , instead of the uniform elliptic structure in the constant viscous flow, the viscosity degenerates when density vanishes, which raises the difficulty of the problem to another level. Recently, some significant progress has been made on the well-posedness of smooth solutions in 3D space. In the study by Li et al. [16], via introducing a "quasi-symmetric hyperbolic"-"degenerate elliptic" coupled structure to control the behavior of the velocity u near the vacuum, they showed that the unique 3D regular solution with vacuum exists locally in time, which has been extended to be a global one with small density but possibly large velocity by Xin and Zhu [31]. Recently, for the case δ 0 1 < < , by using an elaborate elliptic approach on the operators L ρ u δ 1 ( ) − and some initial compatibility conditions, the existence of 3D local regular solution with far field vacuum has also been obtained by Xin and Zhu [32]. Some other interesting results can also be found in [3,5,10].
When the effect of radiation is taken into consideration, studying the radiation hydrodynamics equations becomes more complicated for both inviscid and viscous fluids because of the complexity and mathematical difficulty. For Euler-Boltzmann equations of inviscid fluids, the local well-posedness and finite time blow up of classical solutions away from the vacuum were studied by Jiang and Zhong [33] in multidimensional (M-D) space, and see also the study by Jiang and Wang [12] for a simplified 3D isentropic model. Later, Li and Zhu [18] considered the system discussed in [12] and proved the local existence of a unique regular solution with vacuum by means of the theory of quasi-linear symmetric hyperbolic systems and some technique tools. They also showed that the regular solution will blow up if the initial density vanishes in some local domain. For Navier-Stokes-Boltzmann equations of viscous fluids, Ducomet and Nečasová [7] showed that the global existence theory for the compressible viscous fluids developed in [21] can be generalized to the radiation hydrodynamics. Li and Zhu [17] established the existence of local strong solutions for the isentropic flow in homogeneous Sobolev space for large initial data satisfying the initial compatibility conditions. Ducomet et al. [6] obtained the existence of global weak solutions for some radiation hydrodynamic system, where the velocity u may develop uncontrolled time oscillations on the hypothetical vacuum zones. Some other interesting results for the related radiation hydrodynamics models can also be seen in [2,24,25] and so on.
Our goal in the present paper is to show the local existence and uniqueness of the 3D regular solution with far field vacuum to the Cauchy problem (1.9)-(1.10) for δ 0 1 < < . It is worth pointing out that when vacuum appears, one would encounter some difficulties as follows. First, (1.9) is a system of fluid equations coupled with a nonlinear integro-differential hyperbolic equation, which makes the corresponding calculation very complicated. Second, the degeneracy of the time evolution and elliptic operator in momentum equations caused by the vacuum makes it very difficult to determine and control the behavior of velocity u near the vacuum, which is the key point in the vacuum-related problems (see Xin and Zhu [32]). Finally, besides the difficulties mentioned earlier, additional difficulties in the proof lie in dealing with the nonlinear terms and the coupled cross terms between radiation field and fluid field, which prevent us obtaining the uniform a priori estimates. To this end, we shall employ the delicate energy estimates based on the welldesigned reformulated structure obtained during the linearization and some physically reasonable assumptions on the radiation quantities.
The rest this paper is organized as follows. In Section 2, we first present some physically reasonable assumptions on the radiation quantities and then conclude this section by stating the main result. Section 3 is devoted to establishing the local-in-time well-posedness of regular solution to the Cauchy problem (1.9)-(1.10). Finally, we give an appendix to list some lemmas that are frequently used in our proof.

Hypotheses and main result
In this section, we first make some physical assumptions on the radiation quantities and then state the main result.

Hypotheses on radiation quantities
The general form of radiation coefficients is usually not known because it is difficult to evaluate these physical coefficients in quantum mechanics. Physically speaking, the radiation coefficients can be written as σ ρσ σ ρσ σ ρσ σ ρσ , , , Next we will make some physically reasonable assumptions on the radiation coefficients σ σ , e a , and σ s , which are similar as in [15,17,30].

Main result
Before stating the main result, let us first introduce the definition of regular solution to the Cauchy problem (1.9)-(1.10).
) is called a regular solution to the Cauchy problem (1.9)-(1.10), if I ρ u , , ( ) satisfies this problem in the sense of distributions and If the initial data I ρ u , , and the initial compatibility conditions Moreover, I ρ u , , ( ) is also a classical solution to (1.9)-(1.10) for t T 0, ( ] ∈ * . Remark 2.3. The condition γ 4 3 ( )satisfying the conditions (2.9)-(2.10): As mentioned in the abstract, the compatibility conditions (2.10) are also necessary for the existence of the unique regular solutions obtained in Theorem 2.1. Indeed, the one shown that in the second one of (2.10)1 (resp. (2.10) 2 ) plays a key role in the derivation of u L T L 0, ;

Reformulation
Via introducing the following new variables: then the Cauchy problem (1.9) and (1.10) can be equivalently rewritten as follows: and L u ( ) and Q u ( ) are given by To solve the Cauchy problem (1.9) and (1.10) locally in time, we first establish the local well-posedness of classical solution to the problem (3.2)-(3.4).

Linearization
To prove Theorem 3.1, we begin by considering the following linearized problem: where both ε and η are positive constants and and g are known functions satisfying χ v g t I u ϕ , , 0 , , where T 0 > is an arbitrary constant. Next, by using standard argument ( [4,14,20]), the global well-posedness of classical solutions ) be a classical solution obtained in Lemma 3.1 and the initial data satisfy (3.5) and (3.6). Now, we are going to establish the uniform a priori estimates for , . For this purpose, we first choose a positive constant c 0 independent of η such that As a consequence of Lemma A.6 and (3.11) and (3.12), one can deduce that where C 1 is a positive constant depending on c A α β γ , , Λ, , , 0 , and δ, but independent of ε η , . We assume there exist some time T T 0, ( ] ∈ * and positive constants c i 1, , where T * and c i (i 1, , 7 = … ) will be determined later, and depend only on c 0 and the fixed con- ) and may be different from line to line. Moreover, in the rest of Section 3.3, without causing ambiguity, we simply denote , , , as , , , , , , , as , , , , , , as , , .
The a priori estimates for ϕ and ψ The following two lemmas give the estimates for ϕ and ψ.
Proof. First, the standard arguments for the transport equation and (3.16) yield that for Second, according to the equation (3.8) 2 , one can obtain that for Moreover, one also has that for t T 0 First, let ζ ζ ζ ζ , , integrating over 3 , one obtains | | , it can be estimated as follows: Plugging the aforementioned estimates for Θ ζ 2 | | into (3.18) and using the Gagliardo-Nirenberg inequality, one obtains which, along with the Gronwall inequality, implies that for Second, according to equations (3.17), one has that for t T 0 On the other hand, Lemma A.1 (Appendix A) and (3.16) yield that Finally, it follows from (3.19) and the equation where one has used The proof of Lemma 3.3 is complete.
Proof. First, multiplying (3.8)1 by I 2 and integrating over 3 , one has where one has used the fact σ 0 a ≥ and σ 0 s ′ ≥ .
Second, by applying According to Lemma 3.2, Hölder's inequality, and Gagliardo-Nirenberg inequality, one obtains ≤ , and combining (3.20), one arrives at Integrating (3.22) over S 2 × + , using the hypotheses H1-H2 and the Gronwall inequality, one has that for t T 0 2 ≤ ≤ , According to the equation ( Finally, differentiating (3.8)1 with respect to t, one can similarly obtain The a priori estimates for auxiliary variables φ and f Proof.

The a priori estimates for u
On the basis of estimates obtained in Lemmas 3.2-3.5, now we are ready to give the lower order estimates for u. Proof.
Step 1: Estimate on u 2 | | . First, multiplying (3.8) 3 by u and integrating over 3 , according to Hölder's inequality, Young's inequality, and Gagliardo-Nirenberg inequality, one has Next we estimate the last term R 5 on the right-hand side of (3.29). By Hölder's inequality and the hypotheses H1-H2, one obtains Substituting (3.30) into (3.29), one arrives at which, along with the Gronwall inequality, yields that for Step 2: Estimate on u 2 | | ∇ . Multiplying (3.8) 3 by u t and integrating over 3 , one can obtain ΩdΩd d div d Step 3: Estimate on u D 2 | | . First, it follows from (3.8) 3 and the definition of Lamé operator that Then according to (3.14) for the definition of G, (3.30), and Lemma A.6 (Appendix A), one can obtain that  Second, differentiating the equations (3.8) 3 with respect to t, one has Multiplying (3.38) by u t and integrating over 3 , one arrives at As a consequence of Lemma A.6 and (3.35), one also has Consequently, it follows from The proof of Lemma 3.6 is complete. □ The following lemma gives the higher order estimates for u.  Proof. First, multiplying (3.38) by u tt and integrating over 3 , using Hölder's inequality, Young's inequality, and Gagliardo-Nirenberg inequality, one has It follows from the equations (3.8) 3 that which, along with the hypotheses H1-H2, Letting τ 0 → in (3.48) and using the Gronwall inequality, one concludes that for t T 0 Finally, according to (3.38) and Lemma A.6, one obtains that for t T 0

Passing to the limit → ε 0
With the help of ε η , ( )-independent estimates obtained in (3.62), we will establish the local-in-time existence result for the following linearized problem (3.63) without an artificial viscosity (i.e., ε 0 = ) under the assumption that ϕ η η 0 ≥ . Proof.
Step 1: Existence. First, according to Lemmas 3.1-3.8, one can see that for every fixed ε 0 > and η 0 > , there exist a time T * independent of (ε η , ) and a unique strong solution to the linearized problem (3.8) satisfying the estimates in (3.62), which are independent of (ε η , ). Second, using the characteristic method and standard energy estimates for the equation (3.8) 4 , one can obtain that for Then, based on the uniform estimates in (3.62) independent of (ε η , ), estimates in (3.64) independent of ε, and Lemma A.3 (Appendix A), one concludes that for any R 0 > , there exists a subsequence of solutions , which converges to a limit I ϕ u h , , , η η η η ( )in the following strong sense: where B R is a ball centered at origin with radius R.
On the other hand, the uniform estimates in (3.62) independent of (ε η , ) and estimates in (3.64) independent of ε imply that there exists a subsequence (of subsequence chosen earlier) of solutions (still denoted by I ϕ u h , , , , which converges to a limit I ϕ u h , , , η η η η ( )as ε 0 → in the following weak or weak * sense: which, along with the lower semi-continuity of weak or weak * convergence, imply that I ϕ u h , , , η η η η ( ) satisfies the corresponding estimates in (3.62) and (3.64) except those weighted estimates for u η .
It follows from the strong convergence in (3.65) and the weak or weak * convergence in (3.66) that Moreover, combining (3.65)-(3.67), one can easily show that I ϕ u h , , , )is a weak solution in the sense of distributions to (3.63), satisfying Choosing a time T T 0, ( ] ∈ * small enough such that Step 1: Existence. Let the beginning step of our iteration be χ v g I u h , , , then one can obtain a classical solution I ϕ u h , , ,     then direct calculation shows that problem (3.72) can be rewritten as follows: Then, it follows from (3.74) that are given by Now we are ready to give several estimates, which will be used later. To facilitate the discussion, we will adopt the following notations in the rest of this subsection First, from Remark 3.2 at the end of this subsection, one has Lemma 3.9.
Thus, the corresponding weighted estimates for u η shown in (3.62) still hold for the limit functions.
Step 2: Uniqueness and time continuity. The uniqueness and time continuity follows easily from the same procedure as in Lemma 3.1. Finally, Theorem 3.3 is proved. □ Remark 3.2. We conclude this subsection by giving the proof of Lemma 3.9.
be a truncation function satisfying | | | ) where C 0 ͠ > is a generic constant depending on C and η but independent of R. Similarly, one can also show that f L T H 0,ˆ; . Proof. The proof will be divided into four steps.
Step 1: The locally uniform positivity of ϕ.   where c 0 is a positive constant independent of η. Thus, for the initial data I ϕ u ψ , , , η η η η 0 0 0 0 ( ), the problem (3.69) admits a unique classical solution I ϕ u ψ , , , η η η η ( ) satisfying the local uniform estimates in (3.62) with c 0 replaced by c 0 , and the life span T * is also independent of η.
Moreover, one has the following: where a R0 is independent of η.
Proof. One only needs to consider the case when R 0 is suitable large. ∈ ∞ , which means that the initial vacuum occurs if and only if in the far field. Then there exists a constant C R′ independent of η such that for any R 2 ′ > , Second, let x t x ; 0 ( ) be the solution to the following initial value problem: . Thus, the desired conclusion can be achieved by taking R R 2 0 ′ = and a C C R R 0 = * ′ . □ Step 2: Existence. First, by virtue of the uniform (with respect to η) estimates in (3.62), one can see that there exists a subsequence (still denoted by I ϕ u ψ , , , η η η η ( )) converging to a limit I ϕ u ψ , , , ( ) in the following weak or weak * sense: Conversely, according to the estimates (3.62) except those weighted estimates for u, Lemma 3.10, the weak or weak * convergence in (3.112)  Then the corresponding weighted estimates for u shown in (3.62) still hold for the limit functions. Therefore, one can conclude that I ϕ u ψ , , , ( ) is the weak solution in the sense of distributions to the Cauchy problem (3.2)-(3.4).
Step 3: Uniqueness. The uniqueness can be easily obtained by the same argument as in Theorem 3.3.
Step 4: Time continuity. First, the time continuity of I ϕ u ψ , , , ( ) follows from the similar argument used in Lemma 3.1.
For the velocity u, the uniform estimates obtained earlier and the Sobolev embedding theorem imply that Again the Sobolev embedding theorem implies that u C T 0, 2 3 (( ] ) ∇ ∈ × * . Finally, the proof of Theorem 2.1 is complete. □ Acknowledgments: The authors would like to thank the anonymous referees for their careful reading and several valuable comments. ∈ is a multi-index.
The following lemma is used to obtain the time-weighted estimates of the velocity u.