Global existence of a radiative Euler system coupled to an electromagnetic field

Abstract 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. Assuming smallness hypotheses for the data, we prove that the problem admits a unique global smooth solution and study its asymptotics.


Introduction
In [3], after the studies of Lowrie, Morel and Hittinger [15] and 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 for the evolution of the density ̺ = ̺(t, x), the velocity field u = u(t, x), and the absolute temperature ϑ = ϑ(t, x), and where radiation is described in the limit by an extra temperature T r = T r (t, x).All of these quantities are functions of the time t and the Eulerian spatial coordinate x ∈ R 3 .
In [3] we proved that the associated Cauchy problem admits a unique global smooth solution, provided that the data are small enough perturbations of a constant state.
In [4] we coupled the previous model to the electromagnetic field through the so called magnetohydrodynamic (MHD) approximation, in presence of thermal and radiative dissipation.Hereafter we consider the perfect non-isentropic Euler-Maxwell's system and we also consider a radiative coupling through a pure convective transport equation for the radiation (without dissipation).Then we deal with a pure hyperbolic system with partial relaxation (damping on velocity).More specifically the system of equations to be studied for the unknowns (̺, u, ϑ, E r , B, E) reads ∂ t B + curl x E = 0, (1.5) div x B = 0, (1.7) where ̺ is the density, u the velocity, ϑ the temperature of matter, E = 1 2 | u| 2 + e(̺, ϑ) is the total mechanical energy, 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. Finally E is the electric field and B is the magnetic induction, We assume that the pressure p(̺, ϑ) and the internal energy e(̺, ϑ) are positive smooth functions of their arguments with and we also suppose for simplicity that ν = 1 τ (where τ > 0 is a momentum-relaxation time), µ, σ a and a are positive constants.
A simplification appears if one observes that, provided that equations (1.7) and (1.8) are satisfied at t = 0, they are satisfied for any time t > 0 and consequently they can be discarded from the analysis below.
Notice that the reduced system (1.1)-(1.4) is the non equilibrium regime of radiation hydrodynamics introduced by Lowrie, Morel and Hittinger [15] and more recently by Buet and Després [5], and studied mathematically by Blanc, Ducomet and Nečasová [3].Extending this last analysis, our goal in this work is to prove global existence of solutions for the system (1.1) -(1.8) when data are sufficiently close to an equilibrium state, and study their large time behaviour.
In the following we show that the ideas used by Y. Ueda, S. Wang and S. Kawashima in [19] [20] in the isentropic case can be extended to the (radiative) non isentropic system (1.1-1.6).To this purpose we follow the following plan: in Section 2 we present the main results, then (Section 3) we prove wellposedness of system (1.1-1.6).Finally in Section 4 we prove the large time asymptotics of the solution.

Main results
We are going to prove that system (1.1)-(1.8)has a global smooth solution close to any equilibrium state.Namely we have Theorem 2.1.Let ̺, 0, ϑ, E r , B, 0 be a constant state with ̺ > 0, ϑ > 0 and E r > 0 with compatibility condition E r = aϑ 4 and suppose that d ≥ 3.
There exists ε > 0 such that, for any initial state ̺ 0 , u 0 , ϑ 0 , E 0 r , B 0 , E 0 satisfying there exists a unique global solution ̺, u, ϑ, E r , B, E to (1.1)-(1.8),such that In addition, this solution satisfies the following energy inequality: for some constant C > 0 which does not depend on t.
The large time behaviour of the solution is described as follows Remark 2.1.Note that, due to lack of dissipation by viscous, thermal and radiative fluxes, the Kawashima-Shizuta stability criterion (see [18] and [1]) is not satisfied for the system under study and techniques of [13] relying on the existence of a compensating matrix do not apply.However we will check that radiative sources play the role of relaxation terms for temperature and radiative energy and will lead to global existence for the system.
3 Global existence

A priori estimates
Multiplying (1.2) by u, (1.5) by B, (1.6) by E and adding the result to equations (1.3) and (1.4) we get the total energy conservation law The internal energy equation is and dividing it by ϑ, we get the entropy equation for matter So adding (3.4) and (3.2) we obtain (3.5) Subtracting (3.5) from (3.1) and using the conservation of mass, we get Introducing the Helmholtz functions H ϑ (̺, ϑ) := ̺ e − ϑs and H r,ϑ (T r ) := E r − ϑS r , we check that the quantities ) are non-negative and strictly coercive functions reaching zero minima at the equilibrium state (̺, ϑ, E r ).
Lemma 1.Let ̺ and ϑ = T r be given positive constants.Let O 1 and O 2 be the sets defined by There exist positive constants ) ) Proof: 1. Point 1 is proved in [8] and we only sketch the proof for convenience.According to the decomposition where , one checks that F is strictly convex and reaches a zero minimum at ̺, while G is strictly decreasing for ϑ < ϑ and strictly increasing for ϑ > ϑ, according to the standard thermodynamic stability properties [8].
Computing the derivatives of H ϑ leads directly to the estimate (3.9).

Point 2 follows after properties of
Using the previous entropy properties, we have the energy estimate Proposition 3.1.Let the assumptions of Theorem 2.1 be satisfied with for some t > 0.Then, one gets for a constant C 0 > 0 (3.11) Proof: Defining we multiply (3.5) by ϑ, and subtract the result to (3.1).Integrating over [0, t] × R 3 , we find Applying Lemma 1, we find (3.11).
Defining for any d ≥ 3 the auxiliary quantities , we can bound the spatial derivatives as follows Proposition 3.2.Assume that the hypotheses of Theorem 2.1 are satisfied.Then, we have for a C 0 > 0 Proof: Rewriting the system (1.1)-(1.6) in the form and applying ∂ ℓ x to this system, we get Then taking the scalar product of each of the previous equations respectively by and ∂ ℓ x E and adding the resulting equations, we get where Integrating (3.15) on space, one gets Integrating now with respect to t and summing on ℓ with |ℓ| ≤ d, we get Observing that and that, using commutator estimates (see Moser-type calculus inequalities in [16]) we see that Then integrating with respect to time for any |ℓ| ≤ d.In the same stroke, we estimate for any |ℓ| ≤ d.
The above results, together with (3.11), allow to derive the following energy bound: Corollary 3.1.Assume that the assumptions of Proposition 3.1 are satisfied.Then Our goal is now to derive bounds for the integrals in the right-hand and left-hand sides of equation (3.16).For that purpose we adapt the results of Ueda, Wang and Kawashima [19].
Lemma 2. Under the same assumptions as in Theorem 2.1, and supposing that d ≥ 3, we have the following estimate for any ε > 0

.17)
Proof: We linearize the principal part of the system (1.1)-(1.2)-(1.3)as follows with coefficients and sources where Rearranging the left hand side of (3.24) we get where Integrating (3.25) over space and using Young's inequality, we find In fact one obtains in the same way estimates for the derivatives of V .Namely, applying ∂ ℓ x to the system (3.18-3.23),we get where and x E r .Integrating (3.27) over space and time, we find Observing that and summing (3.28) on ℓ for 1 where we used Corollary 3.1.
Let us estimate the last integral in (3.28).we have Plugging bounds (3.30) into this last inequality gives which ends the proof of Lemma 2.
Finally we check from [19] (see Lemma 4.4) that the following result for the Maxwell's system holds true for our system with a similar proof Lemma 3.Under the same assumptions as in Theorem 2.1, and supposing that d ≥ 3, for any ε > 0 the following estimate (here, we set V = (̺, u, ϑ, E r , B, E) T ) holds where Integrating in space we get Integrating on time and summing for 1 ≤ |ℓ| ≤ d − 2, we have where we used the bound , obtained in the same way as in the proof of Lemma 2, which ends the proof of Lemma 3.
We are now in position to conclude with the proofs of Theorems 2.1 and 2.2.

Proof of Theorem 2.1:
We first point out that local existence for the hyperbolic system (1.1)-(1.6)may be proved using standard fixed-point methods.We refer to [16] for the proof.Now plugging (3.31) into (3.17) with ε small enough, we get Putting this last estimate into (3.31)we find

.33)
Then from (3.17), (3.32) and (3.33) we get , and that, provided that d ≥ 2 one has I(t) ≤ CD(t), for some positive constant C, we see that 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 where N (t) is defined by We are thus reduced to prove that N is bounded.For this purpose, we use the argument used in [3].

Large time behaviour
We have the following analogous of Proposition 3.1 for time derivatives 3) on [0, t], for some t > 0.Then, one gets for a constant C 0 > 0

. 1 )aT 3 r
Introducing the entropy s of the fluid by the Gibbs law ϑds = de + pd1  ̺ and denoting by S r :=4  3 the radiative entropy, equation (1.4) rewrites