GLOBAL WELL-POSEDNESS AND LARGE TIME BEHAVIOR OF CLASSICAL SOLUTIONS TO THE DIFFUSION APPROXIMATION MODEL IN RADIATION HYDRODYNAMICS

. We are concerned with the global well-posedness of the diﬀusion approximation model in radiation hydrodynamics, which describe the com- pressible ﬂuid dynamics taking into account the radiation eﬀect under the non-local thermal equilibrium case. The model consist of the compressible Navier-Stokes equations coupled with the radiative transport equation with non-local terms. Global well-posedness of the Cauchy problem is established in perturbation framework, and rates of convergence of solutions toward equilibrium, which are algebraic in the whole space and exponential on torus, are also obtained under some additional conditions on initial data. The existence of global solution is proved based on the classical energy estimates, which are considerably complicated and some new ideas and techniques are thus required. Moreover, it is shown that neither shock waves nor vacuum and concentration in the solution are developed in a ﬁnite time although there is a complex interaction between photons and matter.


1.
Introduction. The aim of the radiation hydrodynamics is to include the radiation effects into hydrodynamics and the importance of thermal radiation in physical problems increases as the temperature is high. More precisely, the thermal effect usually varies as the fourth power of temperature. In this case, the radiation field significantly affects the dynamics of the field. This gives rise to the theory of radiation hydrodynamics, which is mainly concerned with the propagation of thermal radiation through a field or gas, and the effect of this radiation on the dynamics, see, for example, [1,23,25], and the references cited therein. The theory of radiation hydrodynamics finds a very broad range of applications, such as astrophysical, supernova explosions, laser fusion, and so on (cf. [17,24,29]). As in classical fluid mechanics, the equations of motion in radiation hydrodynamics are derived from the conservation laws for macroscopic quantities. However, due to the presence of radiation, the classical "material" flow has to be coupled with radiation which is an assembly of photons (the photons are massless particles traveling at the speed of light) and need a priori a relativistic treatment. Hence, the whole problem to be considered is then a coupling between the standard hydrodynamics for the matter and a radiative transfer equation for the photon distribution. However, the equation of radiative transfer is very complicated, and hence, the physically valid approximate descriptions of radiative transfer have to be introduced (cf. [1]).
In this paper, we mainly consider the diffusion approximation (also are called the Eddington approximation), which is valid for optically thick regions where the photons emitted by the gas have a high probability of reabsorption within the region. The classical diffusion or Eddington approximation describes the energy flow due to radiative process in a semi-quantitative sense, and is particularly accurate if the specific intensity of radiation is almost isotropic (cf. [25]). Based on the standard hydrodynamics, the governing equations of the diffusion approximation in radiation hydrodynamics for 3-D flow of a viscous polytropic ideal heat-conducting radiative gas, can be written in terms of Euler coordinates as follows (see, Appendix and [1,25]): ρ(u t + u · ∇u) + ∇P = µ∆u + (λ + µ)∇divu, Here, the unknowns are (ρ, u, θ, n), where ρ = ρ(x, t) > 0, θ = θ(x, t) > 0, u = u(x, t) = (u 1 (x, t), u 2 (x, t), u 3 (x, t)), for t ≥ 0, x ∈ Ω denote the mass density, temperature and velocity field of the fluid respectively, and n = n(x, t) ≥ 0 for t ≥ 0, x ∈ Ω denotes the radiation field. The spatial domain Ω = R 3 or T 3 . P = Rρθ is the material pressure, R, c ν , κ are positive constants; λ and µ are the constant viscosity coefficients, µ > 0, 3λ + 2µ ≥ 0; D = D(u) is the deformation tensor In this paper, we are interested in the global existence and asymptotics of smooth solutions of system (1)-(4) with the initial conditions: The system (1)-(4), the conservation laws of mass, momentum, energy, and the radiative transfer equation, describe a non-equilibrium regime where the state of the radiation is determined by the transport equation (4). Since its physical importance, complexity, rich phenomena, and mathematical challenges, there is large of literature on the studies of radiation hydrodynamics from the mathematical/physical point of view, see, for example, [2,4,7,11,27] and [12,13,30]. Let us introduce some related mathematical result in radiation hydrodynamics. In [20], Lin, Coulombel and Goudon considered a situation where the gas is not in thermodynamical equilibrium with the radiation. They showed the existence of smooth traveling waves, called "shock profiles", when the strength of shock is small. The governed system studied in [20] reads as follows: which describes the interaction between an inviscid gas and photons. Here, P = Rρθ and E = e + u 2 /2. While, system (5) can be seen as a inviscid flow with no heatconducting and stationary radiation field case of (1)-(4). In particular, (5) can be simplified to the radiating gases model [15,16] as the following: which is indeed a hyperbolic-elliptic coupled system and can be recasted as: by introducing the convolution operator The thorough study on (6) motivated a lot of works, see, for example, [19,28] and the references cited therein. More models and results of radiation hydrodynamics can see [5,6,26].
Comparing the equations (1)-(4) with the systems (5)-(6), we find that the mathematical model problem (1)-(4) under the present consideration is more physically valid in radiation hydrodynamics, since the state of radiation is now described by the equation of radiative transfer and the effects of radiation on the full dynamics are completely taken into account.
The proof of Theorem 1.1 is based on the careful energy methods and the Fourier multiplier technique. The first key step is to establish the global a priori high order Sobolev's energy estimates in time by using the careful energy methods. The second key step is to obtain the L p − L q time decay rate of the linearized operator for the (7)-(10) by using the Fourier technique. The third step of the proof is to obtain the time decay rate in Theorem 1.1 by combining the previous two steps and apply the energy estimate technique to the nonlinear problem, whose solutions can be represented by the solutions-semigroup operator for the linearized problem by using the Duhamel Principle. We notice that (7)-(10) are Navier-Stokes coupled with parabolic type equations. When we establish both a priori energy estimates in step 1 and L p − L q time decay rate in step 2, different from general compressible Navier-Stokes equations, in our system including some low-order terms like 4Θ 1+ , η 1+ in (9) and 4Θ, η in (10). There terms may lead to the solutions of which H 3 norm can not be control by the initial data. To overcome there difficulties, we shall modify some method motivated by [21,22] and [8,9] to deal with the compressible Navier-Stokes equations and other related models. Moreover, we need to construct some novel functionals which coorporate there low-order terms as a whole new one. We need to obtain more subtle energy estimates. The remainder of this paper is organized as follows. In the next section, we derive the uniform-in-time a priori estimates and then establish the existence of global solution. In Section 3, we prove the rate of convergence of solutions. In Section 4, we adapt our proof to the periodic domain case. Through the paper, we use · to denote norm L 2 (R 3 ), C denotes a positive (generally large) constant and γ a positive (generally small) constant, where both C and γ may take different values in different places. A ∼ B means CA ≤ B ≤ 1 C A for a generic constant C > 0.
2. Global existence. In this section, we shall establish the global existence of classical solutions to Cauchy problem (7)- (11) in the whole space R 3 . At first, we give the uniform a priori estimates.
2.1. A priori estimates. We will show the uniform-in-time a priori estimates in the whole space R 3 under the assumption where 0 < δ < 1 is a generic constant small enough and ( , u, Θ, η) is the smooth solution to the Cauchy problem (7)-(11) on 0 ≤ t < T for T > 0. First, we introduce two useful lemma: There exist a positive constant C, such that for any f, g ∈ H 3 (R 3 ) and any multi-index α with 1 ≤ |α| ≤ 3, where Then we begin to give the priori estimate of , u, Θ, η.
Lemma 2.3. For smooth solutions of the system (7)-(11), we have 1 2 for any 0 ≤ t ≤ T and any T > 0 with C and γ not depending on T .
Proof. Multiplying (7)-(10) by 4 , 4u, 4Θ and η respectively and then taking integration and summation, one has 1 2 Using Hölder's, Sobolev's inequalities, for I 1 to I 11 , we have For the last two terms, under the assumption (13), we have and similarly, we get Then, (14) follows by plugging all estimates above into (15), and hence Lemma 2.3 is proved.
Lemma 2.4. For smooth solutions of the system (7)-(11), we have for any 0 ≤ t ≤ T and any T > 0 with C and γ not depending on T .
For I 1 , I 2 , I 7 , I 8 , I 14 , I 18 , with help of Lemma 14, we obtain For I 10 , we have where the first inequality follows that for β < α, and Sobolev's and Young's inequalities were further used. Similarly, we have Using Hölder's, Soblev's, Young's inequalities and Lemma 13, we easily get the following bounds . For the last two terms, under the assumption (13), we get Plugging these estimates into (17) and taking the sum over 1 ≤ |α| ≤ 3, (16) follows and thus Lemma 2.4 is proved.
At last, we give the estimate of ∇ .
Lemma 2.5. For smooth solutions of the system (7)-(11), we have d dt for any 0 ≤ t ≤ T and any T > 0 with C and γ not depending on T .

Proof of global existence.
It is now immediate to obtain the global a priori estimates. In fact, define the temporal energy functional and the corresponding dissipation rate functional where 0 < τ 1 1 are constant. Notice that since τ 1 > 0 are sufficiently small, under the assumption (13), it holds that H 3 , uniformly for all 0 ≤ t < T . Moreover, by suitably choosing constant τ 1 , the sum of equations (14), (16) for all 0 ≤ t < T . By (13), one has E 1/2 (t) + E(t) ≤ C δ + δ 2 . Thus, as long as 0 < δ < 1 is small enough, the time integration of (22) yields for all 0 ≤ t < T . Besides, (13) can be justified by choosing sufficiently small. For brevity, the proof for local existence of smooth solutions is omitted. Then, the global existence of solutions follows by the obtained global a priori estimates as well the continuity argument, and also (23) holds true for all t ≥ 0.
3. Time-decay of solutions. In this section, so as to obtain the time-decay rates of solutions to the nonlinear system (7)-(10), firstly, we consider the following initial problem on the linearized homogeneous equations corresponding to system (7)-(10): with initial data In this section, we use U (t) = ( (t), u(t), Θ(t), η(t)) to denote the solution of system (24)- (28), and denote U 0 = ( 0 , u 0 , Θ 0 , η 0 ). Define A(t) to be the solution operator of (24)-(28), then, U (t) can be presented as We can now show the following uniform estimates on U (t).
We now need two technical lemmas for the later proof.

4.
The periodic case. In this section, we consider the spatial domain Ω = T 3 . It is easy to check that for smooth solution of the system (1)-(4), the following quantities are conserved: and by the assumption (12), it follows that for all t ≥ 0. Now, we begin to prove Theorem 1.2. Here, we only give the proof of the global a priori estimates. First, we need to find out the zero-order dissipation of ( , u, Θ, η), by using the conservation law (40) and with the help of the Poincaré inequality, we have and Θ + η L 2 ≤ 1 2 (1 + )|u| 2 + + Θ + Θ + η L 2 Let the temporal energy functional E(t) and the corresponding dissipation rate functional D(t) be defined in the same way as in (20), (21) for the case of the whole space Ω = R 3 . Then, the similar process by making the energy estimates lead to Define where 0 < τ 3 , τ 4 1 are constants. Notice , uniformly for all t ≥ 0. Moreover, by choosing 0 < τ 3 , τ 4 1 suitably small, it follows from (41)-(44), we conclude that In fact, E(t) is small enough uniformly in time, which implies for all t ≥ 0. Applying Gronwall's inequality to (45), one has which gives the desired exponential decay of E(t) ∼ , u, Θ, η 2 H 3 . The proof of Theorem 1.2 is complete.
λ and µ are the viscosity coefficients of the fluid satisfying 2λ + µ > 0. E r , F r and P r represent the radiative energy density, radiative flux and the radiative pressure respectively defined by By using (46) and (48), (47) can rewrite as ΩF (x, t, ν, Ω)dΩ, where F (x, t, ν, Ω) = S(ν) − σ a (ν)I(ν, Ω) We notice that (46)-(47) are integro-differential equations, and have complex structure. It is so difficult to solve both numerically and analytically. Practical simplified models are introduced in some physical regions. From the physical and numerical points of view, there models can approximate the general equations of radiation hydrodynamics (46)-(47) very well in some particular physical situations. In particular, non-local thermodynamics equilibrium (non-LTE) assumption and the Eddington approximation or diffusion approximation model are mainly studied in this paper.
First we introduce the non-LTE assumption, we consider the manifestation in the equation of transfer of the quantum statistics (i.e. (46)) obeyed by photons. Since photons are bosons, both the processes of emissions and scattering are enhanced by the number of photons already in the final state following the interaction. This enhancement is generally referred to as resulting from "induced processes". The quantitative statement of this enhancement is simply stated as: If Z represents the basic probability of a photon event (emission or scattering, i.e S(ν) or σ s (ν)) then, due to induced effects, the actual probability Z is given by where ψ is the number of photons in the final state of the transition. In "induced processes" case, ψ = c 2 2hν 3 I(ν, Ω), and thus where h is the Planck constant. To see the effect of the non-LTE assumption on the equation (46), it is convenient to eliminate S, σ a in (46) in favor of B and σ a defined by the relationships S(ν) = σ a B(ν), σ a (ν) = σ a 1 + c 2 B(ν) 2hν 3 , and assume that σ s = 0, where B is Planck function, which is describing the (isotropic) specific intensity of radiation in the case of thermal equilibrium (cf. [11,25]), defined by B(θ) := B(θ, ν) = αν 3 (e βν/θ − 1) −1 , where α and β are some positive physical constants. Thus, from the "induced processes" and the non-LTE assumption together, S(ν), σ a (ν) in (46) can be written as S(ν) = σ a B(ν) 1 + c 2 I(ν, Ω) 2hν 3 , σ a (ν) = σ a 1 + c 2 B(ν) 2hν 3 .
Then, we can rewrite (46) and (49) as Ωσ a (B(ν) − I)dΩ, The basic assumption underlying the classical diffusion or Eddington, descrtption of transfer is that the angular dependence of the specific intensity can be represented by the first two terms in aspherical harmonic expansion. That is, it is assumed that I(x, t, ν, Ω) = 1 4π I 0 (x, t, ν) + 3 4π Ω · I 1 (x, t, ν).
The truncated spherical harmonic representation, (51) is only strictly valid if |I 1 | I 0 , and I 0 , I 1 satisfied the following Fick's Law: where D is the diffusion coefficient. In this paper, we consider a physical case that the energy transfer of photons to plasmas (with subcritical density) completely dominates a process. Therefore, the effect of radiation on the momentum could 2062 PENG JIANG be neglected in the process. Let n := n(x, t) = ∞ 0 I 0 (x, t, ν)dν, assume absorption coefficient σ a , diffusion coefficient D are positive constant, and notice that we integrate the fourth equation of (50) over all solid angle. Then, (50) can rewrite as ρ t + div(ρu) = 0, (ρu) t + ∇P m + div (ρu ⊗ u) = divS, 1 2 ρu 2 + E m t + div ( 1 2 ρu 2 + E m + P m )u = div(Su + κ∇θ) −σθ 4 + σ a n, 1 c n t − D∆n =σθ 4 − σ a n, where σ is a positive constant defined by 0 < σ = 4πσ a αβ −4 ∞ 0 s 3 e s − 1 ds < ∞.
In this paper, we consider only polytropic ideal gases, namely, E m = c ν ρθ, P m = Rρθ.
For simplicity of the presentation and without loss of generality, we assume the positive constants in (52) that c = D =σ = σ a = 1, then, we obtain the diffusion approximation model in radiation hydrodynamics of the form (1)-(4).