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Global existence of a diffusion limit with damping for the compressible radiative Euler system coupled to an electromagnetic field

We study the Cauchy problem for a system of equations corresponding to a singular limit of radiative hydrodynamics, namely the 3D radiative compressible Euler system coupled to an electromagnetic field through the MHD approximation. Assuming the presence of damping together with suitable smallness hypotheses for the data, we prove that this problem admits a unique global smooth solution.


Introduction
In [4], after the study of Buet and Després [5] we considered a singular limit for a compressible inviscid radiative flow where the motion of the fluid is given by the Euler system with damping for the evolution of the density ̺ = ̺(t, x), the velocity field u = u(t, x), and the absolute temperature ϑ = ϑ(t, x) as functions of the time t and the Eulerian spatial coordinate x ∈ R 3 and we proved that the associated Cauchy problems admits a unique global smooth solution, provided that the data are small enough.
In the present work we couple the previous model to an electromagnetic field through the so called magnetohydrodynamic approximation (MHD) [3].
Recall briefly that Maxwell's electromagnetic theory relies on Ampère-Maxwell equation where D = ǫ E is the electric induction and H is the magnetic field, Faraday's law where B = µ H is the magnetic induction. Here, the constant µ > 0 stands for the permeability of free space. The two last laws are Coulomb's law div x D = q, (1.3) where q is the electric charge density, and Gauss's law div x B = 0. (1.4) We first suppose that the electric current density J is related to the electric field E and the macroscopic fluid velocity u via Ohm's law J = σ( E + u × B), (1.5) where σ is the electrical conductivity of the fluid. The magnetic force acting on the fluid (Lorentz's force) f m and the magnetic energy supply E m are given by The MHD approximation consists in neglecting the displacement current ∂ t D (for the electric induction given by D = ǫ E) in Ampère-Maxwell equation and supposing that the charge q is negligible, so we obtain µ J = curl x B, µ > 0, (1.7) where, as mentioned above, the constant µ is the permeability of free space. Accordingly, equation (1.2) can be written [6] in the form where λ = (µσ) −1 is the magnetic diffusivity of the fluid. Finally, from Faraday's law we get Concerning radiation, we consider the non equilibrium diffusion regime where radiation appears through an extra equation of parabolic type for the radiative temperature which is a priori different from the matter temperature. More specifically the system of equations to be studied for the five unknowns (̺, u, ϑ, E r , B) reads where B is a divergence-free vector field, E = 1 2 | u| 2 + e(̺, ϑ), E r is the radiative energy related to the temperature of radiation T r by E r = aT 4 r and p r is the radiative pressure given by p r = 1 3 aT 4 r = 1 3 E r , with a > 0. We have also supposed for simplicity that µ, σ a , σ s , σ and a are positive constants which implies in particular that curl x Extending the analysis of [4] and using stability arguments introduced by Beauchard and Zuazua in [1], our goal is to prove global existence of solutions for the system (1.10) -(1.14) when data are sufficiently close to an equilibrium state.
The plan of the paper is as follows: in Section 2 we state our main result (Theorem 2.1) then in Section 3 we study the MHD model and prove our first result.

Main result
Hypotheses imposed on constitutive relations are motivated by the general existence theory for the Euler-Fourier system developed in [20,21]. Hypotheses on transport coefficients are reasonable physical assumptions for the radiative part [17,19]. We impose that pressure p(̺, ϑ) > 0, internal energy e(̺, ϑ) > 0 and specific entropy s(̺, ϑ) are smooth functions of their arguments. Moreover, we impose the following monotony assumptions: Moreover, in our simplified setting, transport coefficients κ, σ a , σ s and the Planck's coefficient a are supposed to be fixed positive numbers. Finally the damping term with coefficient ν > 0 of Darcy type can be interpreted here as a diffusion of a light gas into a heavy one.
We are going to prove that, under the above structural assumptions on the equation of state, system (1.10)-(1.14) has a global smooth solution close to any equilibrium state.
3 The Euler-MHD system

The linearized Euler-MHD system
Multiplying (1.11) by u and using (1.10) we get Subtracting this relation from (1.12), using the definition C v = ∂ ϑ e and the thermodynamical identity ∂ ̺ e = 1 ̺ 2 (p − ϑ∂ ϑ p) (Maxwell's relation), equation (1.12) can be replaced by an equation for temperature (3.1) Linearizing the system (1.10)(1.11)(3.1)(1.13)(1.14) around the constant state (̺, 0, ϑ, E r , B) with the compatibility condition E r = aϑ 4 and putting ̺ = r + ̺, ϑ = T + ϑ, E r = e r + E r and B = b + B we get (3.6) Using the vector notation U := with In order to apply the Kreiss theory we have to put the system (3.7) in a symmetric form [2]. For that purpose it is sufficient to consider a diagonal symmetrizer Multiplying the first equation (3.7) by A 0 on the left, we get where the matrices A j = A 0 A j are symmetric, for all j = 1, 2, 3. More specifically, The hyperbolic part of system (3.9) is now symmetric while its symmetric dissipative part is given by where one checks the positiveness condition of B t X BX ≥ 0, for any vector X ∈ R 9 .
Applying the Fourier transform in x to (3.9) we get where (3.14) Solving this equation with initial condition U 0 (ξ) we get In the strictly hyperbolic case D = 0, under the Kalman rank condition [12] for the pair (A(ξ), B), it can be proved [1] that Observing the partially parabolic character of the system, one can expect a similar result when D = 0 with a parabolic smoothing effect at low frequencies and an extra damping in the high frequency regime. Taking benefit of the damping, we can use the Shizuta-Kawashima condition (SK) [22] which applies to the previous system. Following the arguments of Beauchard and Zuazua [1], we have Lemma 1. For any ξ ∈ S 2 , a necessary and sufficient condition for the matrices B(ξ) and A(ξ) defined by (3.13) and (3.14) satisfy the Shizuta-Kawashima condition (SK): is that ν > 0. Proof: 1. If ν = 0. One checks that ker B(ξ) is the 1-dimensional subspace spanned by the vector (1, 0, 0, 0, 0, 0, 0, 0, 0).

Entropy properties
Adding equations (1.9), (1.12) and (1.13) we get Introducing the entropy s of the fluid by the Gibbs law ϑds = de + pd 1 ̺ and denoting by S r := 4 3 aT 3 r the radiative entropy, equation (1.13) rewrites Replacing equation (1.12) by the internal energy equation and dividing it by ϑ, we may write an entropy equation for matter
We have the decomposition . Using Gibbs law , which is positive according to (2.1). Hence, F is strictly convex and reaches a zero minimum at ̺. Turning to G, we have, still using Gibbs law, ϑ). Thus, using (2.1) again, we infer that G is strictly decreasing for ϑ < ϑ and strictly increasing for ϑ > ϑ. Computing the derivatives of H ϑ leads directly to the estimate (3.24).

Point 2 follows from the properties of the function
From this simple result, we can obtain an identity leading to energy estimates. In fact, multiplying (3.21) by ϑ, subtracting the result to (3.17) and using the conservation of mass, we get (3.26) In the sequel, we define V = ρ, u, ϑ, E r , B Recall that T r = E 1/4 r a −1/4 . Note also that, since div x B = 0, we have

L ∞ (H d ) estimates
Using these entropy properties, we are going to prove the following result: where the function C is non-decreasing.
Proof: Following the proof of [13, Lemma 3.1] we define We multiply (3.21) by ϑ, and subtract the result to (3.17). Integrating over [0, t] × R 3 , we find Defining and applying Lemma 2, we find that where C : R + → R + is non-decreasing. Equation (3.28) allows to replace curl x B by ∇ x B in the above estimate. Finally, we point out that, since d > 7/2 > 3/2, due to Sobolev embeddings, there exists a universal constante C 0 such that M (t) ≤ C 0 N (t). Since C is non-decreasing, this proves (3.29).
The system satisfied by V may be written formally where It is possible to symmetrize this nonlinear system in the same spirit as what we have done for the linearized system (3.7). However, we do not need to do so here. So we write We first observe that these matrices are Lipschitz continuous with respect to V , away from ̺ = 0 and ϑ = 0 and also that the matrices B and D have, respectively, the same structure as those defined in (3.10). Note also that, since d − 1 > 5/2 = 3/2 + 1, Sobolev embeddings imply that H d−1 R 3 is an algebra. Therefore, we have which proves (3.32).
Next, we bound the spatial derivatives as follows: Proof: Here, we need to symmetrize the nonlinear system. For this purpose, we multiply (3.33) on the right by the matrix This gives to (3.36) then taking the scalar product with the vector ∂ k x V , and integrating over [0, t] × R 3 , we find We estimate separately each term of the right-hand side. First, we have where we have used Sobolev embeddings and the fact that d > 7/2. A similar computation gives We estimate I 3 by applying Cauchy-Schwarz inequality: Then, we apply the same estimate for commutators and composition of functions (see [15, Proposition 2.1]), and |k| ≤ d: Moreover, we have and Hence, I 3 satisfies t 0 Proof: As a first step, we apply the symmetrizer of the linearized system (3.7) (which leads to (3.9)) to the nonlinear system (1.10)-(1.11)-(1.12), which then reads Of course, this system is not symmetric. However, the corresponding linearized system (3.9) is symmetric. Next, we rewrite the nonlinear system by setting U = V − V : Therefore, multiplying this system on the left by A 0 (V ) A 0 (V ) −1 , we find where We apply the Fourier transform to (3.39), and then multiply on the left by −i U * K ξ |ξ| , where * denotes the transpose of the complex conjugate, and K is the compensating matrix (see Proposition 3.1). Taking the real part of the result, we infer for any ε > 0. We choose ε small enough, insert (3.41)-(3.42)-(3.43) into (3.40), and find In order to prove global existence, we argue by contradiction, and assume that T c > 0 is the maximum time existence. Then, we necessarily have lim t→Tc N (t) = +∞, where N (t) is defined by (3.27). We are thus reduced to prove that N is bounded. For this purpose, we use the method of [13], which was also used in [18]. First note that, due to Proposition 3.5 on the one hand, and to Proposition 3.6 on the other hand, we know that there exists a non-decreasing continuous function C : R + → R + such that ∀T ∈ [0, T c ], N (t) 2 ≤ C(N (t)) N (0) 2 + N (t) 3 . (3.47) Hence, setting N (0) = ε, we have N (t) 2 ε 2 + N (t) 3 ≤ C(N (t)), (3.48) Studying the variation of φ(N ) = N 2 / ε 2 + N 3 , we see that φ ′ (0) = 0, that φ is increasing on the interval 0, 2ε 2 1/3 and decreasing on the interval 2ε 2 1/3 , +∞ . Hence, Hence, the function C being independent of ε, we can choose ε small enough to have φ(N ) ≤ C(N ) for all N ∈ [0, N * ], where N * > 0. Since C is continuous, (3.48) implies that N ≤ N * . This is clearly in contradiction with (3.47).