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

Xavier Blanc; Bernard Ducomet; Šárka Nečasová

doi:10.1515/anona-2017-0117

Article Open Access

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

Xavier Blanc , Bernard Ducomet and Šárka Nečasová

Published/Copyright: March 21, 2018

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From the journal Advances in Nonlinear Analysis Volume 8 Issue 1

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.

Keywords: Compressible; Euler; radiation hydrodynamics

MSC 2010: 35Q30; 76N10

1 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 ϱ = ϱ ⁢ ( 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 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 ( ϱ , u → , ϑ , E r , B → , E → ) reads

(1.1) ∂ t ⁡ ϱ + div x ⁡ ( ϱ ⁢ u → ) = 0 ,

(1.2) ∂ t ⁡ ( ϱ ⁢ u → ) + div x ⁡ ( ϱ ⁢ u → ⊗ u → ) + ∇ x ⁡ ( p + p r ) = - ϱ ⁢ ( E → + u → × B → ) - ν ⁢ ϱ ⁢ u → ,

(1.3) ∂ t ⁡ ( ϱ ⁢ E ) + div x ⁡ ( ( ϱ ⁢ E + p ) ⁢ u → ) + u → ⋅ ∇ x ⁡ p r = - σ a ⁢ ( a ⁢ ϑ 4 - E r ) - ϱ ⁢ E → ⋅ u → ,

(1.4) ∂ t ⁡ E r + div x ⁡ ( E r ⁢ u → ) + p r ⁢ div x ⁡ u → = - σ a ⁢ ( E r - a ⁢ ϑ 4 ) ,

(1.5) ∂ t ⁡ B → + curl x ⁡ E → = 0 ,

(1.6) ∂ t ⁡ E → - curl x ⁡ B → = ϱ ⁢ u → ,

(1.7) div x ⁡ B → = 0 ,

(1.8) div x ⁡ E → = ϱ ¯ - ϱ ,

where ϱ is the density, u → the velocity, ϑ the temperature of matter, E = 1 2 ⁢ | u → | 2 + e ⁢ ( ϱ , ϑ ) is the total mechanical energy, E r is the radiative energy related to the temperature of radiation T r by E r = a ⁢ T r 4 , and p r is the radiative pressure given by p r = 1 3 ⁢ a ⁢ T r 4 = 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 ⁢ ( ϱ , ϑ ) and the internal energy e ⁢ ( ϱ , ϑ ) are positive smooth functions of their arguments, with

C v := ∂ ⁡ e ∂ ⁡ ϑ > 0 , ∂ ⁡ p ∂ ⁡ ϱ > 0 ,

and we also suppose for simplicity that ν = 1 τ (where τ > 0 is a momentum-relaxation time) and 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 system (1.1)–(1.8) when the data are sufficiently close to an equilibrium state, and study their large time behavior.

For the sake of completeness, we mention that related non-isentropic Euler–Maxwell systems have been the subject of a number of studies in the recent past. Let us quote the recent works [9, 10, 12, 14, 18, 21].

In the following, we show that the ideas used by Ueda, Wang and Kawashima in [20, 19] in the isentropic case can be extended to the (radiative) non-isentropic system (1.1)–(1.6). To this end, we follow the following plan. In Section 2 we present the main results and then, in Section 3, we prove the well-posedness of system (1.1)–(1.6). Finally, in Section 4, we prove the large time asymptotics of the solution.

2 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 the following theorem.

Theorem 2.1.

Let ( ϱ ¯ , 0 , ϑ ¯ , E r ¯ , B → ¯ , 0 ) be a constant state, with ϱ ¯ > 0 , ϑ ¯ > 0 and E r ¯ > 0 , and compatibility condition E r ¯ = a ⁢ ϑ ¯ 4 , and suppose that d ≥ 3 . There exists ε > 0 such that for any initial state ( ϱ 0 , u → 0 , ϑ 0 , E r 0 , B → 0 , E → 0 ) satisfying

div x ⁡ B → 0 = ϱ 0 - ϱ ¯ , div x ⁡ B → 0 = 0 , ( ϱ 0 - ϱ ¯ , u → 0 , ϑ 0 - ϑ ¯ , E r 0 - E r ¯ , B → 0 - B → ¯ , E → 0 ) ∈ H d

and

∥ ( ϱ 0 , u → 0 , ϑ 0 , E r 0 , B → 0 , E → 0 ) - ( ϱ ¯ , 0 , ϑ ¯ , E r ¯ , B → ¯ , 0 ) ∥ H d ≤ ε ,

there exists a unique global solution ( ϱ , u → , ϑ , E r , B → , E → ) to (1.1)–(1.8) such that

( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E r ¯ , B → - B → ¯ , E → ) ∈ C ⁢ ( [ 0 , + ∞ ) ; H d ) ∩ C 1 ⁢ g ⁢ ( [ 0 , + ∞ ) ; H d - 1 ) .

In addition, this solution satisfies the following energy inequality:

∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E r ¯ , B → - B → ¯ , E → ) ⁢ ( t ) ∥ H d + ∫ 0 t ( ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E r ¯ ) ⁢ ( τ ) ∥ H d 2 + ∥ ∇ x ⁡ B → ⁢ ( τ ) ∥ H d - 2 2 + ∥ E → ⁢ ( τ ) ∥ H d - 1 2 ) ⁢ 𝑑 τ

(2.1) ≤ C ⁢ ∥ ( ϱ 0 - ϱ ¯ , 0 , ϑ 0 - ϑ ¯ , E r 0 - E r ¯ , B → 0 - B → ¯ , E → 0 ) ∥ H d 2

for some constant C > 0 which does not depend on t .

The large time behavior of the solution is described as follows.

Theorem 2.2.

Let d ≥ 3 . The unique global solution ( ϱ , u → , ϑ , E r , B → , E → ) to (1.1)–(1.8), defined in Theorem 2.1, converges to the constant state ( ϱ ¯ , 0 → , ϑ ¯ , E r ¯ , B → ¯ , 0 → ) uniformly in x ∈ R 3 as t → ∞ . More precisely,

∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E r ¯ , E → ) ⁢ ( t ) ∥ W d - 2 , ∞ → 0 as ⁢ t → ∞ .

Moreover, if d ≥ 4 , then

∥ ( B → - B → ¯ ) ⁢ ( t ) ∥ W d - 4 , ∞ → 0 as ⁢ t → ∞ ,

Remark 2.3.

Note that, due to lack of dissipation by viscous, thermal and radiative fluxes, the Kawashima–Shizuta stability criterion (see [17] and [1]) is not satisfied for the system under study, and the 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 the temperature and radiative energy and this will lead to global existence for the system.

3 Global existence

3.1 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

(3.1) ∂ t ⁡ ( 1 2 ⁢ ϱ ⁢ | u → | 2 + ϱ ⁢ e + E r + 1 2 ⁢ ( | B → | 2 + | E → | 2 ) ) + div x ⁡ ( ( ϱ ⁢ E + E r ) ⁢ u → + ( p + p r ) ⁢ u → + E → × B → ) = 0 .

Introducing the entropy s of the fluid by the Gibbs law ϑ ⁢ d ⁢ s = d ⁢ e + p ⁢ d ⁢ ( 1 ϱ ) and denoting by S r := 4 3 ⁢ a ⁢ T r 3 the radiative entropy, equation (1.4) is rewritten as

(3.2) ∂ t ⁡ S r + div x ⁡ ( S r ⁢ u → ) = - σ a ⁢ E r - a ⁢ ϑ 4 T r .

The internal energy equation is

∂ t ⁡ ( ϱ ⁢ e ) + div x ⁡ ( ϱ ⁢ e ⁢ u → ) + p ⁢ div x ⁡ u → - ν ⁢ ϱ ⁢ | u → | 2 = - σ a ⁢ ( a ⁢ ϑ 4 - E r ) ,

and by dividing it by ϑ , we get the entropy equation for matter

(3.3) ∂ t ⁡ ( ϱ ⁢ s ) + div x ⁡ ( ϱ ⁢ s ⁢ u → ) - ν ϑ ⁢ | u → | 2 = - σ a ⁢ a ⁢ ϑ 4 - E r ϑ .

So adding (3.3) and (3.2), we obtain

(3.4) ∂ t ⁡ ( ϱ ⁢ s + S r ) + div x ⁡ ( ( ϱ ⁢ s + S r ) ⁢ u → ) = a ⁢ σ a ϑ ⁢ T r ⁢ ( ϑ - T r ) 2 ⁢ ( ϑ + T r ) ⁢ ( ϑ 2 + T r 2 ) + ν ϑ ⁢ | u → | 2 .

By subtracting (3.4) from (3.1) and using the conservation of mass, we get

∂ t ⁡ ( 1 2 ⁢ ϱ ⁢ | u → | 2 + H ϑ ¯ ⁢ ( ϱ , ϑ ) - ( ϱ - ϱ ¯ ) ⁢ ∂ ϱ ⁡ H ϑ ¯ ⁢ ( ϱ ¯ , ϑ ¯ ) - H ϑ ¯ ⁢ ( ϱ ¯ , ϑ ¯ ) + H r , ϑ ¯ ⁢ ( T r ) + 1 2 ⁢ ( | B → - B → ¯ | 2 + | E → | 2 ) )

= div x ⁡ ( ( ϱ ⁢ E + E r ) ⁢ u → + ( p + p r ) ⁢ u → + ϑ ¯ ⁢ ( ϱ ⁢ s + S r ) ⁢ u → ) - ϑ ¯ ⁢ a ⁢ σ a ϑ ⁢ T r ⁢ ( ϑ - T r ) 2 ⁢ ( ϑ + T r ) ⁢ ( ϑ 2 + T r 2 ) - ν ϑ ⁢ | u → | 2 .

By introducing the Helmholtz functions

H ϑ ¯ ⁢ ( ϱ , ϑ ) := ϱ ⁢ ( e - ϑ ¯ ⁢ s ) and H r , ϑ ¯ ⁢ ( T r ) := E r - ϑ ¯ ⁢ S r ,

we check that the quantities H ϑ ¯ ⁢ ( ϱ , ϑ ) - ( ϱ - ϱ ¯ ) ⁢ ∂ ϱ ⁡ H ϑ ¯ ⁢ ( ϱ ¯ , ϑ ¯ ) - H ϑ ¯ ⁢ ( ϱ ¯ , ϑ ¯ ) and H r , ϑ ¯ ⁢ ( T r ) - H r , ϑ ¯ ⁢ ( T ¯ r ) are non-negative and strictly coercive functions reaching zero minima at the equilibrium state ( ϱ ¯ , ϑ ¯ , E ¯ r ) .

Lemma 3.1.

Let ϱ ¯ and ϑ ¯ = T ¯ r be given positive constants. Let O 1 and O 2 be the sets defined by

𝒪 1 := { ( ϱ , ϑ ) ∈ ℝ 2 : ϱ ¯ 2 < ϱ < 2 ⁢ ϱ ¯ , ϑ ¯ 2 < ϑ < 2 ⁢ ϑ ¯ } , 𝒪 2 := { T r ∈ ℝ : T ¯ r 2 < T r < 2 ⁢ T ¯ r } .

Then there exist positive constants C 1 , 2 ⁢ ( ϱ ¯ , ϑ ¯ ) and C 3 , 4 ⁢ ( T ¯ r ) such that

(3.5) C 1 ⁢ ( | ϱ - ϱ ¯ | 2 + | ϑ - ϑ ¯ | 2 ) ≤ H ϑ ¯ ⁢ ( ϱ , ϑ ) - ( ϱ - ϱ ¯ ) ⁢ ∂ ϱ ⁡ H ϑ ¯ ⁢ ( ϱ ¯ , ϑ ¯ ) - H ϑ ¯ ⁢ ( ϱ ¯ , ϑ ¯ ) ≤ C 2 ⁢ ( | ϱ - ϱ ¯ | 2 + | ϑ - ϑ ¯ | 2 )

for all ( ϱ , ϑ ) ∈ O 1 , and

C 3 ⁢ | T r - T ¯ r | 2 ≤ H r , ϑ ¯ ⁢ ( T r ) - H r , ϑ ¯ ⁢ ( T ¯ r ) ≤ C 4 ⁢ | T r - T ¯ r | 2

for all T r ∈ O 2 .

Proof.

The first assertion is proved in [8], and we only sketch the proof for convenience. According to the decomposition

ϱ → H ϑ ¯ ⁢ ( ϱ , ϑ ) - ( ϱ - ϱ ¯ ) ⁢ ∂ ϱ ⁡ H ϑ ¯ ⁢ ( ϱ ¯ , ϑ ¯ ) - H ϑ ¯ ⁢ ( ϱ ¯ , ϑ ¯ ) = ℱ ⁢ ( ϱ ) + 𝒢 ⁢ ( ϱ ) ,

where

ℱ ⁢ ( ϱ ) = H ϑ ¯ ⁢ ( ϱ , ϑ ¯ ) - ( ϱ - ϱ ¯ ) ⁢ ∂ ϱ ⁡ H ϑ ¯ ⁢ ( ϱ ¯ , ϑ ¯ ) - H ϑ ¯ ⁢ ( ϱ ¯ , ϑ ¯ ) and 𝒢 ⁢ ( ϱ ) = H ϑ ¯ ⁢ ( ϱ , ϑ ) - H ϑ ¯ ⁢ ( ϱ , ϑ ¯ ) ,

one checks that ℱ is strictly convex and reaches a zero minimum at ϱ ¯ , while 𝒢 is strictly decreasing for ϑ < ϑ ¯ and strictly increasing for ϑ > ϑ ¯ , according to the standard thermodynamic stability properties, see [8]. Computing the derivatives of H ϑ ¯ leads directly to estimate (3.5).

The second assertion follows from the properties of

x ↦ H r , ϑ ¯ ⁢ ( x ) - H r , ϑ ¯ ⁢ ( T r ) = a ⁢ x 3 ⁢ ( x - 4 3 ⁢ ϑ ¯ ) + a 3 ⁢ ϑ ¯ 4 . ∎

Using the previous entropy properties, we have the following energy estimate.

Proposition 3.2.

Let the assumptions of Theorem 2.1 be satisfied with

V = ( ϱ , u → , ϑ , E r , B → , E → ) 𝑎𝑛𝑑 V ¯ = ( ϱ ¯ , 0 , ϑ ¯ , E r ¯ , B → ¯ , 0 ) .

Consider a solution ( ϱ , u → , ϑ , E r , B → , E → ) of system (1.1)–(1.3) on [ 0 , t ] , for some t > 0 . Then, for a constant C 0 > 0 , one gets

(3.6) ∥ V ( t ) - V ¯ ∥ L 2 2 + ∫ 0 t ∥ u → ( τ ) ∥ L 2 2 d τ ≤ C 0 ∥ V 0 - V ¯ ∥ L 2 2 .

Proof.

We define

η ⁢ ( t , x ) = H ϑ ¯ ⁢ ( ϱ , ϑ ) - ( ϱ - ϱ ¯ ) ⁢ ∂ ϱ ⁡ H ϑ ¯ ⁢ ( ϱ ¯ , ϑ ¯ ) - H ϑ ¯ ⁢ ( ϱ ¯ , ϑ ¯ ) + H r , ϑ ¯ ⁢ ( T r ) ,

multiply (3.4) by ϑ ¯ , and subtract the result to (3.1). By integrating over [ 0 , t ] × ℝ 3 , we find

∫ ℝ 3 1 2 ⁢ ϱ ⁢ | u → | 2 + η ⁢ ( t , x ) + 1 2 ⁢ | B → - B → ¯ | 2 + 1 2 ⁢ | E → | 2 ⁢ d ⁢ x + ∫ 0 t ∫ ℝ 3 ϑ ¯ ϑ ⁢ ν ⁢ | u → | 2 ≤ ∫ ℝ 3 1 2 ⁢ ϱ 0 ⁢ | u → 0 | 2 ⁢ ( t ) + η ⁢ ( 0 , x ) + 1 2 ⁢ | B → 0 - B → ¯ | 2 + 1 2 ⁢ | E → 0 | 2 ⁢ d ⁢ x .

Applying Lemma 3.1 yields (3.6). ∎

By defining, for any d ≥ 3 , the auxiliary quantities

E ( t ) := sup 0 ≤ τ ≤ t ∥ ( ϱ - ϱ ¯ , u → , B → - B → ¯ , E → ) ( τ ) ∥ W 1 , ∞ ,

F ⁢ ( t ) := sup 0 ≤ τ ≤ t ⁡ ∥ ( V - V ¯ ) ⁢ ( τ ) ∥ H d ,

I 2 ⁢ ( t ) := ∫ 0 t ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L ∞ 2 ⁢ 𝑑 τ

and

D 2 ⁢ ( t ) := ∫ 0 t ( ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ⁢ ( τ ) ∥ H d 2 + ∥ E → ⁢ ( τ ) ∥ H d - 1 2 + ∥ ∂ x ⁡ B → ⁢ ( τ ) ∥ H d - 2 2 ) ⁢ 𝑑 τ ,

we can bound the spatial derivatives as follows.

Proposition 3.3.

Assume that the hypotheses of Theorem 2.1 are satisfied. Then, for C 0 > 0 , we have

∥ ∂ x V ( t ) ∥ H d - 1 2 + ∫ 0 t ∥ ∂ x u → ( τ ) ∥ H d - 1 2 d τ ≤ C 0 ∥ ∂ x V 0 ∥ H d - 1 2 + C 0 ( E ( t ) D ( t ) 2 + F ( t ) I ( t ) D ( t ) ) .

Proof.

By rewriting system (1.1)–(1.6) in the form

(3.7) { ∂ t ⁡ ϱ + u → ⋅ ∇ x ⁡ ϱ + ϱ ⁢ div x ⁡ u → = 0 , ∂ t ⁡ u → + ( u → ⋅ ∇ x ) ⁢ u → + p ϱ ϱ ⁢ ∇ x ⁡ ϱ + p ϑ ϱ ⁢ ∇ x ⁡ ϑ + 1 3 ⁢ a ⁢ ϱ ⁢ ∇ x ⁡ E r + E → + u → × B → ¯ + ν ⁢ u → = - u → × ( B → - B → ¯ ) , ∂ t ⁡ ϑ + ( u → ⋅ ∇ x ) ⁢ ϑ + ϑ ⁢ p ϑ ϱ ⁢ C v ⁢ div x ⁡ u → = - σ a ϱ ⁢ C v ⁢ ( a ⁢ ϑ 4 - E r ) , ∂ t ⁡ E r + ( u → ⋅ ∇ x ) ⁢ E r + 4 3 ⁢ E r ⁢ div x ⁡ u → = - σ a ⁢ ( E r - a ⁢ ϑ 4 ) , ∂ t ⁡ B → + curl x ⁡ E → = 0 , ∂ t ⁡ E → - curl x ⁡ B → - ϱ ¯ ⁢ u → = ( ϱ - ϱ ¯ ) ⁢ u → ,

and applying ∂ x ℓ to this system, we get

∂ t ⁡ ( ∂ x ℓ ⁡ ϱ ) + ( u → ⋅ ∇ x ) ⁢ ∂ x ℓ ⁡ ϱ + ϱ ⁢ div x ⁢ ∂ x ℓ ⁡ u → = F 1 ℓ ,

∂ t ⁡ ( ∂ x ℓ ⁡ u → ) + ( u → ⋅ ∇ x ) ⁢ ∂ x ℓ ⁡ u → + p ϱ ϱ ⁢ ∇ x ⁡ ∂ x ℓ ⁡ ϱ + p ϑ ϱ ⁢ ∇ x ⁡ ∂ x ℓ ⁡ ϑ + 1 3 ⁢ a ⁢ ϱ ⁢ ∇ x ⁡ ∂ x ℓ ⁡ E r + ∂ x ℓ ⁡ E → + ∂ x ℓ ⁡ u → × B → ¯ + ν ⁢ ∂ x ℓ ⁡ u → = - ∂ x ℓ ⁡ [ u → × ( B → - B → ¯ ) ] + F 2 ℓ ,

∂ t ⁡ ( ∂ x ℓ ⁡ ϑ ) + ( u → ⋅ ∇ x ) ⁢ ∂ x ℓ ⁡ ϑ + ϑ ⁢ p ϑ ϱ ⁢ C v ⁢ div x ⁢ ∂ x ℓ ⁡ u → = - ∂ x ℓ ⁡ [ σ a ϱ ⁢ C v ⁢ ( a ⁢ ϑ 4 - E r ) ] + F 3 ℓ ,

∂ t ⁡ ( ∂ x ℓ ⁡ E r ) + ( u → ⋅ ∇ x ) ⁢ ∂ x ℓ ⁡ E r + 4 3 ⁢ E r ⁢ div x ⁢ ∂ x ℓ ⁡ u → = - ∂ x ℓ ⁡ [ σ a ⁢ ( E r - a ⁢ ϑ 4 ) ] + F 4 ℓ ,

∂ t ⁡ ( ∂ x ℓ ⁡ B → ) + curl x ⁢ ∂ x ℓ ⁡ E → = 0 ,

∂ t ⁡ ( ∂ x ℓ ⁡ E → ) - curl x ⁢ ∂ x ℓ ⁡ B → - ϱ ¯ ⁢ ∂ x ℓ ⁡ u → = ∂ x ℓ ⁡ [ ( ϱ - ϱ ¯ ) ⁢ u → ] ,

where

F 1 ℓ := - [ ∂ x ℓ , u → ⋅ ∇ x ] ⁢ u → - [ ∂ x ℓ , ϱ ⁢ div x ] ⁢ u → ,

F 2 ℓ := - [ ∂ x ℓ , u → ⋅ ∇ x ] ⁢ u → - [ ∂ x ℓ , p ϱ ϱ ⁢ ∇ x ] ⁢ ϱ - [ ∂ x ℓ , p ϑ ϱ ⁢ ∇ x ] ⁢ ϑ - [ ∂ x ℓ , 1 3 ⁢ a ⁢ ϱ ⁢ ∇ x ] ⁢ E r ,

F 3 ℓ := - [ ∂ x ℓ , u → ⋅ ∇ x ] ⁢ ϑ - [ ∂ x ℓ , ϑ ⁢ p ϑ ϱ ⁢ C v ⁢ div x ] ⁢ u → ,

F 4 ℓ := - [ ∂ x ℓ , u → ⋅ ∇ x ] ⁢ E r - [ ∂ x ℓ , 4 3 ⁢ E r ⁢ div x ] ⁢ u → .

Then, by taking the scalar product of each of the previous equations, respectively, by

p ϱ ϱ 2 ⁢ ∂ x ℓ ⁡ ϱ , ∂ x ℓ ⁡ u → , C v ϑ ⁢ ∂ x ℓ ⁡ ϑ , 1 4 ⁢ a ⁢ ϱ ⁢ E r ⁢ ∂ x ℓ ⁡ E r , ∂ x ℓ ⁡ B → and ∂ x ℓ ⁡ E → ,

and adding the resulting equations, we get

(3.8) ∂ t ⁡ ℰ ℓ + div x ⁡ ℱ → ℓ + ν ⁢ ( ∂ x ℓ ⁡ u → ) 2 = ℛ ℓ + 𝒮 ℓ ,

where

ℰ ℓ := 1 2 ⁢ ( ∂ x ℓ ⁡ u → ) 2 + 1 2 ⁢ p ϱ ϱ ⁢ ( ∂ x ℓ ⁡ ϱ ) 2 + 1 2 ⁢ C v ϑ ⁢ ( ∂ x ℓ ⁡ ϑ ) 2 + 1 2 ⁢ 1 4 ⁢ a ⁢ ϱ ⁢ E r ⁢ ( ∂ x ℓ ⁡ E r ) 2 + 1 2 ⁢ ( ∂ x ℓ ⁡ E → ) 2 + 1 2 ⁢ ( ∂ x ℓ ⁡ B → ) 2 ,

ℱ → ℓ := ( p ϱ ϱ ⁢ ∂ x ℓ ⁡ ϱ + p ϑ ϱ ⁢ ∂ x ℓ ⁡ ϑ + 1 3 ⁢ a ⁢ ϱ ⁢ ∂ x ℓ ⁡ E r ) ⁢ ∂ x ℓ ⁡ u → + 1 2 ⁢ ( ( ∂ x ℓ ⁡ u → ) 2 + p ϱ ϱ ⁢ ( ∂ x ℓ ⁡ ϱ ) 2 + C v ϑ ⁢ ( ∂ x ℓ ⁡ ϑ ) 2 + 1 4 ⁢ a ⁢ ϱ ⁢ E r ⁢ ( ∂ x ℓ ⁡ E r ) 2 ) ⁢ u → ,

ℛ ℓ := 1 2 ⁢ [ p ϱ ϱ 2 ] t ⁢ ( ∂ x ℓ ⁡ ϱ ) 2 + 1 2 ⁢ [ C v ϑ ] t ⁢ ( ∂ x ℓ ⁡ ϑ ) 2 + 1 2 ⁢ [ 1 4 ⁢ a ⁢ ϱ ⁢ E r ] t ⁢ ( ∂ x ℓ ⁡ E r ) 2 + 1 2 ⁢ div x ⁡ ( p ϱ ϱ 2 ⁢ u → ) ⁢ ( ∂ x ℓ ⁡ ϱ ) 2

+ 1 2 ⁢ div x ⁡ u → ⁢ ( ∂ x ℓ ⁡ u → ) 2 + 1 2 ⁢ div x ⁡ ( C v ϑ ⁢ u → ) ⁢ ( ∂ x ℓ ⁡ ϑ ) 2 + 1 2 ⁢ div x ⁡ ( 1 4 ⁢ a ⁢ ϱ ⁢ E r ⁢ u → ) ⁢ ( ∂ x ℓ ⁡ E r ) 2

+ ∇ x ⁡ ( p ϱ ϱ ) ⁢ ∂ x ℓ ⁡ ϱ ⁢ ∂ x ℓ ⁡ u → + ∇ x ⁡ ( p ϑ ϱ ) ⁢ ∂ x ℓ ⁡ ϑ ⁢ ∂ x ℓ ⁡ u → + ∇ x ⁡ ( 1 3 ⁢ a ⁢ ϱ ) ⁢ ∂ x ℓ ⁡ E r ⁢ ∂ x ℓ ⁡ u →

+ p ϱ ϱ 2 ⁢ ∂ x ℓ ⁡ ϱ ⁢ F 1 ℓ + ∂ x ℓ ⁡ u → ⁢ F 2 ℓ + C v ϑ ⁢ ∂ x ℓ ⁡ ϑ ⁢ F 3 ℓ + ∂ x ℓ ⁡ E r ⁢ F 4 ℓ + ϱ ¯ ⁢ ∂ x ℓ ⁡ E → ⋅ ∂ x ℓ ⁡ u → ,

𝒮 ℓ := - ∂ x ℓ ⁡ u → ⋅ ∂ x ℓ ⁡ [ u → × ( B → - B → ¯ ) ] - C v ϑ ⁢ ∂ x ℓ ⁡ ϑ ⁢ ∂ x ℓ ⁡ [ σ a ϱ ⁢ C v ⁢ ( a ⁢ ϑ 4 - E r ) ] - 1 4 ⁢ a ⁢ ϱ ⁢ E r ⁢ ∂ x ℓ ⁡ E r ⁢ ∂ x ℓ ⁡ [ σ a ⁢ ( E r - a ⁢ ϑ 4 ) ] + ∂ x ℓ ⁡ E → ⁢ ∂ x ℓ ⁡ [ ( ϱ - ϱ ¯ ) ⁢ u → ] .

By integrating (3.8) on space, one gets

∂ t ⁢ ∫ ℝ 3 ℰ ℓ ⁢ 𝑑 x + ∥ ∂ x ℓ ⁡ u → ∥ L 2 2 ≤ ∫ ℝ 3 ( | ℛ ℓ | + | 𝒮 ℓ | ) ⁢ 𝑑 x .

Integrating now with respect to t and summing on ℓ , with | ℓ | ≤ d , yields

∥ ∂ x V ( t ) ∥ H d - 1 2 + ∫ 0 t ∥ ∂ x u → ( τ ) ∥ H d - 1 2 d τ ≤ C 0 ∥ ∂ x V 0 ∥ H d - 1 2 + C 0 ∑ | ℓ | = 1 d ∫ ℝ 3 ( | ℛ ℓ | + | 𝒮 ℓ | ) d x .

By observing that

| ∂ t ⁡ ϱ | ≤ C ⁢ | ∂ x ⁡ ϱ | , | ∂ t ⁡ ϑ | ≤ C ⁢ ( | ∂ x ⁡ ϱ | + | ∂ x ⁡ ϑ | + | ∂ x ⁡ E r | ⁢ | Δ ⁢ ϑ | ) and | ∂ t ⁡ E r | ≤ C ⁢ ( | ∂ x ⁡ ϱ | + | ∂ x ⁡ ϑ | + | ∂ x ⁡ E r | ) ,

and using the commutator estimates (see the Moser-type calculus inequalities in [16])

∥ ( F 1 ℓ , F 2 ℓ , F 3 ℓ , F 4 ℓ ) ∥ L 2 ≤ ∥ ∂ x ⁡ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L ∞ ⁢ ∥ ∂ x ℓ ⁡ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L 2 2 ,

we see that

| ℛ ℓ | ≤ C ⁢ ( ∥ ∂ x ⁡ ϱ ∥ L ∞ + ∥ ∂ x ⁡ u → ∥ L ∞ + ∥ ∂ x ⁡ ϑ ∥ L ∞ + ∥ ∂ x ⁡ E r ∥ L ∞ ) ⁢ ∥ ∂ x ℓ ⁡ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L 2 2 .

Then integrating with respect to time gives

∫ 0 t | ℛ ℓ ( τ ) | d τ ≤ C sup 0 ≤ τ ≤ t { ∥ ∂ x ϱ ∥ L ∞ + ∥ ∂ x u → ∥ L ∞ + ∥ ∂ x ϑ ∥ L ∞ + ∥ ∂ x E r ∥ L ∞ } ∫ 0 t ∥ ∂ x ℓ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L 2 2 d τ

≤ C ⁢ E ⁢ ( t ) ⁢ D 2 ⁢ ( t )

for any | ℓ | ≤ d . Similarly, we estimate

| 𝒮 ℓ | ≤ C ⁢ ∥ ∂ x ℓ ⁡ u → ∥ L 2 2 ⁢ ∥ ∂ x ℓ ⁡ [ u → × ( B → - B → ¯ ) ] ∥ L 2 2 + C ⁢ ∥ ∂ x ℓ ⁡ ϑ ∥ L 2 2 ⁢ ∥ ∂ x ℓ ⁡ [ σ a ϱ ⁢ C v ⁢ ( a ⁢ ϑ 4 - E r ) ] ∥ L 2 2

+ C ⁢ ∥ ∂ x ℓ ⁡ E r ∥ L 2 2 ⁢ ∥ ∂ x ℓ ⁡ [ σ a ⁢ ( E r - a ⁢ ϑ 4 ) ] ∥ L 2 2 + C ⁢ ∥ ∂ x ℓ ⁡ E → ∥ L 2 2 ⁢ ∥ ∂ x ℓ ⁡ [ ( ϱ - ϱ ¯ ) ⁢ u → ] ∥ L 2 2 .

Then we get

| 𝒮 ℓ | ≤ C ⁢ ∥ B → - B → ¯ ∥ L ∞ ⁢ ∥ ∂ x ℓ ⁡ u → ∥ L 2 2 + C ⁢ ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L ∞ ⁢ ∥ ∂ x ℓ ⁡ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L 2 ⁢ ∥ ∂ x ℓ ⁡ ( B → , E → ) ∥ L ∞

+ C ⁢ ( ∥ ∂ x ⁡ ϱ ∥ L ∞ + ∥ ∂ x ⁡ u → ∥ L ∞ + ∥ ∂ x ⁡ ϑ ∥ L ∞ + ∥ ∂ x ⁡ E r ∥ L ∞ ) ⁢ ∥ ∂ x ℓ ⁡ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L 2 2 .

Then integrating with respect to time yields

∫ 0 t | 𝒮 ℓ ( τ ) | d τ ≤ C sup 0 ≤ τ ≤ t ∥ ( B → - B → ¯ ) ( τ ) ∥ L ∞ ∫ 0 t ∥ ∂ x ℓ u → ( τ ) ∥ L 2 2 d τ + C sup 0 ≤ τ ≤ t ∥ ∂ x ℓ ( B → , E → ) ( τ ) ∥ L 2

× ∫ 0 t ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ( τ ) ∥ L ∞ ∥ ∂ x ℓ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ( τ ) ∥ L 2 d τ

+ C sup 0 ≤ τ ≤ t { ∥ ∂ x ϱ ∥ L ∞ + ∥ ∂ x u → ∥ L ∞ + ∥ ∂ x ϑ ∥ L ∞ + ∥ ∂ x E r ∥ L ∞ ( τ ) } ∫ 0 t ∥ ∂ x ℓ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L 2 2 d τ

≤ C ⁢ ( E ⁢ ( t ) ⁢ D 2 ⁢ ( t ) + F ⁢ ( t ) ⁢ I ⁢ ( t ) ⁢ D ⁢ ( t ) )

for any | ℓ | ≤ d . ∎

The above results, together with (3.6), allow us to derive the following energy bound.

Corollary 3.4.

Assume that the assumptions of Proposition 3.2 are satisfied. Then

(3.9) ∥ ( V - V ¯ ) ( t ) ∥ H d 2 + ∫ 0 t ∥ u → ( τ ) ∥ H d 2 d τ ≤ C ∥ ( V - V ¯ ) ( 0 ) ∥ H d 2 + C ( E ( t ) D ( t ) 2 + F ( t ) I ( t ) D ( t ) ) .

Our goal is now to derive bounds for the integrals in the right- and left-hand sides of equation (3.9). For this purpose we adapt the results of Ueda, Wang and Kawashima [20].

Lemma 3.5.

Under the assumptions of Theorem 2.1, and supposing that d ≥ 3 , we have the following estimate for any ε > 0 :

∫ 0 t ( ∥ ( ϱ - ϱ ¯ , ϑ - ϑ ¯ , E r - E ¯ r ) ⁢ ( τ ) ∥ H d 2 + ∥ E → ⁢ ( τ ) ∥ H d - 1 2 ) ⁢ 𝑑 τ

(3.10) ≤ ε ∫ 0 t ∥ ∂ x B → ( τ ) ∥ H d - 2 2 d τ + C ε { ∥ V 0 - V ¯ ∥ H d - 1 2 + E ( t ) D ( t ) 2 + F ( t ) I ( t ) D ( t ) } .

Proof.

We linearize the principal part of system (1.1)–(1.3) as follows:

(3.11) ∂ t ⁡ ϱ + ϱ ¯ ⁢ div x ⁡ u → = g 1 ,

(3.12) ∂ t ⁡ u → + a ¯ 1 ⁢ ∇ x ⁡ ϱ + a ¯ 2 ⁢ ∇ x ⁡ ϑ + a ¯ 3 ⁢ ∇ x ⁡ E r + E → + u → × B → ¯ + ν ⁢ u → = g 2 ,

(3.13) ∂ t ⁡ ϑ + b ¯ 1 ⁢ div x ⁡ u → + b ¯ 2 ⁢ ( ϑ - ϑ ¯ ) = g 3 ,

(3.14) ∂ t ⁡ E r + c ¯ 1 ⁢ div x ⁡ u → + c ¯ 3 ⁢ ( E r - E ¯ r ) = g 4 ,

(3.15) ∂ t ⁡ B → + curl x ⁡ E → = 0 ,

(3.16) ∂ t ⁡ E → - curl x ⁡ B → - ϱ ¯ ⁢ u → = g 5 ,

with coefficients

a 1 ⁢ ( ϱ , ϑ ) = p ϱ ϱ , ⁢ a 2 ⁢ ( ϱ , ϑ ) = p ϑ ϱ , ⁢ a 3 ⁢ ( ϱ , ϑ ) = 1 3 ⁢ ϱ , ⁢ a ¯ j = a j ⁢ ( ϱ ¯ , ϑ ¯ ) ,

b 1 ⁢ ( ϱ , ϑ ) = ϑ ⁢ p ϑ ϱ ⁢ C v , ⁢ b 2 ⁢ ( ϱ , ϑ , E r ) = a ⁢ σ a ϱ ⁢ C v ⁢ ( ϑ 2 + ϑ ¯ 2 ) ⁢ ( ϑ + ϑ ¯ ) , ⁢ b 3 ⁢ ( ϱ , ϑ , E r ) = a ⁢ σ a ϱ ⁢ C v , ⁢ b ¯ j = b j ⁢ ( ϱ ¯ , ϑ ¯ ) ,

c 1 ⁢ ( ϱ , ϑ , E r ) = 4 3 ⁢ E r , ⁢ c 2 ⁢ ( ϱ , ϑ , E r ) = a ⁢ σ a ⁢ ( ϑ 2 + ϑ ¯ 2 ) ⁢ ( ϑ + ϑ ¯ ) , ⁢ c 3 ⁢ ( ϱ , ϑ , E r ) = σ a , ⁢ c ¯ j = c j ⁢ ( ϱ ¯ , ϑ ¯ ) ,

and sources

g 1 := - { u → ⋅ ∇ x ⁡ ϱ + ( ϱ - ϱ ¯ ) ⁢ div x ⁡ u → } ,

g 2 := - { ( u → ⋅ ∇ x ) ⁢ u → + ( a 1 - a ¯ 1 ) ⁢ ∇ x ⁡ ϱ + ( a 2 - a ¯ 2 ) ⁢ ∇ x ⁡ ϑ + ( a 3 - a ¯ 3 ) ⁢ ∇ x ⁡ E r + u → × ( B → - B → ¯ ) } ,

g 3 := - { ( u → ⋅ ∇ x ) ⁢ ϑ + ( b 1 - b ¯ 1 ) ⁢ div x ⁡ u → + ( b 2 - b ¯ 2 ) ⁢ ( ϑ - ϑ ¯ ) + b 3 ⁢ ( E r - E ¯ r ) } ,

g 4 := - { ( u → ⋅ ∇ x ) ⁢ E r + ( c ¯ 1 - c 1 ) ⁢ div x ⁡ u → + c 2 ⁢ ( ϑ - ϑ ¯ ) + ( c 3 - c ¯ 3 ) ⁢ ( E r - E ¯ r ) }

and

g 5 = ( ϱ - ϱ ¯ ) ⁢ u → .

By multiplying (3.11) by - a ¯ 1 ⁢ div x ⁡ u → , (3.12) by a ¯ 1 ⁢ ∇ x ⁡ ϱ + a ¯ 2 ⁢ ∇ x ⁡ ϑ + a ¯ 3 ⁢ ∇ x ⁡ E r + E → , (3.13) by - a ¯ 2 ⁢ div x ⁡ u → + ϑ - ϑ ¯ , (3.14) by - a ¯ 3 ⁢ div x ⁡ u → + E r - E ¯ r , (3.15) by 1 , (3.16) by u → and summing up, we get

a ¯ 1 ⁢ ( ∇ x ⁡ ϱ ⁢ u → t - ϱ t ⁢ div x ⁡ u → ) + a ¯ 2 ⁢ ( ∇ x ⁡ ϑ ⁢ u → t - ϑ t ⁢ div x ⁡ u → ) + a ¯ 3 ⁢ ( ∇ x ⁡ E r ⁢ u → t - ( E r ) t ⁢ div x ⁡ u → )

+ E → ⁢ u → t + E → t ⁢ u → + { 1 2 ⁢ [ ( ϑ - ϑ ¯ ) 2 + ( E r - E ¯ r ) 2 ] } t + ( a ¯ 1 ⁢ ∇ x ⁡ ϱ + a ¯ 2 ⁢ ∇ x ⁡ ϑ + a ¯ 3 ⁢ ∇ x ⁡ E r + E → ) 2

+ ( a ¯ 1 ⁢ ∇ x ⁡ ϱ + a ¯ 2 ⁢ ∇ x ⁡ ϑ + a ¯ 3 ⁢ ∇ x ⁡ E r + E → ) ⁢ ( u → × B → ¯ + ν ⁢ u → ) + b ¯ 2 ⁢ ( ϑ - ϑ ¯ ) 2 + c ¯ 3 ⁢ ( E r - E ¯ r ) 2

+ b ¯ 1 ⁢ ( ϑ - ϑ ¯ ) ⁢ div x ⁡ u → + c ¯ 1 ⁢ ( E r - E ¯ r ) ⁢ div x ⁡ u → + ( a ¯ 3 ⁢ c ¯ 2 - a ¯ 2 ⁢ b ¯ 2 ) ⁢ ( ϑ - ϑ ¯ ) ⁢ div x ⁡ u →

(3.17) + ( a ¯ 2 ⁢ b ¯ 3 - a ¯ 3 ⁢ c ¯ 3 ) ⁢ ( E r - E ¯ r ) ⁢ div x ⁡ u → - u → ⁢ curl x ⁡ B → - ϱ ¯ ⁢ u → 2 - ( div x ⁡ u → ) 2 ⁢ [ a ¯ 1 + a ¯ 2 + a ¯ 3 ] = G 1 0 ,

where

G 1 0 := - a ¯ 1 ⁢ g 1 ⁢ div x ⁡ u → + [ a ¯ 1 ⁢ ∇ x ⁡ ϱ + a ¯ 2 ⁢ ∇ x ⁡ ϑ + a ¯ 3 ⁢ ∇ x ⁡ E r + E → ] ⁢ g 2 - [ a ¯ 2 + ϑ - ϑ ¯ ] ⁢ div x ⁡ u → ⁢ g 3 - [ a ¯ 3 + E r - E ¯ r ] ⁢ div x ⁡ u → ⁢ g 4 + g 5 ⁢ u → .

By rearranging the left-hand side of (3.17), we get

(3.18) { H 1 0 } t + div x ⁡ F → 1 0 + D 1 0 = M 1 0 + G 1 0 ,

where

H 1 0 = - [ a ¯ 1 ⁢ ( ϱ - ϱ ¯ ) + a ¯ 2 ⁢ ( ϑ - ϑ ¯ ) + a ¯ 3 ⁢ ( E r - E ¯ r ) ] ⁢ div x ⁡ u → + E → ⋅ u → + 1 2 ⁢ [ ( ϑ - ϑ ¯ ) 2 + ( E r - E ¯ r ) 2 ] ,

F → 1 0 = [ a ¯ 1 ⁢ ( ϱ - ϱ ¯ ) + a ¯ 2 ⁢ ( ϑ - ϑ ¯ ) + a ¯ 3 ⁢ ( E r - E ¯ r ) ] ⁢ u → t - 2 ⁢ [ a ¯ 1 ⁢ ( ϱ - ϱ ¯ ) + a ¯ 2 ⁢ ( ϑ - ϑ ¯ ) + a ¯ 3 ⁢ ( E r - E ¯ r ) ] ⁢ E →

+ ( a ¯ 3 ⁢ c ¯ 2 - a ¯ 2 ⁢ b ¯ 2 + b ¯ 1 ) ⁢ ( ϑ - ϑ ¯ ) ⁢ u → + ( a ¯ 2 ⁢ b ¯ 3 - a ¯ 3 ⁢ c ¯ 3 + c ¯ 1 ) ⁢ ( E r - E ¯ r ) ⁢ u → ,

D 1 0 = a ¯ 1 2 ⁢ | ∇ x ⁡ ϱ | 2 + a ¯ 2 2 ⁢ | ∇ x ⁡ ϑ | 2 + a ¯ 3 2 ⁢ | ∇ x ⁡ E r | 2 + | E → | 2 + 2 ⁢ a ¯ 1 ⁢ ( ϱ - ϱ ¯ ) 2 + b ¯ 2 ⁢ ( ϑ - ϑ ¯ ) 2 + c ¯ 3 ⁢ ( E r - E ¯ r ) 2 ,

M 1 0 = - { 2 a ¯ 1 a ¯ 2 ∇ x ϱ ⋅ ∇ x ϑ + 2 a ¯ 1 a ¯ 3 ∇ x ϱ ⋅ ∇ x E r + 2 a ¯ 2 a ¯ 3 ∇ x ϑ ⋅ ∇ x E r + 2 a ¯ 2 ( ϱ - ϱ ¯ ) ( ϑ - ϑ ¯ )

+ 2 ⁢ a ¯ 2 ⁢ ( ϱ - ϱ ¯ ) ⁢ ( E r - E ¯ r ) + ( a ¯ 1 ⁢ ∇ x ⁡ ϱ + a ¯ 2 ⁢ ∇ x ⁡ ϑ + a ¯ 3 ⁢ ∇ x ⁡ E r + E → ) ⁢ ( u → × B → ¯ + ν ⁢ u → ) - u → ⁢ curl x ⁡ B → - ϱ ¯ ⁢ u → 2

- ( div x u → ) 2 [ a ¯ 1 + a ¯ 2 + a ¯ 3 ] - ( a ¯ 3 c ¯ 2 - a ¯ 2 b ¯ 2 + b ¯ 1 ) ∇ x ϑ ⋅ u → - ( a ¯ 2 b ¯ 3 - a ¯ 3 c ¯ 3 + c ¯ 1 ) ∇ x E r ⋅ u → } .

Integrating (3.18) over space and using Young’s inequality yields

d d ⁢ t ⁢ ∫ ℝ 3 H 1 0 ⁢ 𝑑 x + C ⁢ ( ∥ ϱ ∥ L 2 2 + ∥ ∇ x ⁡ ϑ ∥ L 2 2 + ∥ ∇ x ⁡ E r ∥ L 2 2 + ∥ E → ∥ L 2 2 + ∥ ϱ - ϱ ¯ ∥ L 2 2 )

≤ ε ∥ ∂ x B → ∥ L 2 2 + C ε ( ∥ u → ∥ H 1 2 + ∥ ϑ - ϑ ¯ ∥ H 1 2 + ∥ E r - E ¯ r ∥ H 1 2 ) + ∫ ℝ 3 | G 1 0 | d x .

In fact, in the same way one obtains estimates for the derivatives of V . Namely, applying ∂ x ℓ to system (3.11)–(3.16), we get

(3.19) { H 1 ℓ } t + div x ⁡ F → 1 ℓ + D 1 ℓ = M 1 ℓ + G 1 ℓ ,

where

H 1 ℓ = - [ a ¯ 1 ⁢ ∂ x ℓ ⁡ ( ϱ - ϱ ¯ ) + a ¯ 2 ⁢ ∂ x ℓ ⁡ ( ϑ - ϑ ¯ ) + a ¯ 3 ⁢ ∂ x ℓ ⁡ ( E r - E ¯ r ) ] ⁢ div x ⁢ ∂ x ℓ ⁡ u → + ∂ x ℓ ⁡ E → ⋅ ∂ x ℓ ⁡ u → + 1 2 ⁢ [ ( ∂ x ℓ ⁡ ϑ ) 2 + ( ∂ x ℓ ⁡ E r ) 2 ] ,

F → 1 ℓ = [ a ¯ 1 ⁢ ∂ x ℓ ⁡ ( ϱ - ϱ ¯ ) + a ¯ 2 ⁢ ∂ x ℓ ⁡ ( ϑ - ϑ ¯ ) + a ¯ 3 ⁢ ∂ x ℓ ⁡ ( E r - E ¯ r ) ] ⁢ u → t

+ ( a ¯ 3 ⁢ c ¯ 2 - a ¯ 2 ⁢ b ¯ 2 + b ¯ 1 ) ⁢ ∂ x ℓ ⁡ ϑ ⁢ ∂ x ℓ ⁡ u → + ( a ¯ 2 ⁢ b ¯ 3 - a ¯ 3 ⁢ c ¯ 3 + c ¯ 1 ) ⁢ ∂ x ℓ ⁡ E r ⁢ ∂ x ℓ ⁡ u →

- 2 ⁢ [ a ¯ 1 ⁢ ∂ x ℓ ⁡ ( ϱ - ϱ ¯ ) + a ¯ 2 ⁢ ∂ x ℓ ⁡ ( ϑ - ϑ ¯ ) + a ¯ 3 ⁢ ∂ x ℓ ⁡ ( E r - E ¯ r ) ] ⁢ ∂ x ℓ ⁡ E → + ∂ x ℓ ⁡ u → × ∂ x ℓ ⁡ ( B → - B → ¯ ) ,

D 1 ℓ = a ¯ 1 2 ⁢ | ∇ x ⁡ ∂ x ℓ ⁡ ϱ | 2 + a ¯ 2 2 ⁢ | ∂ x ℓ ⁡ ∇ x ⁡ ϑ | 2 + a ¯ 3 2 ⁢ | ∂ x ℓ ⁡ ∇ x ⁡ E r | 2 + | ∂ x ℓ ⁡ E → | 2 + 2 ⁢ a ¯ 1 ⁢ ( ∂ x ℓ ⁡ ( ϱ - ϱ ¯ ) ) 2 + b ¯ 2 ⁢ ( ∂ x ℓ ⁡ ϑ ) 2 + c ¯ 3 ⁢ ( ∂ x ℓ ⁡ E r ) 2 ,

M 1 ℓ = - { 2 a ¯ 1 a ¯ 2 ∇ x ∂ x ℓ ϱ ⋅ ∇ x ∂ x ℓ ϑ + 2 a ¯ 1 a ¯ 3 ∇ x ∂ x ℓ ϱ ⋅ ∇ x ∂ x ℓ E r + 2 a ¯ 2 a ¯ 3 ∇ x ∂ x ℓ ϑ ⋅ ∇ x ∂ x ℓ E r + 2 a ¯ 2 ∂ x ℓ ( ϱ - ϱ ¯ ) ∂ x ℓ ( ϑ - ϑ ¯ )

+ 2 ⁢ a ¯ 2 ⁢ ∂ x ℓ ⁡ ( ϱ - ϱ ¯ ) ⁢ ∂ x ℓ ⁡ ( E r - E ¯ r ) + ( a ¯ 1 ⁢ ∇ x ⁡ ∂ x ℓ ⁡ ϱ + a ¯ 2 ⁢ ∇ x ⁡ ∂ x ℓ ⁡ ϑ + a ¯ 3 ⁢ ∇ x ⁡ ∂ x ℓ ⁡ E r + ∂ x ℓ ⁡ E → ) ⁢ ( ∂ x ℓ ⁡ u → × B → ¯ + ν ⁢ ∂ x ℓ ⁡ u → )

- ( a ¯ 3 ⁢ c ¯ 2 - a ¯ 2 ⁢ b ¯ 2 + b ¯ 1 ) ⁢ ∇ x ⁡ ∂ x ℓ ⁡ ϑ ⋅ ∂ x ℓ ⁡ u → - ( a ¯ 2 ⁢ b ¯ 3 - a ¯ 3 ⁢ c ¯ 3 + c ¯ 1 ) ⁢ ∇ x ⁡ ∂ x ℓ ⁡ E r ⋅ ∂ x ℓ ⁡ u →

- curl x ∂ x ℓ u → ∂ x ℓ ( B → - B → ¯ ) - ϱ ¯ ( ∂ x ℓ u → ) 2 - ( div x ∂ x ℓ u → ) 2 [ a ¯ 1 + a ¯ 2 + a ¯ 3 ] } ,

G 1 ℓ = - a ¯ 1 ⁢ ∂ x ℓ ⁡ g 1 ⁢ div x ⁢ ∂ x ℓ ⁡ u → + [ a ¯ 1 ⁢ ∇ x ⁡ ∂ x ℓ ⁡ ϱ + a ¯ 2 ⁢ ∇ x ⁡ ∂ x ℓ ⁡ ϑ + a ¯ 3 ⁢ ∇ x ⁡ ∂ x ℓ ⁡ E r + ∂ x ℓ ⁡ E → ] ⁢ ∂ x ℓ ⁡ g 2

- a ¯ 2 ⁢ ∂ x ℓ ⁡ g 3 ⁢ div x ⁢ ∂ x ℓ ⁡ u → - a ¯ 3 ⁢ ∂ x ℓ ⁡ g 4 ⁢ div x ⁢ ∂ x ℓ ⁡ u → + ∂ x ℓ ⁡ g 5 ⁢ ∂ x ℓ ⁡ u → + ∂ x ℓ ⁡ g 3 ⁢ ∂ x ℓ ⁡ ϑ + ∂ x ℓ ⁡ g 4 ⁢ ∂ x ℓ ⁡ E r .

Integrating (3.19) over space and time yields

∫ ℝ 3 H 1 ℓ ⁢ ( t ) ⁢ 𝑑 x - ∫ ℝ 3 H 1 ℓ ⁢ ( 0 ) ⁢ 𝑑 x + C ⁢ ∫ 0 t ( ∥ ∇ x ⁡ ∂ x ℓ ⁡ ϱ ∥ L 2 2 + ∥ ∇ x ⁡ ∂ x ℓ ⁡ ϑ ∥ L 2 2 + ∥ ∇ x ⁡ ∂ x ℓ ⁡ E r ∥ L 2 2 + ∥ ∂ x ℓ ⁡ E → ∥ L 2 2 ) ⁢ 𝑑 τ

+ C ⁢ ∫ 0 t ( ∥ ∂ x ℓ ⁡ ( ϱ - ϱ ¯ ) ∥ L 2 2 + ∥ ∂ x ℓ ⁡ ( ϑ - ϑ ¯ ) ∥ L 2 2 + ∥ ∂ x ℓ ⁡ ( E r - E ¯ r ) ∥ L 2 2 ) ⁢ 𝑑 τ

(3.20) ≤ ε ∫ 0 t ∥ ∂ x ℓ ( B → - B → ¯ ) ∥ L 2 2 d τ + C ε ∫ 0 t ( ∥ ∂ x ℓ u → ∥ H 1 2 + ∥ ∂ x ℓ ( ϑ - ϑ ¯ ) ∥ H 1 2 + ∥ ∂ x ℓ ( E r - E ¯ r ) ∥ H 1 2 ) d τ + ∫ 0 t ∫ ℝ 3 | G 1 ℓ | d x d τ .

By observing that

| ∫ ℝ 3 H 1 ℓ ⁢ ( t ) ⁢ 𝑑 x | ≤ C ⁢ ( ∥ ∂ x ℓ ⁡ ( ϱ - ϱ ¯ ) ∥ L 2 2 + ∥ ∂ x ℓ ⁡ ( ϑ - ϑ ¯ ) ∥ L 2 2 + ∥ ∂ x ℓ ⁡ ( E r - E ¯ r ) ∥ L 2 2 + ∥ ∂ x ℓ ⁡ u → ∥ H 1 2 ) ,

and summing (3.20) on ℓ for 1 ≤ ℓ ≤ d - 1 , we get

∫ 0 t ( ∥ ( ϱ - ϱ ¯ , ϑ - ϑ ¯ , E r - E ¯ r ) ( τ ) ∥ H d 2 + ∥ E → ( τ ) ∥ H d - 1 2 ) d τ

≤ C ε ∥ ( V - V ¯ ) ( 0 ) ∥ H d 2 + ε ∫ 0 t ∥ ∂ x B → ( τ ) ∥ H d - 2 2 d τ + C ε ( E ( t ) D 2 ( t ) + F ( t ) I ( t ) D ( t ) ) + ∑ | ℓ | = 1 d - 1 ∫ 0 t ∫ ℝ 3 | G 1 ℓ ( τ ) | d x d τ ,

where we used Corollary 3.4.

Let us estimate the last integral in (3.20). We have

(3.21) { ∥ ∂ x ℓ ⁡ g 1 ∥ L 2 ≤ C ⁢ ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L ∞ ⁢ ∥ ∂ x ℓ + 1 ⁡ ( ϱ , u → ) ∥ L 2 , ∥ ∂ x ℓ ⁡ g 2 ∥ L 2 ≤ C ⁢ ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L ∞ ⁢ ∥ ∂ x ℓ + 1 ⁡ ( ϱ , u → ) ∥ L 2 + C ⁢ ∥ B → - B → ¯ ∥ L ∞ ⁢ ∥ ∂ x ℓ ⁡ u → ∥ L 2 + C ⁢ ∥ ∂ x ℓ ⁡ ( B → - B → ¯ ) ∥ L 2 ⁢ ∥ u → ∥ L ∞ , ∥ ∂ x ℓ ⁡ g 3 ∥ L 2 ≤ C ⁢ ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L ∞ ⁢ ∥ ∂ x ℓ + 1 ⁡ ( ϱ , u → ) ∥ L 2 + C ⁢ ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L ∞ ⁢ ∥ ∂ x ℓ + 2 ⁡ ( ϑ , E r ) ∥ L 2 , ∥ ∂ x ℓ ⁡ g 4 ∥ L 2 ≤ C ⁢ ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L ∞ ⁢ ∥ ∂ x ℓ + 1 ⁡ ( ϱ , u → ) ∥ L 2 + C ⁢ ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L ∞ ⁢ ∥ ∂ x ℓ + 2 ⁡ ( ϑ , E r ) ∥ L 2 , ∥ ∂ x ℓ ⁡ g 5 ∥ L 2 ≤ C ⁢ ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ L ∞ ⁢ ∥ ∂ x ℓ ⁡ ( ϱ , u → ) ∥ L 2

for 1 ≤ | ℓ | ≤ d - 1 . Then

∫ 0 t ∫ ℝ 3 | G 1 ℓ ( τ ) | d x d τ ≤ C ∥ ∂ x ℓ + 1 u → ∥ L 2 ∥ ∂ x ℓ g 1 ∥ L 2 + C ( ∥ ∂ x ℓ + 1 ϱ ∥ L 2 + ∥ ∂ x ℓ + 1 ϑ ∥ L 2 + ∥ ∂ x ℓ + 1 E r ∥ L 2 + ∥ ∂ x ℓ E → ∥ L 2 ) ∥ ∂ x ℓ g 2 ∥ L 2

+ C ⁢ ∥ ∂ x ℓ + 1 ⁡ u → ∥ L 2 ⁢ ∥ ∂ x ℓ ⁡ g 3 ∥ L 2 + C ⁢ ∥ ∂ x ℓ + 1 ⁡ u → ∥ L 2 ⁢ ∥ ∂ x ℓ ⁡ g 4 ∥ L 2 + C ⁢ ∥ ∂ x ℓ ⁡ u → ∥ L 2 ⁢ ∥ ∂ x ℓ ⁡ g 5 ∥ L 2 .

Plugging bounds (3.21) into the last inequality gives

∑ | ℓ | = 1 d - 1 ∫ 0 t ∫ ℝ 3 | G 1 ℓ ( τ ) | d x d τ ≤ C E ( t ) D 2 ( t ) ,

which completes the proof of Lemma 3.5. ∎

Finally, we check from [20, Lemma 4.4] that the following result for the Maxwell’s system holds true for our system with a similar proof.

Lemma 3.6.

Under the assumptions of Theorem 2.1, and supposing that d ≥ 3 , for any ε > 0 , the following estimate (here, we set V = ( ϱ , u → , ϑ , E r , B → , E → ) T ) holds:

(3.22) ∫ 0 t ∥ ∂ x B → ( τ ) ∥ H s - 2 2 d τ ≤ C ∥ V 0 - V ¯ ∥ H s - 1 2 + C ∫ 0 t ∥ ∂ x E → ( τ ) ∥ H s - 2 2 d τ + C ( E ( t ) D ( t ) 2 + F ( t ) I ( t ) D ( t ) ) .

Proof.

By applying ∂ x ℓ to (1.5) and (1.6), multiplying, respectively, by - curl x ⁢ ∂ x ℓ ⁡ B → and curl x ⁢ ∂ x ℓ ⁡ E → , and adding the resulting equations, we get

- ( ∂ x ℓ E → ⋅ curl x ∂ x ℓ B → ) t + | curl x ∂ x ℓ B → | 2 - div x ( ∂ x ℓ E → × ∂ x ℓ B → t ) = M 2 ℓ + G 2 ℓ ,

where

M 2 ℓ = - ϱ ¯ ∂ x ℓ u → ⋅ curl x ∂ x ℓ B → + | curl x ∂ x ℓ E → | 2

and

G 2 ℓ = - ∂ x ℓ ⁡ ( ( ϱ - ϱ ¯ ) ⁢ u → ) ⋅ curl x ⁢ ∂ x ℓ ⁡ B → .

Integrating in space gives

- d d ⁢ t ∫ ℝ 3 ∂ x ℓ E → ⋅ curl x ∂ x ℓ B → d x + C ∥ curl x ∂ x ℓ B → ∥ L 2 2 ≤ ∥ curl x ∂ x ℓ E → ∥ L 2 2 + ∥ ∂ x ℓ u → ∥ L 2 2 + ∫ ℝ 3 | G 2 ℓ | d x .

By integrating on time and summing for 1 ≤ | ℓ | ≤ d - 2 , we have

∫ 0 t ∥ ∂ x B → ∥ H d - 2 2 d t ≤ C ∥ ( V - V ¯ ) ( t ) ∥ H d - 1 + C ∥ ( V - V ¯ ) ( 0 ) ∥ H d - 1 + C ∫ 0 t ∥ ∂ x E → ∥ H d - 2 2 d t

+ C ∫ 0 t ∥ u → ∥ H d - 2 2 d t + C ∑ | ℓ | = 0 d - 2 ∫ ℝ 3 | G 2 ℓ ( τ ) | d x d τ

≤ C ∥ ( V - V ¯ ) ( 0 ) ∥ H d - 1 + C ∫ 0 t ∥ ∂ x E → | H d - 2 2 d t + C ( E ( t ) D ( t ) 2 + F ( t ) I ( t ) D ( t ) ) ,

where we used the bound

∑ | ℓ | = 1 d - 1 ∫ 0 t ∫ ℝ 3 | G 2 ℓ ( τ ) | d x d τ ≤ C E ( t ) D 2 ( t ) ,

obtained in the same way as in the proof of Lemma 3.5. The proof of Lemma 3.6 is completed. ∎

We are now in position to conclude with the proofs of Theorems 2.1 and 2.2.

3.2 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, by plugging (3.22) into (3.10) with ε small enough, we get

(3.23) ∫ 0 t ( ∥ ( ϱ - ϱ ¯ , ϑ - ϑ ¯ , E r - E ¯ r ) ∥ H d 2 + ∥ E → ⁢ ( τ ) ∥ H d - 1 2 ) ⁢ 𝑑 τ ≤ C ⁢ { ∥ V 0 - V ¯ ∥ H d - 1 2 + E ⁢ ( t ) ⁢ D ⁢ ( t ) 2 + F ⁢ ( t ) ⁢ I ⁢ ( t ) ⁢ D ⁢ ( t ) } .

Putting this last estimate into (3.22) yields

(3.24) ∫ 0 t ∥ ∂ x B → ( τ ) ∥ H s - 2 2 d τ ≤ C ∥ V 0 - V ¯ ∥ H s 2 + C ( E ( t ) D ( t ) 2 + F ( t ) I ( t ) D ( t ) ) .

Then, from (3.10), (3.23) and (3.24), we get

∥ ( V - V ¯ ) ( t ) ∥ H d 2 + ∫ 0 t ( ∥ ( ϱ - ϱ ¯ , ϑ - ϑ ¯ , E r - E ¯ r ) ( τ ) ∥ H d 2 + ∥ E → ( τ ) ∥ H d - 1 2 + ∥ ∂ x B → ( τ ) ∥ H d - 2 2 ) d τ

≤ C ⁢ ∥ V 0 - V ¯ ∥ H d 2 + C ⁢ ( E ⁢ ( t ) ⁢ D ⁢ ( t ) 2 + F ⁢ ( t ) ⁢ I ⁢ ( t ) ⁢ D ⁢ ( t ) )

or, equivalently,

F ⁢ ( t ) 2 + D ⁢ ( t ) 2 ≤ C ⁢ ∥ V 0 - V ¯ ∥ H d 2 + C ⁢ ( E ⁢ ( t ) ⁢ D ⁢ ( t ) 2 + F ⁢ ( t ) ⁢ I ⁢ ( t ) ⁢ D ⁢ ( t ) ) .

Now, by observing that, provided d ≥ 3 , one has ∥ ( V - V ¯ ) ⁢ ( t ) ∥ H d ≤ E ⁢ ( t ) ≤ C ⁢ F ⁢ ( t ) , and, provided d ≥ 2 , one has I ⁢ ( t ) ≤ C ⁢ D ⁢ ( t ) for some positive constant C , we see that

F ⁢ ( t ) 2 + D ⁢ ( t ) 2 ≤ C ⁢ ∥ V 0 - V ¯ ∥ H d 2 + C ⁢ F ⁢ ( t ) ⁢ D ⁢ ( t ) 2 .

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 → T c ⁡ N ⁢ ( t ) = + ∞ ,

where N ⁢ ( t ) is defined by

N ⁢ ( t ) := ( F ⁢ ( t ) 2 + D ⁢ ( t ) 2 ) 1 / 2 .

Thus, we are left to prove that N is bounded. For this purpose, we use the argument used in [3]. After the previous calculation, we have

(3.25) N ⁢ ( t ) 2 ≤ C ⁢ ( ∥ V 0 - V ¯ ∥ H d 2 + N ⁢ ( t ) 3 ) for all ⁢ T ∈ [ 0 , T c ] .

Hence, setting ∥ V 0 - V ¯ ∥ H d = ε , we have

N ⁢ ( t ) 2 ε 2 + N ⁢ ( t ) 3 ≤ C .

By studying the variation of ϕ ⁢ ( N ) = N 2 / ( ε 2 + N 3 ) , we see that ϕ ′ ⁢ ( 0 ) = 0 , and that ϕ is increasing on the interval [ 0 , ( 2 ⁢ ε 2 ) 1 / 3 ] and decreasing on the interval [ ( 2 ⁢ ε 2 ) 1 / 3 , + ∞ ) . Hence,

max ⁡ ϕ = ϕ ⁢ ( ( 2 ⁢ ε 2 ) 1 / 3 ) = 1 3 ⁢ ( 2 ε ) 2 / 3 .

Hence, we can choose ε small enough to have ϕ ⁢ ( N ) ≤ C for all N ∈ [ 0 , N * ] , where N * > 0 , and we see that N ≤ N * , which contradicts (3.25).

4 Large time behavior

We have the following analogue of Proposition 3.2 for time derivatives.

Corollary 4.1.

Let the assumptions of Theorem 2.1 be satisfied, and consider the solution V := ( ϱ , u → , ϑ , E r , B → , E → ) of system (1.1)–(1.3) on [ 0 , t ] , for some t > 0 . Then, for a constant C 0 > 0 , one gets

(4.1) ∥ ∂ t ⁡ V ⁢ ( t ) ∥ H d - 1 2 + ∫ 0 t ( ∥ ∂ t ⁡ ( ϱ , u → , ϑ , E r ) ⁢ ( τ ) ∥ H d - 1 2 + ∥ ∂ t ⁡ ( B → , E → ) ⁢ ( τ ) ∥ H d - 2 2 ) ⁢ 𝑑 τ ≤ C 0 ⁢ ∥ V 0 - V ¯ ∥ H d 2 .

Proof.

By using system (3.7), we see that

∥ ∂ t ⁡ V ∥ H d - 1 ≤ C ⁢ ∥ V - V ¯ ∥ H d ,

∥ ∂ t ⁡ ( ϱ , u → , ϑ , E r ) ∥ H d - 1 ≤ ∥ ∂ x ⁡ ( ϱ , u → , ϑ , E r , B → , E → ) ∥ H d - 1 + C ⁢ ∥ ( ϱ , u → , ϑ , E r , B → , E → ) ∥ H d - 1

and

∥ ∂ t ⁡ ( B → , E → ) ∥ H d - 2 ≤ ∥ ∂ x ⁡ ( B → , E → ) ∥ H d - 2 + C ⁢ ∥ u → ∥ H d - 1 .

Then, for d ≥ 3 , using the uniform estimate ∥ V - V ¯ ∥ H d 2 ≤ C of Theorem 2.1, we get estimate (4.1). ∎

4.1 Proof of Theorem 2.2

By using Corollary 4.1, we get

∫ 0 ∞ | d d ⁢ t ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ( t ) ∥ H d - 1 | d t ≤ 2 ∫ 0 ∞ ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ( t ) ∥ H d - 1 ∥ ∂ t ( ϱ , u → , ϑ , E r ) ( t ) ∥ H d - 1 d t

≤ ∫ 0 ∞ ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ( t ) ∥ H d - 1 2 + ∥ ∂ t ( ϱ , u → , ϑ , E r ) ( t ) ∥ H d - 1 2 d t

≤ C 0 ⁢ ∥ V 0 - V ¯ ∥ H d 2 .

This implies that

t ↦ ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ⁢ ( t ) ∥ H d - 1 2 ∈ L 1 ⁢ ( 0 , ∞ )

and

t ↦ d d ⁢ t ⁢ ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ⁢ ( t ) ∥ H d - 1 ∈ L 1 ⁢ ( 0 , ∞ ) ,

and then

∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ⁢ ( t ) ∥ H d - 1 → 0 when t → ∞ .

Now, by applying the Gagliardo–Nirenberg inequality and (2.1), we get

∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ⁢ ( t ) ∥ W d - 2 , ∞ ≤ ∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ⁢ ( t ) ∥ H d - 2 1 / 4 ⁢ ∥ ∂ x 2 ⁡ ( ϱ , u → , ϑ , E r ) ⁢ ( t ) ∥ H d - 2 3 / 4 .

∥ ( ϱ - ϱ ¯ , u → , ϑ - ϑ ¯ , E r - E ¯ r ) ⁢ ( t ) ∥ W d - 2 , ∞ → 0 when ⁢ t → ∞ .

Similarly,

t ↦ ∥ E → ⁢ ( t ) ∥ H d - 1 2 ∈ L 1 ⁢ ( 0 , ∞ ) and t ↦ d d ⁢ t ⁢ ∥ E → ⁢ ( t ) ∥ H d - 1 ∈ L 1 ⁢ ( 0 , ∞ ) ,

and then

∥ E → ⁢ ( t ) ∥ W d - 1 , ∞ → 0 when ⁢ t → ∞ .

Finally,

t ↦ ∥ ∂ x ⁡ B → ⁢ ( t ) ∥ H d - 3 2 ∈ L 1 ⁢ ( 0 , ∞ ) and t ↦ d d ⁢ t ⁢ ∥ ∂ x ⁡ B → ⁢ ( t ) ∥ H d - 3 ∈ L 1 ⁢ ( 0 , ∞ ) .

Then, arguing as before,

∥ ( B → - B → ¯ ) ⁢ ( t ) ∥ W d - 4 , ∞ ≤ ∥ ( B → - B → ¯ ) ⁢ ( t ) ∥ H d - 4 1 / 4 ⁢ ∥ ∂ x 2 ⁡ B → ⁢ ( t ) ∥ H d - 3 3 / 4 .

∥ ( B → - B → ¯ ) ∥ W d - 4 , ∞ → 0 when ⁢ t → ∞ ,

which completes the proof.

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: ANR-15-CE40-0011

Funding source: Grantová Agentura České Republiky

Award Identifier / Grant number: 201-16-03230S

Funding statement: Šárka Nečasová acknowledges the support of the GAČR (Czech Science Foundation) project 16-03230S in the framework of RVO: 67985840. Bernard Ducomet is partially supported by the ANR project INFAMIE (ANR-15-CE40-0011).

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Received: 2017-05-22

Accepted: 2018-02-28

Published Online: 2018-03-21

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/anona-2017-0117

Keywords for this article

Compressible; Euler; radiation hydrodynamics

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