Yue-Jun Peng
email: yue-jun.peng@uca.fr

Liang Zhao

GLOBAL CONVERGENCE TO COMPRESSIBLE FULL NAVIER-STOKES EQUATIONS BY APPROXIMATION WITH OLDROYD-TYPE CONSTITUTIVE LAWS

Keywords: Full Navier-Stokes equations, Non-Newtonian fluid, Relaxed Euler systems, Oldroyd derivative, Global convergence AMS Subject Classification (2010) : 35B25, 35L60, 35Q30, 76A05

We consider smooth solutions to a relaxed Euler system with Oldroyd-type constitutive laws. This system is derived from the one-dimensional compressible full Navier-Stokes equations for a Newtonian fluid by using the Cattaneo-Christov model and the Oldroyd-B model. In a neighborhood of equilibrium states, we construct an explicit symmetrizer and show that the system is symmetrizable hyperbolic with partial dissipation. Moreover, by establishing uniform estimates with respect to the relaxation times, we prove the uniform global existence of smooth solutions and the global-in-time convergence of the system towards the full Navier-Stokes equations.

Introduction

Euler and Navier-Stokes equations are two important models in fluid dynamics. There are rich literatures on the mathematical analysis around these equations. We refer to [START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF][START_REF] Lions | Compressible Models[END_REF][START_REF] Majda | Vorticity and incompressible flow[END_REF][START_REF] Feireisl | Dynamics of viscous compressible fluids[END_REF][START_REF] Masmoudi | Examples of singular limits in hydrodynamics[END_REF] and references therein for mathematical results. There are deep relationships between Euler and Navier-Stokes equations. It is well known that the Euler equations can be derived from Navier-Stokes equations through the vanishing viscosity limit. Meanwhile, the Navier-Stokes equations can also be approximated by first-order partial differential equations using different kinds of constitutive laws for non-Newtonian fluids. These approximate equations are referred to as relaxed Euler systems or hyperbolic Navier-Stokes equations, see for instance [START_REF] Racke | Hyperbolic Navier-Stokes equations I: Local well-posedness[END_REF]34,[START_REF] Hu | Hyperbolic compressible Navier-Stokes equations[END_REF][START_REF] Peng | Relaxed Euler systems and convergence to Navier-Stokes equations[END_REF][START_REF] Coulaud | Hyperbolic quasilinear Navier-Stokes equations in R 2[END_REF].

In the paper, we study the global-in-time convergence from relaxed Euler-type equations with Oldroyd's constitutive laws to compressible full (non-isentropic) Navier-Stokes equations by letting relaxation times tend to zero. Let t ≥ 0 be the time variable and x = (x 1 , • • • , x d ) ∈ R d be the space variable. The compressible full Navier-Stokes equations are of the form

           ∂ t ρ + div(ρu) = 0,
∂ t (ρu) + div(ρu ⊗ u) + ∇p = divτ, ∂ t (ρE) + div(ρuE + up) + divq = div(uτ ), (1.1) in R + × R d , where ρ > 0 is the density, u = (u 1 , • • • , u d ) ∈ R d is the velocity, θ is the temperature, p is the pressure function, q ∈ R d is the heat flux, τ is the stress tensor and

E = e + 1 2
|u| 2 is the total energy per unit mass with e being the specific internal energy. The symbols and ⊗ represent the transpose and the tensor product, respectively. In (1.1), ρ, u and θ are independent variables, e and p are functions of (ρ, θ). In particular, for the ideal fluid, we have

e = c v θ, p = Rρθ, (1.2)
which satisfy the thermodynamic equation

ρ 2 e ρ = p -θp θ , (1.3)
where R and c v are positive constants, and e ρ = ∂e ∂ρ etc. Generally speaking, the heat flux q satisfies the following Fourier's law q = -κ∇θ, (1.4) where κ > 0 is the heat conduction constant. For Newtonian viscous fluids, the stress tensor τ takes the form

τ = µσ(u) + λ(divu)I d , (1.5)
with

σ(u) = ∇u + (∇u) - 2 d (divu)I d .
Here I d denotes the unit matrix of order d. The parameters µ > 0 and λ > 0 are the shear and bulk viscosity coefficients, respectively, which are all assumed to be constants.

For the full Navier-Stokes equations (1.1) with constitutive laws (1.4) and (1.5), the construction of the corresponding relaxed Euler systems depends on the way how to decompose the second-order derivatives divτ, div(uτ ) and divq into first-order derivative terms. Clearly, there are lots of ways to do it, among which the most natural one is to replace (1.4) and (1.5) by the following Maxwell's constitutive laws [START_REF] Maxwell | IV. on the dynamical theory of gases[END_REF] ε 2 1 ∂ t q + q = -κ∇θ, (1.6)

ε 2 2 ∂ t τ + τ = µσ(u) + λ(divu)I d , (1.7)
where ε 1 > 0 and ε 2 > 0 are the relaxation times. Equations in (1.1) together with (1.6)-

(1.7) form a relaxed Euler system. Formally, letting (ε 1 , ε 2 ) → 0 recovers the Navier-Stokes equations (1.1) with (1.4) and (1.5). This idea is not recent. It dates back to 1860s, see for instance [START_REF] Maxwell | IV. on the dynamical theory of gases[END_REF][START_REF] Cattaneo | Sulla conduzione del calore[END_REF][START_REF] Cattaneo | Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée[END_REF]. These approximations have not only the mathematical sense but also physical interpretations. Relation (1.6), also known as the Cattaneo's law, gives rise to heat waves with finite propagation speed. Relation (1.7) describes motions of viscoelastic fluids.

The laws are combinations of the Newtonian's law of viscosity and the Hooke's law of elasticity.

The existence of smooth solutions to system (1.1) with Maxwell's constitutive laws (1.6) and

(1.7) and their convergence to the classical non-isentropic Navier-Stokes equations (1.1) with

(1.4) and (1.5) have been studied in previous works. In the case where ε 1 = 0, the authors of [START_REF] Hu | Compressible Navier-Stokes equations with revised Maxwell's law[END_REF] proved the global existence of smooth solutions near constant equilibrium states for fixed ε 2 > 0 and the local-in-time convergence towards the Navier-Stokes equations as ε 2 → 0.

Similar results are obtained in [START_REF] Hu | Compressible Navier-Stokes equations with hyperbolic heat conduction[END_REF] in the case where ε 2 = 0 and ε 1 > 0. In these results, only one of the constitutive laws within (1.6) and (1.7) is used. Hence the systems studied in [START_REF] Hu | Compressible Navier-Stokes equations with hyperbolic heat conduction[END_REF][START_REF] Hu | Compressible Navier-Stokes equations with revised Maxwell's law[END_REF] are of mixed hyperbolic-parabolic type in the sense of Shizuta-Kawashima [START_REF] Kawashima | Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics[END_REF][START_REF] Shizuta | Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation[END_REF].

The local-in-time convergence is based on the error estimates between the original system and the limiting system. For the isentropic Navier-Stokes equations with constitutive law (1.7), the author of [START_REF] Yong | Newtonian limit of Maxwell fluid flows[END_REF] obtained the local existence and the local convergence to the classical isentropic Navier-Stokes equations under condition tr(τ ) = 0, where tr(τ ) means the trace of matrix τ . In [START_REF] Peng | Relaxed Euler systems and convergence to Navier-Stokes equations[END_REF], the first author of the present paper constructed approximate systems with vector variables instead of tensor variables by using Hurwitz-Radon matrices in both compressible and incompressible cases. He proved the uniform (with respect to ε 1 and ε 2 ) global existence of smooth solutions near constant equilibrium state and the global-in-time convergence of the systems towards classical isentropic Navier-Stokes equations. He also obtained similar results for the isentropic Navier-Stokes equations with constitutive law (1.7) without condition tr(τ ) = 0.

For the approximation of incompressible isentropic Navier-Stokes equations with constitutive law (1.7), we also refer to [START_REF] Racke | Hyperbolic Navier-Stokes equations I: Local well-posedness[END_REF]34,[START_REF] Schöwe | A quasilinear delayed hyperbolic Navier-Stokes system: global solution, asymptotics and relaxation limit[END_REF][START_REF] Schöwe | Blow-up results to certain hyperbolic model problems in fluid mechanics[END_REF][START_REF] Coulaud | Hyperbolic quasilinear Navier-Stokes equations in R 2[END_REF].

However, these two constitutive laws (1.6) and (1.7) have drawbacks as they do not ensure Galilean invariance. In other words, these laws lead to paradoxical evolution of thermal waves in a moving frame, see [START_REF] Christov | Heat conduction paradox involving second-sound propagation in moving media[END_REF]. To overcome it, the Oldroyd's upper-convected time derivative (or simply Oldroyd derivative) should be considered. In this paper, we consider the following two constitutive laws. The first one is the Cattaneo-Christov model introduced in [START_REF] Christov | On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction[END_REF],

ε 2 1 ∂ t q + u • ∇q -q • ∇u + (divu)q = -q -κ∇θ, (1.8)
in which the terms on the left-hand side are the Oldroyd derivative. It is proved in [START_REF] Christov | On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction[END_REF] that the constitutive law (1.8) is Galilean invariant. The second one is the following Oldroyd-B model for the tensor variable τ (see for instance [START_REF] Saut | Some remarks on the limit of viscoelastic fluids as the relaxation time tends to zero[END_REF][START_REF] Renardy | Mathematical problems in viscoelasticity[END_REF][START_REF] Molinet | Newtonian limit for weakly viscoelastic fluid flows of Oldroyd type[END_REF][START_REF] Bresch | Newtonian limit for weakly viscoelastic fluid flows[END_REF] and the references therein)

ε 2 2 (∂ t τ + u • ∇τ + g(τ, ∇u)) + τ = µσ(u) + λ(divu)I d , (1.9)
where

g(τ, ∇u) = τ W (u) -W (u)τ, with W (u) = 1 2 ∇u -(∇u) .
Hence, the relaxed Euler system for (1.1) with constitutive laws (1.8) and (1.9) is of the form

                               ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p = divτ, ∂ t (ρE) + div(ρuE + up) + divq = div(uτ ), ε 2 1 ∂ t q + u • ∇q -q • ∇u + (divu)q = -q -κ∇θ, ε 2 2 (∂ t τ + u • ∇τ + g(τ, ∇u)) + τ = µ ∇u + ∇u - 2 d (divu)I d + λ(divu)I d , (1.10) in R + × R d .
System (1.10) is very complicated. One can observe that it contains d 2 + 2d + 2 equations for fluids in space R d . So far the symmetrizable hyperbolicity for (1.10) is unknown in cases d ≥ 2. This makes it hard to establish the existence results. See classical theories [START_REF] Lax | Hyperbolic systems of conservation laws and the mathematical theory of shock waves[END_REF][START_REF] Kato | The Cauchy problem for quasi-linear symmetric hyperbolic systems[END_REF][START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF].

Moreover, when considering the constitutive laws (1.8) and (1.9) at the same time, there are no apparent dissipative structures for ∇u and ∇θ due to the loss of the elliptic structures for u and θ. This is different from the situation for the systems treated in [START_REF] Hu | Compressible Navier-Stokes equations with hyperbolic heat conduction[END_REF][START_REF] Hu | Compressible Navier-Stokes equations with revised Maxwell's law[END_REF].

In a recent paper [START_REF] Hu | Hyperbolic compressible Navier-Stokes equations[END_REF], the authors considered (1.10) in one space dimension. In this case, system (1.10) is reduced to the following form

                       ∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 + p -τ ) = 0, ∂ t (ρE) + ∂ x (ρuE + up + q -uτ ) = 0, ε 2 1 (∂ t q + u∂ x q) + κ∂ x θ = -q, ε 2 2 (∂ t τ + u∂ x τ ) -λ∂ x u = -τ, (1.11) in R + × R, with the initial data (ρ, u, θ, q, τ )| t=0 = (ρ ε 0 , u ε 0 , θ ε 0 , q ε 0 , τ ε 0 )(x), (1.12)
where ε = (ε 1 , ε 2 ). In the limit as ε → 0, we have formally q = -κ∂ x θ, τ = λ∂ x u.

Substituting these relations into (1.11), we recover the following one-dimensional non-isentropic Navier-Stokes equations

           ∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 ) + ∂ x p = λ∂ xx u, ∂ t (ρE) + ∂ x (ρuE + up) = κ∂ xx θ + λ∂ x (u∂ x u), (1.13)
in R + × R. Equations (1.13) have been widely studied. We refer to [START_REF] Kanel | A model system of equations for the one-dimensional motion of a gas[END_REF][START_REF] Kazhikhov | Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas[END_REF][START_REF] Hoff | Global existence for 1d, compressible, isentropic Navier-Stokes equations with large initial data[END_REF][START_REF] Matsumura | The initial value problem for the equations of motion of viscous and heatconductive gases[END_REF] for the global existence of smooth solutions. See also [START_REF] Lions | Compressible Models[END_REF][START_REF] Feireisl | Dynamics of viscous compressible fluids[END_REF] for related topics and references therein.

In (1.11), ρ, u, θ, q and τ are independent variables, and both the internal energy e and the pressure function p are functions of (ρ, θ, q, τ ). In [START_REF] Hu | Hyperbolic compressible Navier-Stokes equations[END_REF], the authors used the following state equations for e and p :

e = c v θ + ε 2 1 κρθ q 2 + ε 2 2 2λρ τ 2 , (1.14) p = Rρθ - ε 2 1 2κθ q 2 - ε 2 2 2λ τ 2 .
(1.15)

Equation (1.14) is based on the results in [START_REF] Coleman | On the thermodynamics of second sound in dielectric crystals[END_REF], where the authors rigorously proved that the constitutive law (1.8) is consistent with the second law of thermodynamics if and only if the dependence of e on q is quadratic. Similarly, the quadratic dependence of e on τ also implies 2) as ε → 0. For more explanations, see [START_REF] Chen | On second sound in materials with memory[END_REF][START_REF] Coleman | Stability of equilibrium for a nonlinear hyperbolic system describing heat propagation by second sound in solids[END_REF][START_REF] Tarabek | On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound[END_REF][START_REF] Hu | Hyperbolic compressible Navier-Stokes equations[END_REF].

Let V e = (1, 0, 1, 0, 0) be an equilibrium state for (1.11). Due to the explicit expressions 2) as ε → 0. Remark that in these results condition ε 1 = ε 2 is not needed and system (1.11) is not included in the class of systems studied in [START_REF] Peng | Parabolic limit with differential constraints of first-order quasilinear hyperbolic systems[END_REF][START_REF] Peng | Uniform global existence and parabolic limit for partially dissipative hyperbolic systems[END_REF][START_REF] Li | Convergence rate from hyperbolic systems of balance laws to parabolic systems[END_REF][START_REF] Zhao | Convergence rate from systems of balance laws to isotropic parabolic systems, a periodic case[END_REF]. The proof of the results is based on uniform estimates with respect to the time and the relaxation parameters.

We use the strictly convex entropy given in [START_REF] Hu | Hyperbolic compressible Navier-Stokes equations[END_REF] for the L 2 estimate. A key step is to find an appropriate symmetrizer of system (1.11) for higher-order estimates. It is well-known that the second-order derivative of a strictly convex entropy provides a symmetrizer for a system of conservation laws [START_REF] Godunov | An interesting class of quasi-linear systems (Russian)[END_REF][START_REF] Boillat | Sur l'existence et la recherche d'équations de conservation supplémentaires pour les systèmes hyperboliques[END_REF]. However, this result cannot be applied to (1.11) because it is a non-conservative system.

It is worth mentioning that the global existence result in the present paper is different from that obtained in [START_REF] Hu | Hyperbolic compressible Navier-Stokes equations[END_REF], which is not uniform with respect to ε 1 and ε 2 . More precisely, there are terms ∂ t q and ∂ t τ in the definition of the energy in [START_REF] Hu | Hyperbolic compressible Navier-Stokes equations[END_REF]. Because of boundary layers in the limit as ε → 0, such an energy cannot be uniformly bounded with respect to the relaxation parameters. In order to avoid this situation, in the proof of our results, the energy contains only terms of derivative of solutions with respect to x.

Theorem 1.1. (Uniform global existence) Let s ≥ 2 be an integer and

(ρ ε 0 -1, u ε 0 , θ ε 0 -1, q ε 0 , τ ε 0 ) ∈ H s (R).
There exist two positive constants δ and C, independent of ε 1 and ε 2 , such that if

ρ ε 0 -1 s + u ε 0 s + θ ε 0 -1 s + ε 1 q ε 0 s + ε 2 τ ε 0 ≤ δ, then for all ε 1 , ε 2 ∈ (0, 1], the Cauchy problem (1.11)-(1.12) together with (1.14)-(1.15) admits a unique global smooth solution (ρ ε , u ε , θ ε , q ε , τ ε ) satisfying ρ ε -1, u ε , θ ε -1, q ε , τ ε ∈ C(R + ; H s (R)) ∩ C 1 (R + ; H s-1 (R)), and
ρ ε (t) -1 2 s + u ε (t) 2 s + θ ε (t) -1 2 s + ε 2 1 q ε (t) 2 s + ε 2 2 τ ε (t) 2 s + t 0 ∂ x ρ ε (t ) 2 s-1 + ∂ x u ε (t ) 2 s-1 + ∂ x θ ε (t ) 2 s-1 + q ε (t ) 2 s + τ ε (t ) 2 s dt ≤ C ρ ε 0 -1 2 s + u ε 0 2 s + θ ε 0 -1 2 s + ε 2 1 q ε 0 2 s + ε 2 2 τ ε 0 2 s , ∀ t ≥ 0, (1.16)
where

• k denotes the usual norm of H k (R). Theorem 1.2. (Global convergence) Let ε = (ε 1 , ε 2 ) and (ρ ε , u ε , θ ε , q ε , τ ε ) be the global solution obtained in Theorem 1.1, then there exist functions (ρ, ū, θ) ∈ L ∞ (R + ; H s (R)) and (q, τ ) ∈ L 2 (R + ; H s (R)), such that, as ε → 0 up to subsequences, (ρ ε , u ε , θ ε ) (ρ, ū, θ) weakly- * in L ∞ (R + ; H s (R)), (1.17) (q ε , τ ε ) (q, τ ) weakly in L 2 (R + ; H s (R)), (1.18)
where (ρ, ū, θ) is the solution to the one-dimensional full compressible Navier-Stokes equations (1.13) for the ideal fluid (1.2), with initial value (ρ 0 , ū0 , θ0 ) which is the weak limit of (ρ ε 0 , u ε 0 , θ ε 0 ) up to subsequences. Moreover, q = -κ∂ x θ, τ = λ∂ x ū.

The rest of the paper is organized as follows. In the next section, we prove results on the hyperbolic structure for system (1.11). These results are crucial in the proof of the above theorems. Section 3 is devoted to uniform global estimates. The proof of the theorems are completed in the last section.

Symmetrizable hyperbolicity

In what follows, s ≥ 2 denotes an integer. Let ε = (ε 1 , ε 2 ) and C be a generic positive constant independent of ε and any time. We assume that

ε 1 , ε 2 ∈ (0, 1]. For a integer k ≥ 1, we denote by • k , • and • ∞ the norms of the usual Sobolev spaces H k (R), L 2 (R) and
L ∞ (R), respectively. The inner product in L 2 (R) is denoted by •, • . In the proof we will frequently use the fact that the embedding from

H l (R) to L ∞ (R) is continuous for all integers l ≥ 1.
For simplicity, the dependence of solution on the parameters ε 1 and ε 2 is not expressed explicitly. We want to write (1.11) into a first-order quasilinear system of variables (ρ, u, θ, q, τ ).

First, it is clear that by using (1.15) and the first equation in (1.11), the second equation in (1.11) is equivalent to

ρ(∂ t u + u∂ x u) + p ρ ∂ x ρ + p θ ∂ x θ + p q ∂ x q + (p τ -1)∂ x τ = 0.
Similarly, by using the first two equations in (1.11), the third equation in (1.11) is equivalent to

ρ∂ t e + ρu∂ x e + (p -τ )∂ x u + ∂ x q = 0. (2.1)
This equation can be further treated by using equations (1.3), (1.14) and (1.15). Indeed,

ρ∂ t e = ρe θ ∂ t θ + ρe ρ ∂ t ρ + ρe q ∂ t q + ρe τ ∂ t τ = ρe θ ∂ t θ + ρe ρ ∂ t ρ + ρ 2q κθρ (-ε 2 1 u∂ x q -q -κ∂ x θ) + ρ τ λρ (-ε 2 2 u∂ x τ -τ + λ∂ x u) = ρe θ ∂ t θ + ρe ρ ∂ t ρ - 2q θ ∂ x θ - 2ε 2 1 uq κθ ∂ x q - 2q 2 κθ - ε 2 2 uτ λ ∂ x τ - τ 2 λ + τ ∂ x u, and
ρu∂ x e = ρue θ ∂ x θ + ρue ρ ∂ x ρ + ρue q ∂ x q + ρue τ ∂ x τ = ρue θ ∂ x θ + ρue ρ ∂ x ρ + 2ε 2 1 uq κθ ∂ x q + ε 2 2 uτ λ ∂ x τ,
hence,

ρ∂ t e + ρu∂ x e = ρe θ ∂ t θ + ρue θ - 2q θ ∂ x θ + ρe ρ (∂ t ρ + u∂ x ρ) - 2q 2 κθ - τ 2 λ + τ ∂ x u.
Combining the last equation with (1.3), (2.1) and the first equation in (1.11), we have

ρe θ ∂ t θ + θp θ ∂ x u + ρue θ - 2q θ ∂ x θ + ∂ x q = 2q 2 κθ + τ 2 λ .
Therefore, system (1.11) is equivalent to

                       ∂ t ρ + ∂ x (ρu) = 0, ρ(∂ t u + u∂ x u) + p ρ ∂ x ρ + p θ ∂ x θ + p q ∂ x q + (p τ -1)∂ x τ = 0, ρθe θ ∂ t θ + θ 2 p θ ∂ x u + (ρuθe θ -2q)∂ x θ + θ∂ x q = 2q 2 κ + θτ 2 λ , ε 2 1 (∂ t q + u∂ x q) + κ∂ x θ = -q, ε 2 2 (∂ t τ + u∂ x τ ) -λ∂ x u = -τ.
(2.2)

Let D 0 (ε) be a diagonal matrix defined by

D 0 (ε) = diag 1, 1, 1, ε 2 1 , ε 2 2 .
Then system (2.2) is written as

D 0 (ε)V t + A(V )∂ x V + BV = F (V ), (2.3)
where

V = (ρ, u, θ, q, τ ) , F (V ) = 0, 0, 1 ρθe θ 2q 2 κ + θτ 2 λ , 0, 0 , and
A(V ) =          u ρ 0 0 0 pρ ρ u p θ ρ pq ρ pτ -1 ρ 0 θp θ ρe θ u -2q ρθe θ 1 ρe θ 0 0 0 κ ε 2 1 u 0 0 -λ 0 0 ε 2 2 u          , B =          0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1          .
We define

A 0 (V ) = Ã0 (V )D 0 (ε), Ã(V ) = Ã0 (V )A(V ), with Ã0 (V ) =          θ 2 p ρ p θ 0 0 0 0 0 ρ 2 θ 2 p θ 0 0 0 0 0 ρ 2 θe θ p θ -1 κ (ρ 2 qe θ ) 0 0 0 - ε 2 1 κ ρ 2 qe θ 1 κ (ρθp θ + 2ρqp q ) 0 0 0 0 0 1 λ ρθ 2 p θ (1 -p τ )          .
Then straightforward calculations give

A 0 (V ) =          θ 2 p ρ p θ 0 0 0 0 0 ρ 2 θ 2 p θ 0 0 0 0 0 ρ 2 θe θ p θ - ε 2 1 κ (ρ 2 qe θ ) 0 0 0 - ε 2 1 κ (ρ 2 qe θ ) ε 2 1 κ ρθp θ + 2ρqp q 0 0 0 0 0 ε 2 2 λ ρθ 2 p θ (1 -p τ )          , Ã(V ) =          uθ 2 p ρ p θ ρθ 2 p ρ p θ 0 0 0 ρθ 2 p ρ p θ ρ 2 uθ 2 p θ ρθ 2 p 2 θ ρθ 2 p θ p q ρθ 2 p θ (p τ -1) 0 ρθ 2 p 2 θ J 1 ρθp θ + ρ 2 uθe θ p q 0 0 ρθ 2 p θ p q ρθp θ + ρ 2 uθe θ p q J 2 0 0 ρθ 2 p θ (p τ -1) 0 0 ε 2 2 λ ρuθ 2 p θ (1 -p τ )         
, where

J 1 = ρ 2 uθe θ p θ -2ρqp θ -ρ 2 qe θ , J 2 = ε 2 1 κ ρu(θp θ + 2qp q ) -ρq .
It is clear that A 0 (V ) and Ã(V ) are symmetric. Let T > 0 and

N = ρ -1, Θ = θ -1, W = N, u, Θ, ε 1 q, ε 2 τ .
We denote

E (t) = W (t, •) 2 s , E T = sup 0≤t≤T E (t). (2.4)
In Theorem 1.1, estimate (1.16) implies that E T is uniformly sufficiently small when E (0) is, although the L ∞ (0, T ; H s (R)) norm of q and τ may not be uniformly small with respect to ε 1 and ε 2 . The following result shows that A 0 (V ) is positive definite when E T is sufficiently small. (1.12). Then there exist constants δ > 0 and c 1 > 0, independent of ε 1 and ε 2 , such that if

E 1/2 T ≤ δ, we have c 1 ≤ ρ, θ, p ρ , p θ , e θ ≤ C, (2.5)
p q s ≤ Cδε 1 , p τ s ≤ Cδε 2 , qp q s ≤ Cδ, (2.6)
and A 0 (V ) is positive definite and then system (2.2) is symmetrizable hyperbolic.

Proof. When E T is sufficiently small, ρ and θ are sufficiently close to 1, then they have uniform positive upper and lower bounds. From the state equations (1.14) and (1.15), we have

p ρ = Rθ, p θ = Rρ + ε 2 1 2κθ 2 q 2 , e θ = c v - ε 2 1 κρθ 2 q 2 , (2.7)
which imply (2.5) since ε 1 q ∞ ≤ E T which is sufficiently small. Besides,

p q = - ε 2 1 κθ q, p τ = - ε 2 2 λ τ, qp q = - ε 2 1 κθ q 2 , (2.8)
imply (2.6).

Moreover,

ρ 2 θe θ p θ - ε 2 1 κ (ρ 2 qe θ ) - ε 2 1 κ (ρ 2 qe θ ) ε 2 1 κ ρθp θ + 2ρqp q = ε 2 1 ρ 3 e θ κ θ 2 p 2 θ - ε 2 1 q 2 κ (2p θ + ρe θ ) .
Then (2.5) implies that the above determinate is positive for sufficiently small E T . From (2.6) and ε 2 ≤ 1, we also have p τ s ≤ Cδ. Therefore, A 0 (V ) is positive definite from its explicit expression.

Applying the theory on the symmetrizable hyperbolic system, Lemma

Uniform global estimates

Let T > 0 and W = (N, u, Θ, ε 1 q, ε 2 τ ) be the unique local smooth solution to (2.2) with

(1.14)-(1.15) and (1.12), defined on [0, T ]. We assume that E T defined in (2.4) is sufficiently small. This gives rise to the rational assumption that

|ρ -1| ≤ 1 2 , |θ -1| ≤ 1 2 , |p τ | ≤ 1 2 . ( 3.1)
We want to establish the uniform global estimate for W with respect to the parameters ε 1 and ε 2 and the time T . For this purpose, we introduce the following dissipative energy

D(t) = ∂ x N (t) 2 s-1 + ∂ x u(t) 2 s-1 + ∂ x Θ(t) 2 s-1 + q(t) 2 s + τ (t) 2 s .
3.1. L 2 -estimate. The L 2 estimate relies on the existence of a strictly convex entropy and its corresponding entropy flux.

Lemma 3.1. For system (2.2), there exists a strictly convex entropy η and its corresponding entropy flux Ψ satisfying

∂ t η(W ) + ∂ x Ψ(W ) + q 2 κθ 2 + τ 2 λθ = 0, (3.2)
where

η(W ) = R(ρ ln ρ -ρ + 1) + 1 2 ρu 2 + c v ρ(θ -ln θ -1) + ε 2 1 κθ 1 - 1 2θ q 2 + ε 2 2 2λ τ 2 ,
and

Ψ(W ) = Rρ(ln ρ -1)u + 1 2 ρu 3 + c v ρu(θ -ln θ -1) + (p -τ )u + 1 - 1 θ q + ε 2 1 u κθ 1 - 1 2θ q 2 + ε 2 2 2λ uτ 2 .
In addition, if E T is sufficiently small, the following L 2 -energy estimate holds

(ρ -1, u, θ -1, ε 1 q, ε 2 τ )(t) 2 + t 0 q(t ) 2 + τ (t ) 2 dt ≤ C ρ 0 -1 2 + u 0 2 + θ 0 -1 2 + ε 2 1 q 0 2 + ε 2 2 τ 0 2 , ∀ t ∈ [0, T ]. (3.3)
Proof. The entropy-entropy flux identity (3.2) was established in [START_REF] Hu | Hyperbolic compressible Navier-Stokes equations[END_REF]. We now prove (3.3). By the Taylor's expansion at (ρ, θ) = (1, 1) and (3.1), we have

c v ρ(θ -ln θ -1) = c v ρ 2 θ2 Θ 2 ≥ 1 9 c v Θ 2 , R(ρ ln ρ -ρ + 1) = R 2ρ N 2 ≥ R 3 N 2 ,
where ρ is between ρ and 1, and θ is between θ and 1. By Lemma 2.1, this implies that there exists a constant c 2 > 0, such that

c 2 |W | 2 ≤ η(W ) ≤ C|W | 2 ,
where

|•| is the Euclidean norm in R 5 . Integrating (3.2) over [0, t] for t ∈ [0, T ] implies (3.3).
3.2. Higher order estimates. Let l be an integer with 1 ≤ l ≤ s. Applying ∂ l x to both sides of the equation (2.3) yields

D 0 (ε)∂ t (∂ l x V ) + A(V )∂ x (∂ l x V ) + ∂ l x (BV ) = ∂ l x F (V ) + K l ,
where

K l = A(V )∂ l+1 x V -∂ l x (A(V )∂ x V ). Then, A 0 (V )∂ t (∂ l x V ) + Ã(V )∂ x (∂ l x V ) + Ã0 (V )∂ l x (BV ) = Ã0 (V )∂ l x F (V ) + Ã0 (V )K l .
Taking the inner product of the above system with ∂ l x V in L 2 (R), we have

d dt A 0 (V )∂ l x V, ∂ l x V + 2 Ã0 (V )∂ l x (BV ), ∂ l x V (3.4) = ∂ t A 0 (V )∂ l x V, ∂ l x V + ∂ x Ã(V )∂ l x V, ∂ l x V + 2 Ã0 (V )∂ l x F (V ), ∂ l x V + 2 Ã0 (V )K l , ∂ l x V .
We deal with (3.4) term by term in a series of lemmas as follows.

Lemma 3.2. There exists positive constants c 3 and c 4 such that

2 Ã0 (V )∂ l x (BV ), ∂ l x V ≥ c 3 ∂ l x q 2 + c 4 ∂ l x τ 2 -CE 1/2 T D(t). (3.5)
Proof. By the definition of Ã0 (V ) and B together with (2.8), we have

Ã0 (V )∂ l x (BV ) = 0, 0, 0, - ρ 2 qe θ κ ∂ l x q, ρθp θ + 2ρqp q κ ∂ l x q, ρθ 2 p θ (1 -p τ ) λ ∂ l x τ , which implies that Ã0 (V )∂ l x (BV ), ∂ l x V = - ρ 2 qe θ κ ∂ l x q, ∂ l x Θ + ρθp θ + 2ρqp q κ ∂ l x q, ∂ l x q ρθ 2 p θ (1 -p τ ) λ ∂ l x τ, ∂ l x τ .
Obviously,

ρ 2 qe θ κ ∂ l x q, ∂ l x Θ ≤ C ∂ l x Θ ∞ q ∂ l x q ≤ CE 1/2 T D(t).
For the second term, noticing that ρθp θ κ has a uniform lower bound when E T is sufficiently small. Besides, the last estimate in (2.6) implies that 2ρqp q κ is small and consequently, there exists a positive constant c 3 such that

2 ρθp θ + 2ρqp q κ ∂ l x q, ∂ l x q ≥ c 3 ∂ l x q 2 .
Similarly, (3.1) implies that 1 -p τ ≥ 1 2 . Therefore, there exists a positive constant c 4 such that

2 ρθ 2 p θ (1 -p τ ) λ ∂ l x τ, ∂ l x τ ≥ c 4 ∂ l x τ 2 .
Combining all these estimates yields (3.5).

Lemma 3.3. It holds

∂ t A 0 (V )∂ l x V, ∂ l x V ≤ CE 1/2 T D(t). (3.6)
Proof. We denote

V = V 1 V 2 , V 1 =     N u Θ     , V 2 = q τ , and
A 0 (V ) = A 11 0 (V ) A 12 0 (V ) A 21 0 (V ) A 22 0 (V ) , Ã(V ) = Ã11 (V ) Ã12 (V ) Ã21 (V ) Ã22 (V ) , where                                      A 11 0 (V ) =     θ 2 p ρ p θ 0 0 0 ρ 2 θ 2 p θ 0 0 0 ρ 2 θe θ p θ     , A 12 0 (V ) = A 21 0 (V ) =     0 0 0 0 - ε 2 1 κ ρ 2 qe θ 0     , A 22 0 (V ) = ε 2 1 κ (ρθp θ + 2ρqp q ) 0 0 ε 2 2 λ ρθ 2 p θ (1 -p τ )
.

Obviously,

∂ t A 0 (V )∂ l x V, ∂ l x V = ∂ t A 11 0 (V )∂ l x V 1 , ∂ l x V 1 + 2 ∂ t A 12 0 (V )∂ l x V 2 , ∂ l x V 1 + ∂ t A 22 0 (V )∂ l x V 2 , ∂ l x V 2 .
From (1.14) and (1.15), we see that A 11 0 (V ) is a smooth function of (N, Θ, ε 2 1 q 2 ). Moreover, from (2.2) together with the Cauchy-Schwarz inequality, we have

∂ t N ∞ + ∂ t Θ ∞ ≤ CE 1/2 T + C q s + q 2 s + τ 2 s , ε 2 1 ∂ t q ∞ ≤ CE 1/2 T + C q s , ε 2 1 ∂ t (q 2 ) ∞ ≤ CE 1/2 T + C q s + C q 2 s .
Hence,

∂ t A 11 0 (V ) ∞ ≤ CE 1/2 T + C q s + q 2 s + τ 2 s . Moreover, 2 q s ∂ l x V 1 2 ≤ ∂ l x V 1 2 + q 2 s ∂ l x V 1 ≤ CE 1/2 T D(t)
. Since E T is uniformly small, we also have

q 2 s + τ 2 s ∂ l x V 1 2 ≤ CE T D(t) ≤ CE 1/2 T D(t).
Therefore,

∂ t A 11 0 (V )∂ l x V 1 , ∂ l x V 1 ≤ CE 1/2
T D(t). Next, a straightforward calculation yields

∂ t A 12 0 (V )∂ l x V 2 , ∂ l x V 1 = - ε 2 1 κ ∂ t (ρ 2 qe θ )∂ l x q, ∂ l x Θ .
From (2.7), we have

ρ 2 qe θ = c ν ρ 2 q - ε 2 1 ρq 3 κθ 2 . Then (2.
2) together with the bound of E T yields

ε 2 1 κ ∂ t ρ 2 qe θ ∞ ≤ CE 1/2 T + C q s + Cε 1 τ 2 s .
Therefore,

∂ t A 12 0 (V )∂ l x V 2 , ∂ l x V 1 ≤ ε 2 1 κ ∂ t ρ 2 qe θ ∞ ∂ l x q ∂ l x Θ ≤ CE 1/2 T D(t) + CE 1/2 T q 2 s + C(ε 1 ∂ l x q ∂ l x Θ ) τ 2 s ≤ CE 1/2 T D(t).
Finally,

∂ t A 22 0 (V )∂ l x V 2 , ∂ l x V 2 = ε 2 1 κ ∂ t (ρθp θ + 2ρqp q )∂ l x q, ∂ l x q + ε 2 2 λ ∂ t ρθ 2 p θ (1 -p τ ) ∂ l x τ, ∂ l x τ .
Similarly to the estimates above, we have

ε 2 1 ∂ t (ρθp θ + 2ρqp q ) ∞ ≤ CE 1/2 T + Cε 2 1 q s + q 2 s + τ 2 s , and
ε 2 2 ∂ t ρθ 2 p θ (1 -p τ ) ∞ ≤ CE 1/2 T + Cε 2 2 q s + q 2 s + τ s + τ 2 s ,
imply that

∂ t A 22 0 (V )∂ l x V 2 , ∂ l x V 2 ≤ CE 1/2 T D(t).
This proves the lemma.

Lemma 3.4. It holds

∂ x Ã(V )∂ l x V, ∂ l x V ≤ CE 1/2 T D(t). (3.7)
Proof. From (1.14) and (1.15) and the explicit expression of Ã(V ), we see that all elements of Ã(V ) are smooth functions of (ρ, u, θ, ε 2 1 q 2 , ε 2 2 τ 2 ) except J 1 . Noticing that J 1 appears at the position of the third line and the third column of Ã(V ) and

∂ x J 1 ∞ ≤ C( ∂ x W s-1 + q s ), we obtain ∂ x Ã(V )∂ l x V, ∂ l x V ≤ C ∂ x W ∞ ∂ l x V 2 + ∂ x J 1 ∂ l x Θ, ∂ l x Θ ≤ CE 1/2 T D(t) + C ∂ x J 1 ∞ ∂ l x Θ 2 ≤ CE 1/2 T D(t) + C q s ∂ l x Θ 2 ≤ CE 1/2 T D(t).
This proves (3.7).

Lemma 3.5. It holds

Ã0 (V )∂ l x F (V ), ∂ l x V ≤ CE 1/2 T D(t). (3.8)
Proof. By the definition of Ã0 (V ) and F , we have

Ã0 (V )∂ l x F (V ) = 0, 0, ρ 2 θe θ p θ ∂ l x 1 ρθe θ 2q 2 κ + θτ 2 λ , - ε 2 1 ρ 2 qe θ κ ∂ l x 1 ρθe θ 2q 2 κ + θτ 2 λ , 0 . It follows that Ã0 (V )∂ l x F (V ), ∂ l x V ≤ ρ 2 θe θ p θ ∂ l x 1 ρθe θ 2q 2 κ + θτ 2 λ , ∂ l x Θ + ε 2 1 ρ 2 qe θ κ ∂ l x 1 ρθe θ 2q 2 κ + θτ 2 λ , ∂ l x q ≤ C( ∂ l x Θ ∞ + ε 1 ∂ l x q ε 1 q ∞ )( q 2 s + τ 2 s ) ≤ CE 1/2 T D(t),
which proves (3.8).

Lemma 3.6. It holds

Ã0 (V )K l , ∂ l x V ≤ CE 1/2 T D(t). (3.9)
Proof. Recall that

K l = A(V )∂ l+1 x V -∂ l x (A(V )∂ x V ).
Similarly to the proof of Lemma 3.4, it is easy to see that all elements of A(V ) are smooth functions of (ρ, u, θ, ε 2 1 q 2 , ε 2 2 τ 2 ) except the element at the position of the third line and the third column. We denote by J 3 this element. Then

J 3 = u - 2q ρθe θ , and
∂ x J 3 s-1 ≤ C( ∂ x W s-1 + q s ).
The only two terms that contain J 3 are the following

∂ l x Θ, ρ 2 θe θ p θ ∂ l x (J 3 ∂ x Θ) -J 3 ∂ l+1 x Θ and -∂ l x q, ε 2 1 ρ 2 qe θ κ ∂ l x (J 3 ∂ x Θ) -J 3 ∂ l+1 x Θ ,
which can be treated similarly to the proof of Lemma 3.4 and are obviously bounded by

CE 1/2 T D(t).
On the other hand, each element of Ã0 (V ) is uniformly bounded in L ∞ ([0, T ] × R) except for the element at the position of the third line and the fourth column. This element is -1 κ (ρ 2 qe θ ) which only touches the fourth line of K l in the product Ã0 (V )K l . Now the only nonzero element on the fourth line of K l is ε 2 1 (u∂ l+1 x q -∂ l x (u∂ x q)). Then the fourth component of Ã0 (V )K l is -

ε 2 1 κ (ρ 2 qe θ )(u∂ l+1
x q -∂ l x (u∂ x q)). By the Moser-type calculus inequalities (see [START_REF] Majda | Compressible fluid flow and systems of conservation laws in several space variables[END_REF] for instance), we have

ε 2 1 ρ 2 qe θ (u∂ l+1 x q -∂ l x (u∂ x q)), ∂ l x Θ ≤ CE 1/2
T D(t). The estimates for the other terms can be easily obtained. This proves (3.9).

ε 2 1 s-1 m=0 d dt ∂ m x q, ∂ m+1 x Θ + κ 4 ∂ x Θ 2 s-1 ≤ Cν ∂ x u 2 s-1 + C q 2 s + CE 1/2 T D(t), (3.10)
where ν > 0 is a small positive constant to be determined later.

Proof. Let m be an integer with 0 ≤ m ≤ s -1. Applying ∂ m x to the fourth equation in (2.2) and making the inner product of the resulting equation with

∂ m+1 x Θ in L 2 (R), we have κ ∂ m+1 x Θ 2 = -ε 2 1 d dt ∂ m x q, ∂ m+1 x Θ + ε 2 1 ∂ t ∂ m+1 x Θ, ∂ m x q -ε 2 1 ∂ m+1 x Θ, ∂ m x (u∂ x q) -∂ m x q, ∂ m+1 x Θ .
Obviously,

ε 2 1 ∂ m+1 x Θ, ∂ m x (u∂ x q) + ∂ m x q, ∂ m+1 x Θ ≤ κ 2 ∂ m+1 x Θ 2 + C q 2 s . ε 2 1 d dt ∂ m x q, ∂ m+1 x Θ + κ 2 ∂ m+1 x Θ 2 ≤ ε 2 1 ∂ t ∂ m+1 x Θ, ∂ m x q + C q 2 s . (3.11)
By using the third equation in (2.2) and an integration by parts, we have

ε 2 1 ∂ t ∂ m+1 x Θ, ∂ m x q ≤ ε 1 ∂ m x u∂ x θ + θp θ ρe θ ∂ x u - 2q ρθe θ ∂ x θ + 1 ρe θ ∂ x q , ∂ m+1 x (ε 1 q) + ε 1 ∂ m x 2q 2 κρθe θ + τ 2 λρe θ , ∂ m+1 x (ε 1 q) .
Obviously,

ε 1 ∂ m x u∂ x θ + θp θ ρe θ ∂ x u - 2q ρθe θ ∂ x θ + 1 ρe θ ∂ x q , ∂ m+1 x (ε 1 q) ≤ κ 4 ∂ x Θ 2 s-1 + ν ∂ x u 2 s-1 + C q 2 s + CE 1/2 T D(t),
and

ε 1 ∂ m x 2q 2 κρθe θ + τ 2 λρe θ , ∂ m+1 x (ε 1 q) ≤ Cε 1 ε 1 q s q 2 s + Cε 1 ε 1 q s τ 2 s ≤ CE 1/2 T D(t).
Combining these estimates, we arrive at

ε 2 1 ∂ t ∂ m+1 x Θ, ∂ m x q ≤ κ 4 ∂ x Θ 2 s-1 + ν ∂ x u 2 s-1 + C q 2 s + CE 1/2 T D(t).
This together with (3.11) yields

ε 2 1 d dt ∂ m x q, ∂ m+1 x Θ + κ 4 ∂ m+1 x Θ 2 ≤ ν ∂ x u 2 s-1 + C q 2 s + CE 1/2 T D(t).
Adding this inequality for all 0 ≤ m ≤ s -1 yields (3.10).

Lemma 3.8. (Dissipative estimates for ∂

x u) It holds -ε 2 2 s-1 m=0 d dt ∂ m x τ, ∂ m+1 x u + λ 2 ∂ x u 2 s-1 ≤ Cν( ∂ x N 2 s-1 + ∂ x u 2 s-1 + ∂ x Θ 2 s-1 ) + C q 2 s + τ 2 s + CE 1/2 T D(t), (3.12)
where ν > 0 is a small positive constant to be determined later.

Proof. For 0 ≤ m ≤ s -1, applying ∂ m x to the fifth equation in (2.2) and making the inner product of the resulting equation with ∂

m+1 x u in L 2 (R), we have λ ∂ m+1 x u 2 = ε 2 2 d dt ∂ m x τ, ∂ m+1 x u -ε 2 2 ∂ t ∂ m+1 x u, ∂ m x τ + ε 2 2 ∂ m+1 x u, ∂ m x (u∂ x τ ) + ∂ m+1 x u, ∂ m x τ .
Obviously,

ε 2 2 ∂ m+1 x u, ∂ m x (u∂ x τ ) + ∂ m+1 x u, ∂ m x τ ≤ λ 2 ∂ m+1 x u 2 + C τ 2 s + CE 1/2 T D(t). Therefore, -ε 2 2 d dt ∂ m x τ, ∂ m+1 x u + λ 2 ∂ m+1 x u 2 ≤ -ε 2 2 ∂ t ∂ m+1 x u, ∂ m x τ + C τ 2 s + CE 1/2 T D(t). (3.13)
By using the second equation in (2.2) and an integration by parts, we have

ε 2 2 ∂ t ∂ m+1 x u, ∂ m x τ ≤ ε 2 2 ∂ m x ρu∂ x u + p ρ ∂ x ρ + p θ ∂ x θ ρ , ∂ m+1 x τ +ε 2 2 ∂ m x p q ∂ x q + (p τ -1)∂ x τ ρ , ∂ m+1 x τ ≤ ν ∂ x N 2 s-1 + ∂ x u 2 s-1 + ∂ x Θ 2 s-1 + C q 2 s + τ 2 s .
Substituting this estimate into (3.13) and adding the resulting equation for all 0 ≤ m ≤ s -1 yield (3.12).

Lemma 3.9. (Dissipative estimates for ∂

x N ) It holds s-1 m=0 d dt ∂ m x u, ∂ m+1 x N + R 6 ∂ x N 2 s-1 ≤ C ∂ x u 2 s-1 + ∂ x Θ 2 s-1 + C q 2 s + τ 2 s + CE 1/2 T D(t). (3.14)
Proof. We first write the second equation in (2.2) as

∂ t u + Rθ ρ ∂ x N + 1 ρ ρu∂ x u + p θ ∂ x θ + p q ∂ x q + (p τ -1)∂ x τ = 0.
For m ≤ s-1, applying ∂ m x to the above equation and making the inner product of the resulting equation with ∂ m+1

x N in L 2 (R), we have

∂ m+1 x N, Rθ ρ ∂ m+1 x N = - d dt ∂ m x u, ∂ m+1 x N + ∂ m x u, ∂ t ∂ m+1 x N -∂ m x ρu∂ x u + p θ ∂ x θ + p q ∂ x q + (p τ -1)∂ x τ ρ , ∂ m+1 x N -∂ m x Rθ ρ ∂ x N - Rθ ρ ∂ m+1 x N, ∂ m+1 x N .
Noticing (3.1), we have

∂ m+1 x N, Rθ ρ ∂ m+1 x N ≥ R 3 ∂ m+1 x N 2 .
It is clear that

∂ m x ρu∂ x u + p θ ∂ x θ + p q ∂ x q + (p τ -1)∂ x τ ρ , ∂ m+1 x N ≤ R 6 ∂ m+1 x N 2 + C ∂ x u 2 s-1 + ∂ x Θ 2 s-1 + C q 2 s + τ 2 s ,
and by the Moser-type calculus inequalities,

∂ m x Rθ ρ ∂ x N - Rθ ρ ∂ m+1 x N, ∂ m+1 x N ≤ C ∂ x θ s-1 ∂ x N 2 s-1 + C ∂ x N 3 s-1 CE 1/2 T D(t).
Moreover, by the mass equation in (2.2) and the integration by parts, we obtain

∂ m x u, ∂ t ∂ m+1 x N = ∂ m+1 x u, ∂ m+1 x (ρu) ≤ C ∂ x u s-1 ( ∂ m x (∂ x N u + ρ∂ x u) ) ≤ C ∂ x u 2 s-1 + CE 1/2 T D(t).
Combining all these estimates, we arrive at

d dt ∂ m x u, ∂ m+1 x N + R 6 ∂ m+1 x N 2 ≤ C ∂ x u 2 s-1 + ∂ x Θ 2 s-1 + C q 2 s + τ 2 s + CE 1/2 T D(t).
Adding the above estimate for all m ≤ s -1 yields (3.14). Since A 0 (V ) is positive definite with respect to W , there exists a constant c 5 > 0, independent of ε 1 and ε 2 , such that

A 0 (V )∂ l x V, ∂ l x V ≥ c 5 ∂ l x W 2 .
Integrating On the other hand, combining the estimates in Lemmas 3.7-3.9, we have

s-1 m=0 d dt ε 2 1 ∂ m x q, ∂ m+1 x Θ -ε 2 2 ∂ m x τ, ∂ m+1 x u + α ∂ m x u, ∂ m+1 x N + Rα 6 ∂ x N 2 s-1 + λ 2 ∂ x u 2 s-1 + κ 4 ∂ x Θ 2 s-1 ≤ Cν ∂ x N 2 s-1 + C(ν + α)( ∂ x u 2 s-1 + ∂ x Θ 2 s-1 ) + C q 2 s + τ 2 s + CE 1/2 T D(t),
where α > 0 is a constant. We choose ν > 0 and α > 0 sufficiently small such that 2Cν ≤ Rα 6 , 8C(ν + α) ≤ min(λ, κ).

It follows that there is a constant a 0 > 0, independent of ε 1 and ε 2 , such that

s-1 m=0 d dt ε 2 1 ∂ m x q, ∂ m+1 x Θ -ε 2 2 ∂ m x τ, ∂ m+1 x u + α ∂ m x u, ∂ m+1 x N + a 0 ∂ x N 2 s-1 + ∂ x u 2 s-1 + ∂ x Θ 2 s-1 ≤ C q 2 s + τ 2 s + CE 1/2
T D(t). (4.4)

It is clear that

ε 2 1 ∂ m x q, ∂ m+1 x Θ -ε 2 2 ∂ m x τ, ∂ m+1 x u + α ∂ m x u, ∂ m+1 x N ≤ C W 2 s .
Integrating (4.4) over [0, T ] yields Proof of Theorem 1.2. From (1.11), (ρ ε , u ε , θ ε , q ε , τ ε ) satisfies the following system

-C W (t)
                       ∂ t ρ ε + ∂ x (ρ ε u ε ) = 0, ∂ t (ρ ε u ε ) + ∂ x (ρ ε (u ε ) 2 + p ε -τ ε ) = 0, ∂ t (ρ ε E ε ) + ∂ x (ρ ε u ε E ε + u ε p ε + q ε -u ε τ ε ) = 0, ε 2 1 (∂ t q ε + u ε ∂ x q ε ) + κ∂ x θ ε = -q ε , ε 2 2 (∂ t τ ε + u ε ∂ x τ ε ) -λ∂ x u ε = -τ ε , (4.5)
in R + × R, where

               e ε = c v θ ε + ε 2 1 κρ ε θ ε (q ε ) 2 + ε 2 2 2λρ ε (τ ε ) 2 , p ε = Rρ ε θ ε - ε 2 1 2κθ ε (q ε ) 2 - ε 2 2 2λ (τ ε ) 2 , E ε = e ε + 1 2 (u ε ) 2 .
(4.6)

The uniform estimate (1.16) implies that the sequences (ρ ε -1) ε , (u ε ) ε and (θ ε -1) ε are bounded in L ∞ (R + ; H s (R)) and the sequence (q ε ) ε and (τ ε ) ε are bounded in L 2 (R + ; H s (R)). It

  the compatibility with the second law of thermodynamics. The choice of equation (1.15) makes it consistent with (1.3) and the state equations (1.14)-(1.15) yield formally those for the ideal fluid (1.

( 1 .

 1 [START_REF] Hu | Compressible Navier-Stokes equations with hyperbolic heat conduction[END_REF] and(1.15), the authors of[START_REF] Hu | Hyperbolic compressible Navier-Stokes equations[END_REF] constructed a strictly convex entropy for system(1.11), see Lemma 3.1 in[START_REF] Hu | Hyperbolic compressible Navier-Stokes equations[END_REF] or Lemma 3.1 below. Based on this, they established the global existence of smooth solutions near V e for fixed ε 1 > 0 and ε 2 > 0, and the local convergence of the system towards the classical full Navier-Stokes equations as ε 1 = ε 2 → 0. However, the convergence for large time has not been investigated. The purpose of this paper is to study the global convergence of system (1.11) with state equations (1.14) and (1.15). The main results of the paper are contained in the following two theorems. The first theorem shows the uniform global existence of smooth solutions to the Cauchy problem (1.11)-(1.12) near V e . The second one concerns the global convergence of the solution of (1.11) to that of the one-dimensional full Navier-Stokes equations (1.13) for the ideal fluid (1.

Lemma 2 . 1 .

 21 Let W ∈ C([0, T ]; H s (R)) be the smooth solution to (2.2) with (1.14)-(1.15) and

3. 3 .

 3 Dissipative estimates for ∂ x N , ∂ x u and ∂ x Θ. Lemma 3.7. (Dissipative estimates for ∂ x Θ) It holds

4 . 0 D+ c 3 ∂ x q 2 s- 1 + c 4 ∂ x τ 2 s

 4032142 Proof of Theorems 1.1-1.2 Lemma 4.1. (Final energy estimate) If E T is sufficiently small, thenE (t) + t (t )dt ≤ CE (0), ∀ t ∈ [0, T ]. (4.1)Proof. Combining (3.4) and Lemmas 3.2-3.6, and adding for all 1 ≤ l ≤ s, we arrive at

Proof of Theorem 1 . 1 .

 11 The estimate in Lemma 4.1 shows that the smooth solution W is uniformly bounded in L ∞ ([0, T ]; H s (R)) with respect to ε and T . By the bootstrap principle, it yields uniformly global solution. In particular, this estimate gives(1.16).

  (4.2) over [0, T ] together with (3.3) yields

	t		t
	W (t) 2 s +	q(t ) 2 s + τ (t ) 2 s dt ≤ CE (0) + CE	1/2 T	D(t )dt .	(4.3)
	0		0

that there exist functions (ρ, ū, θ) ∈ L ∞ (R + ; H s (R)) and (q, τ ) ∈ L 2 (R + ; H s (R)), such that (1.17)-(1.18) hold. In addition, as ε → 0,

Moreover, from the first three equations in (4.5), it is easy to see that (

. Hence, by a classical compactness theorem [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF], for all T > 0, (ρ ε ) ε , (u ε ) ε and (θ ε ) ε are relatively compact in C([0, T ]; H s-1 loc (R)). As a consequence, as ε → 0, up to subsequences,

This is sufficient to pass the limit in (4.5)-(4.6) in the sense of distributions and to obtain the Navier-Stokes equations for the ideal fluid. This ends the proof of Theorem 1.2.


Denise Aregba-Driollet

S T Éphane Brull

Yue-Jun J Peng
email: yue-jun.peng@uca.fr

S Brull

GLOBAL EXISTENCE OF SMOOTH SOLUTIONS FOR A NON-CONSERVATIVE BITEMPERATURE EULER MODEL

Keywords: non-conservative hyperbolic system -partial dissipation -symmetrization -energy estimates -Euler type model for plasmas AMS Subject Classification (2000) : 35L60, 35F55, 35Q31, 76N10, 76W05

The bitemperature Euler model describes a crucial step of Inertial Confinement Fusion (ICF) when the plasma is quasineutral while ionic and electronic temperatures remain distinct. The model is written as a first-order hyperbolic system in non-conservative form with partially dissipative source terms. We consider the polytropic case for both ions and electrons with different γ-law pressures. The system does not fulfill the Shizuta-Kawashima condition and the physical entropy, which is a strictly convex function, doses not provide a symmetrizer of the system. In this paper we exhibit a symmetrizer to apply the result on the local existence of smooth solutions in several space dimensions. In the one-dimensional case we establish energy and dissipation estimates leading to global existence for small perturbations of equilibrium states.

Introduction

This paper is devoted to the study of the global existence of smooth solutions near constant equilibrium states for a bitemperature Euler system. This fluid model describes the interaction of a mixture of one species of ions and one species of electrons in thermal nonequilibrium, with applications in the field of Inertial Confinement Fusion (ICF). It was derived from a kinetic model by using a hydrodynamic limit and the Boltzmann entropy. For this kinetic model, a Discrete Velocity Model (DVM) method with an asymptotic preserving discretization toward Euler equations was obtained. The kinetic approach also allows to design numerical schemes for the bitemperature Euler equations. See [START_REF] Aregba-Driollet | Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF][START_REF] Brull | A kinetic approach of the bi-temperature Euler model[END_REF].

We denote by ρ e and ρ i the electronic and ionic densities, ρ = ρ e + ρ i the total density, m e and m i the related masses, c e and c i the mass fractions. These variables satisfy (1.1) ρ e = m e n e = c e ρ, ρ i = m i n i = c i ρ, m e > 0, m i > 0, c e + c i = 1.

Quasineutrality is assumed, so that the ionization ratio Z = n e /n i is a constant. This implies that the electronic and ionic mass fractions are constant and given by (1.2) c e = Zm e m i + Zm e , c i = m i m i + Zm e .

We suppose that the ionic and electronic velocities are equal: u e = u i = u, and the pressure of each species satisfies a gamma-law with its own γ exponent :

p e = (γ e -1)ρ e ε e = n e k B T e , p i = (γ i -1)ρ i ε i = n i k B T i , γ e > 1, γ i > 1, (1.3) where k B is the Boltzmann constant (k B > 0), ε α and T α represent respectively the internal specific energy and the temperature of species α for α = e, i.

Denoting by | • | the Euclidean norm in R d , the total energies for the particles are defined by (1.4)

E α = ρ α ε α + 1 2 ρ α |u| 2 = c α ρε α + 1 2 ρ|u| 2 , α = e, i.
We denote by ν ≥ 0 the interaction coefficient between the electronic and ionic temperatures. Physically this coefficient is a complicated function of the electronic and ionic temperatures and of ρ, see the NRL plasma formulary [START_REF] Huba | NRL plasma formulary, Supported by the Office of Naval Research[END_REF]. A rigorous derivation of ν is obtained via a kinetic underlying formulation [START_REF] Aregba-Driollet | Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF]. It gives ν(ρ) = Kρ where K is a positive constant. This expression of ν implies that more dense is the plasma, faster it reaches the thermal equilibrium. In order to simplify the notation, we assume that ν is a sufficiently smooth function of ρ, denoted by ν = ν(ρ), and satisfies ν(ρ) > 0 for ρ > 0. In particular, it suffices to assume that ν(1) > 0 in the study of the global existence of smooth solutions for ρ near 1. From the proof of the main theorem, we will see easily that global existence still holds when ν is a smooth function of (T e , T i , ρ) and remains positive at an equilibrium point (T e , T i , ρ) = ( T , T , 1) for a positive constant T . The model consists of two conservative equations for mass and momentum and two nonconservative equations for each energy:

(1.5)

           ∂ t ρ + div(ρu) = 0,
∂ t (ρu) + div(ρu ⊗ u) + ∇(p e + p i ) = 0, ∂ t E e + div u(E e + p e ) -u • (c i ∇p e -c e ∇p i ) = ρν(T i -T e ),

∂ t E i + div u(E i + p i ) + u • (c i ∇p e -c e ∇p i ) = -ρν(T i -T e ),
where "•" stands for the inner product in R d . This is a non-conservative hyperbolic system which can be written in the synthetic form

(1.6) ∂ t W + d j=1 C j (W)∂ x j W = F (W). Now we introduce (1.7)      η(ρ, ρu, E e , E i ) = - α=e,i ρ α b α ln (γ α -1)ρ α ε α ρ γα α , φ(ρ, ρu, E e , E i ) = η(ρ, ρu, E e , E i )u,
where

(1.8) b α = (γ α -1)m α k B > 0, α = e, i.
It was proved in [START_REF] Aregba-Driollet | Modelling and numerical approximation for the nonconservative bitemperature Euler model[END_REF] (see Theorem 2.9) that the functions (η, φ) defined in (1.7) are a pair of entropy-entropy flux of (1.5), and η is strictly convex in the set of state space Ω given by

Ω = (ρ, u, ε e , ε i ) ∈ R d+3 ρ > 0, ε e > 0, ε i > 0 .
Moreover, any smooth solution of the system satisfies the entropy equality (1.9)

∂ t η(ρ, ρu, E e , E i ) + div φ(ρ, ρu, E e , E i ) = - νρ T e T i (T e -T i ) 2 ,
which is a partially dissipative condition of the system. It is known that the second-order derivative of a strictly convex entropy provides a symmetrizer of a hyperbolic system in conservative form (see [START_REF] Godunov | An interesting class of quasi-linear systems (Russian)[END_REF][START_REF] Boillat | Sur l'existence et la recherche d'équations de conservation supplémentaires pour les systèmes hyperboliques[END_REF]). Unfortunately, the equations for E e and E i in (1.5) are not in conservative form. As already noticed in [START_REF] Aregba-Driollet | A viscous approximation of the bitemperature Euler system[END_REF], η (W) is not a symmetrizer of system (1.5). For the sake of completeness we prove this result in the Appendix of the present article.

According to the theory on the symmetrizable hyperbolic system [START_REF] Lax | Hyperbolic systems of conservation laws and the mathematical theory of shock waves[END_REF][START_REF] Kato | The Cauchy problem for quasilinear symmetric hyperbolic systems[END_REF][START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF], the existence of a symmetrizer is very important to study smooth solutions in Sobolev spaces. Such a symmetrizer for (1.5) is constructed in Section 2 in any space dimension. It implies the local existence of smooth solutions. See B 0 (V) defined in (2.10) and Proposition 2.1.

In order to study global existence, we may introduce the total energy E and the total pressure p defined by

E = E e + E i , p = p e + p i . From (1.3)-(1.5), we have E = p e γ e -1 + p i γ i -1 + 1 2 ρ|u| 2 , p = ρ (γ e -1)c e ε e + (γ i -1)c i ε i and (1.10)      ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p = 0, ∂ t E + div u(E + p) = 0, t > 0, x ∈ R d .
The last equation in (1.10) shows that the total energy is a conservative variable. If γ e = γ i , we introduce a total internal specific energy ε by ε = c e ε e + c i ε i . Then

E = ρε + 1 2 ρ|u| 2 , p = (γ e -1)ρε.
Therefore, (1.10) becomes the gas dynamics equations. In this case, system (1.5) is decoupled and contains (1.10). It is known that smooth solutions to the gas dynamics equations blow up in finite time [START_REF] Lax | Development of singularities of solutions of nonlinear hyperbolic partial differential equations[END_REF][START_REF] Sideris | Formation of singularities in three-dimensional compressible fluids[END_REF]. Hence, global existence is not expected. In physically realistic situations, one can define a weak solution containing shocks. Existence and uniqueness of weak entropy solutions is rather well understood for one-dimensional strictly hyperbolic systems of conservation laws, see [START_REF] Bressan | Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem[END_REF] and references therein. For systems with non-conservative products, the authors of [START_REF] Maso | Definition and weak stability of nonconservative products[END_REF] gave a definition of shocks, but to our knowledge there is no result on the existence of such solutions for (1.5).

In what follows, we consider the Cauchy problem for (1.5) near constant equilibrium states in case γ e = γ i . Let us introduce V = ρ, u T , ε e , ε i T .

An equilibrium state V is a constant solution of (1.5). We consider in particular the equilibrium state with zero velocity. Let V = (1, 0, εe , εi ) T , which is such an equilibrium state with εe > 0 and εi > 0.

System (1.5) is supplemented by an initial condition

(1.11) t = 0 : V = V 0 (x) def = ρ 0 (x), u T 0 (x), ε e0 (x), ε i0 (x) T , x ∈ R d .
For a positive integer m we denote by H m (R d ) the usual Sobolev space equipped with the norm • m . The result of the global existence of solutions holds in one space dimension and can be stated as follows.

Theorem 1.1. Let d = 1 and m ≥ 2. Assume V 0 -V ∈ H m (R) and γ e = γ i . There are two positive constants c and κ 0 such that if V 0 -V m ≤ κ 0 , then the Cauchy problem (1.5) and (1.11) admits a unique global solution

V satisfying V -V ∈ C(R + ; H m (R)) ∩ C 1 (R + ; H m-1 (R)). Moreover, (1.12) sup t∈R + V(t, •) -V m ≤ c V 0 -V m .
For conservative hyperbolic systems with source terms, the global existence of smooth solutions near constant equilibrium states was proved in [START_REF] Hanouzet | Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy[END_REF][START_REF] Yong | Entropy and Global Existence for Hyperbolic Balance Laws[END_REF] in a general framework under two main conditions. A typical example in this framework can be seen in [START_REF] Sideris | Long time behavior of solutions to the 3D compressible Euler equations with damping[END_REF][START_REF] Coulombel | The strong relaxation limit of the multidimensional isothermal Euler equations[END_REF] for the gas dynamics equations with damping. The first condition required in [START_REF] Hanouzet | Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy[END_REF][START_REF] Yong | Entropy and Global Existence for Hyperbolic Balance Laws[END_REF] is an entropy dissipation near an equilibrium state. It implies in particular an L 2 energy estimate of solutions. The second one is the classical Shizuta-Kawashima condition (SK) at the equilibrium state [START_REF] Shizuta | Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation[END_REF]. Unfortunately, these two conditions are not satisfied by system (1.5). The first condition fails because (1.5) is not a conservative system. However, it is known that condition (SK) is not necessary for the global existence of smooth solutions. There do exist conservative systems for which global existence holds without this condition. We refer the reader to [START_REF] Zeng | Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation[END_REF][START_REF] Carbou | Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation[END_REF][START_REF] Peng | Relaxation limit and global existence of smooth solutions of compressible Euler-Maxwell equations[END_REF][START_REF] Mori | Global existence and optimal decay rates for the Timoshenko system: the case of equal wave speeds[END_REF] for examples in which different techniques are employed to avoid condition (SK).

Thus, it is important to establish a global existence result for a class of systems including at least one of these examples. In [START_REF] Mascia | On relaxation hyperbolic systems violating the Shizuta-Kawashima condition[END_REF] the authors studied energy estimates of smooth solutions near non-constant equilibrium states for conservative systems. In one space dimension, they obtained global existence for systems violating condition (SK) but admitting a very special structure. This allows them to give a proof of global existence by using only a partially dissipative condition via an entropy dissipation. This situation is different from that of the present paper. In the proof of Theorem 1.1, we not only need a partially dissipative condition but also a dissipation estimate for other variables (see Lemma 3.5). In [START_REF] Qu | Global classical solutions to partially dissipative hyperbolic systems violating the Kawashima condition[END_REF] the authors tried to explore a link between the linear degeneracy of characteristic fields and condition (SK) for conservative systems. Under restrictive conditions, they obtained time-decay estimates of solutions which imply global existence. One can check that the conditions in [START_REF] Mascia | On relaxation hyperbolic systems violating the Shizuta-Kawashima condition[END_REF] and [START_REF] Qu | Global classical solutions to partially dissipative hyperbolic systems violating the Kawashima condition[END_REF] are not fulfilled by (1.5) and the systems in [START_REF] Zeng | Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation[END_REF][START_REF] Carbou | Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation[END_REF][START_REF] Peng | Relaxation limit and global existence of smooth solutions of compressible Euler-Maxwell equations[END_REF][START_REF] Mori | Global existence and optimal decay rates for the Timoshenko system: the case of equal wave speeds[END_REF].

Up to our knowledge, Theorem 1.1 gives a first result on the global existence of smooth solutions for a non-conservative nonlinear hyperbolic system with source terms without condition (SK). The proof of this theorem is based on the local existence of solutions and uniform energy estimates with respect to time through Lagrangian coordinates. It consists of three steps. The first step concerns an L 2 energy estimate. For this purpose, the entropy equality (1.9) is not sufficient because the system is not in conservative form. We need further to prove equilibrium conditions between the system and the entropy η given in (1.7) at the equilibrium state. The verification of these conditions is very complicated and tedious for (1.5). To avoid this, we turn to consider the Cauchy problem in Lagrangian coordinates where these conditions can be easily checked (see Lemma 3.1). The second step is to establish higher-order energy estimates with a dissipation estimate for T e -T i . This is a classical step which is done by choosing an appropriate symmetrizer of the system (see Lemma 3.4). In the last step, we prove a dissipation estimate for (∇u, ∇p) (see Lemma 3.5). In view of special structures of the system, these estimates are sufficient to obtain the global existence of solutions in Lagrangian coordinates. Then Theorem 1.1 follows from the equivalence result for the solutions between Eulerian and Lagrangian coordinates. Remark that in the proof of Theorem 1.1, we need to use different independent unknown variables in different energy estimates. The difficulty on the lack of condition (SK) for system (1.5) is overcome by choosing appropriate variables connected by C ∞ -diffeomorphisms.

Finally, we point out that there exists a result on the global existence of solutions for partially dissipative hyperbolic systems in non-conservative form which satisfy condition (SK). However, the space dimension is required to be bigger than 3 [START_REF] Peng | Uniform global existence and parabolic limit for partially dissipative hyperbolic systems[END_REF] (see Theorem 2.4). System (1.5) is not included in this framework since it does not satisfy condition (SK). So far, global existence in several space dimensions is an open problem for (1.5).

This paper is organized as follows. In the next section, we first exhibit a symmetrizer to apply the result on the local existence of smooth solutions in several space dimensions. Then we study the structure of the system in one space dimension in Euler and Lagrangian coordinates. In particular, we show that system (1.5) does not satisfy condition (SK). We also state a result on the global existence of solutions for the system in Lagrangian coordinates (see Theorem 2.1). Section 3 is devoted to the proof of the energy estimates in the three steps mentioned above. In the last section, we give the proof of Theorem 2.1 and then the proof of Theorem 1.1 by using a result on the equivalence of solutions for the Cauchy problem in Euler and Lagrangian coordinates.

Study of the bitemperature Euler model

           ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p = 0, ∂ t E e + div u(E e + p e ) -u • (c i ∇p e -c e ∇p i ) = ν(ρ)v, ∂ t E i + div u(E i + p i ) + u • (c i ∇p e -c e ∇p i ) = -ν(ρ)v, t > 0, x ∈ R d , with relations (1.1), (1.2), (1.3), (1.4), (1.8) and v = ρ(T i -T e ), T α = b α ε α , α = e, i.
Now we write the system with variables (ρ, u, ε e , ε i ). We first remark that

(2.2) div(ρu ⊗ u) = ρ(u • ∇)u + u div(ρu), p = p e + p i .
Then, for ρ > 0, the first two equations in (2.1) give

(2.3) ∂ t u + (u • ∇)u + ρ -1 ∇p = 0.
By the definition of E α and the first two equations in (2.1) together with (2.3), we have

1 c α ∂ t E α + div(uE α ) = 1 2 ρu • ∂ t u + 1 2 u • ∂ t (ρu) + ρ∂ t ε α + ε α ∂ t ρ + div 1 2 ρ|u| 2 u + ρuε α = - 1 2 ρu • (u • ∇)u + ρ -1 ∇p - 1 2 u • div(ρu ⊗ u) + ∇p + ρ∂ t ε α -ε α div(ρu) + div 1 2 ρ|u| 2 u + ρuε α . Since -ρu • (u • ∇)u] = - 1 2 ρu • ∇(|u| 2 ), div 1 2 ρ|u| 2 u = 1 2 |u| 2 div(ρu) + 1 2 ρu • ∇(|u| 2 ), using (2.
2), we obtain

- 1 2 ρu • (u • ∇)u + ρ -1 ∇p - 1 2 u • div(ρu ⊗ u) + ∇p + div 1 2 ρ|u| 2 u = - 1 2 ρu • ∇(|u| 2 ) -u • ∇p - 1 2 |u| 2 div(ρu) + div 1 2 ρ|u| 2 u) = -u • ∇p.
We also have

-ε α div(ρu) + div(ρuε α ) = ρu • ∇ε α .
These equalities imply that 1

c α ∂ t E α + div(uE α ) = ρ∂ t ε α + ρu • ∇ε α -u • ∇p. Moreover, (2.4) div(up e ) -u • (c i ∇p e -c e ∇p i ) = p e div u + c e u • ∇p, div(up i ) + u • (c i ∇p e -c e ∇p i ) = p i div u + c i u • ∇p.
It follows that 1 c e ∂ t E e + div u(E e + p e ) -u • (c i ∇p e -c e ∇p i ) = ρ∂ t ε e + ρu • ∇ε e + 1 c e p e div u,

1 c i ∂ t E i + div u(E i + p i ) + u • (c i ∇p e -c e ∇p i ) = ρ∂ t ε i + ρu • ∇ε i + 1 c i p i div u.
Finally, by the expression of p α and the last two equations in (2.1), we obtain

∂ t ε e + u • ∇ε e + (γ e -1)ε e div u = ν(ρ)(c e ρ) -1 v, ∂ t ε i + u • ∇ε i + (γ i -1)ε i div u = -ν(ρ)(c i ρ) -1 v,
which are the equations for ε e and ε i . Thus, system (2.1) is equivalent to

(2.5)            ∂ t ρ + div(ρu) = 0, ∂ t u + (u • ∇)u + ρ -1 ∇p = 0, ∂ t ε e + u • ∇ε e + (γ e -1)ε e div u = ν(ρ)(c e ρ) -1 v, ∂ t ε i + u • ∇ε i + (γ i -1)ε i div u = -ν(ρ)(c i ρ) -1 v, t > 0, x ∈ R d , where (2.6) p = ρ[c e (γ e -1)ε e + c i (γ i -1)ε i ], v = ρ(b i ε i -b e ε e ). Let V = (ρ, u T , ε e , ε i ) T , ε 1 = c e (γ e -1)ε e + c i (γ i -1)ε i ,
where the superscript T denotes the transpose of a vector. Since p = ρε 1 and

ρ -1 ∇p = ρ -1 ε 1 ∇ρ + c e (γ e -1)∇ε e + c i (γ i -1)∇ε i , system (2.5) is written in the form (2.7) ∂ t V + d j=1 B j (V)∂ x j V = H(V), t > 0, x = (x 1 , • • • , x d ) ∈ R d , where (2.8) B j (V) =      u j ρe T j 0 0 ρ -1 ε 1 e j u j I d c e (γ e -1)e j c i (γ i -1)e j 0 (γ e -1)ε e e T j u j 0 0 (γ i -1)ε i e T j 0 u j     
, and

(2.9)

H(V) =      0 0 ν(ρ)(c e ρ) -1 v -ν(ρ)(c i ρ) -1 v      , with u = (u 1 , • • • , u d ) T , I d being the unit matrix and (e 1 , • • • , e d ) being the standard basis of R d .
By a symmetrizer B 0 (V) for system (2.7) we mean that B 0 (V) is a symmetric positive definite matrix such that B 0 (V)B j (W) is symmetric for all j ∈ {1, 2, • • • , d} (see [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF]). Now we introduce a diagonal matrix (2.10)

B 0 (V) = diag ε 1 ε e ε i , ρ 2 ε e ε i I d , c e ρ 2 ε i , c i ρ 2 ε e .
Obviously, B 0 (V) is symmetric positive definite in Ω. Moreover,

B 0 (V)B j (V) =      u j ε 1 ε e ε i ρε 1 ε e ε i e T j 0 0 ρε 1 ε e ε i e j ρ 2 u j ε e ε i I d c e (γ e -1)ρ 2 ε e ε i e j c i (γ i -1)ρ 2 ε e ε i e j 0 c e (γ e -1)ρ 2 ε e ε i e T j c e ρ 2 u j ε i 0 0 c i (γ i -1)ρ 2 ε e ε i e T j 0 c i ρ 2 u j ε e     
, which is a symmetric matrix. Therefore, B 0 (V) is a symmetrizer and system (2.7) is symmetrizable hyperbolic in the sense of Friedrichs. According to Lax [START_REF] Lax | Hyperbolic systems of conservation laws and the mathematical theory of shock waves[END_REF] or Kato [START_REF] Kato | The Cauchy problem for quasilinear symmetric hyperbolic systems[END_REF] (see also Majda [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF]), for smooth initial data, the Cauchy problem for (2.1) admits a unique smooth solution, locally in time. This result is stated as follows and it holds in any space dimension.

Proposition 2.1. Let m > d/2 + 1 be an integer and εe > 0 and εi > 0 be two constants. We suppose that

V 0 -V ∈ H m (R d ) and
(2.11) inf

x∈R d ρ 0 (x) > 0, inf x∈R d ε e0 (x) > 0, inf x∈R d ε i0 (x) > 0.
Then, there exist T > 0 and a unique smooth solution V to the Cauchy problem (1.5) and (1.11). This solution satisfies

V -V ∈ C([0, T ]; H m (R d )) ∩ C 1 ([0, T ]; H m-1 (R d )) and inf (t,x)∈[0,T ]×R d ρ(t, x) > 0, inf (t,x)∈[0,T ]×R d ε e (t, x) > 0, inf (t,x)∈[0,T ]×R d ε i (t, x) > 0. Remark 2.1. Condition V 0 -V m ≤ κ 0 in Theorem 1.
1 with κ 0 being sufficiently small implies (2.11).

2.2. The system in one space dimension. In one space dimension, systems (2.1) and (2.5) are written as :

(2.12)

           ∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 + p) = 0, ∂ t E e + ∂ x u(E e + p e ) -u(c i ∂ x p e -c e ∂ x p i ) = ν(ρ)v, ∂ t E i + ∂ x u(E i + p i ) + u(c i ∂ x p e -c e ∂ x p i ) = -ν(ρ)v, t > 0, x ∈ R and (2.13)            ∂ t ρ + ∂ x (ρu) = 0, ∂ t u + u∂ x u + ρ -1 ∂ x p = 0, ∂ t ε e + u∂ x ε e + (γ e -1)ε e ∂ x u = ν(ρ)(c e ρ) -1 v, ∂ t ε i + u∂ x ε i + (γ i -1)ε i ∂ x u = -ν(ρ)(c i ρ) -1 v, t > 0, x ∈ R,
respectively. From (2.6) and (2.13), we further obtain

       ∂ t (ρ 2 ε e ) + ∂ x (ρ 2 ε e u) + γ e ρ 2 ε e ∂ x u = 1 c e ν(ρ)ρv, ∂ t (ρ 2 ε i ) + ∂ x (ρ 2 ε i u) + γ i ρ 2 ε i ∂ x u = - 1 c i ν(ρ)ρv, which imply that (2.14)    ∂ t (ρp) + ∂ x (ρpu) + ρµ 1 ∂ x u = (γ e -γ i )ν(ρ)ρv, ∂ t (ρv) + ∂ x (ρvu) + ρµ 2 ∂ x u = - b i c i + b e c e ν (
ε e + c i γ i (γ i -1)ε i , µ 2 = ρ b i γ i ε i -b e γ e ε e .
By (2.6) and the expression of µ 2 above, we see that µ 1 and µ 2 can further be expressed as linear functions of p and v as (2.16)

µ 1 µ 2 = M p v ,
where M is a constant invertible matrix given by (2.17)

M = 1 (m e c i + m i c e )k B k B (m e c i γ i + m i c e γ e ) c e c i k 2 B (γ i -γ e ) m e m i (γ i -γ e ) k B (m e c i γ e + m i c e γ i )
.

By the expression of B j given in (2.8), we can calculate the eigenvalues λ i and the eigenvectors r i of (2.13). They are given by

λ 1 (V) = u -a, λ 2 (V) = λ 3 (V) = u, λ 4 (V) = u + a, r 1 (V) =     ρ -a (γ e -1)ε e (γ i -1)ε i     , r 2 (V) =     0 0 -(γ i -1)c i (γ e -1)c e     , r 3 (V) =     -ρ 0 ε e ε i     , r 4 (V) =     ρ a (γ e -1)ε e (γ i -1)ε i     ,
where a(ε e , ε i ) = c e γ e (γ e -1)ε e + c i γ i (γ i -1)ε i .

Moreover, by (2.9), we have

H ( V) =      0 0 0 0 0 0 0 0 0 0 -be ce ν(1) b i ce ν(1) 0 0 be c i ν(1) -b i c i ν(1)      .
It is known that condition (SK) is invariant under a change of unknown variables by a C 1diffeomorphism [START_REF] Hanouzet | Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy[END_REF]. This condition shows a coupling property between the eigenvectors and the source terms of the system. At a given equilibrium state V, it means that H ( V)r i ( V) = 0 for all i = 1, 2, 3, 4. From (3.3), we see easily that H ( V)r 3 ( V) = 0. This shows that condition (SK) is not satisfied for system (2.13).

2.3. The system in Lagrangian coordinates.

Let (ρ, u) ∈ C 1 (R + × R) satisfying ρ ≥ const > 0 in R + × R and (2.18) ∂ t ρ + ∂ x (ρu) = 0.
The Euler-Lagrange change of variables from (t, x) to (t , y) is defined by

t = t, dy = ρdx -ρudt,
or equivalently for y :

y = x X 1 (t) ρ(t, ξ)dξ, with X 1 (t) = u(t, X 1 (t)).
It is clear that this change of variables is a diffeomorphism from R + × R to itself. For simplicity, we use the same notation for unknown variables in Euler coordinates (t, x) and in Lagrangian coordinates (t, y). Consider smooth solutions for (2.12). Let

(2.19) τ = ρ -1 , E α = τ E α = 1 2 c α u 2 + c α ε α , α = e, i.
Given a first-order partial differential equation (2.20)

∂ t w + ∂ x z 1 + b∂ x z 2 = f.
By (2.18), in Lagrangian coordinates this equation is written equivalently as

(2.21) ∂ t (τ w) + ∂ y (z 1 -wu) + b∂ y z 2 = τ f.
Applying this to (2.12), we obtain

(2.22)          ∂ t τ -∂ y u = 0, ∂ t u + ∂ y p = 0, ∂ t E e + p e ∂ y u + c e u∂ y p = ντ v, ∂ t E i + p i ∂ y u + c i u∂ y p = -ντ v.
Similarly to (2.14), by (2.4), we obtain

(2.23)    ∂ t p + τ -1 µ 1 ∂ y u = (γ e -γ i )νv, ∂ t v + τ -1 µ 2 ∂ y u = - b i c i + b e c e νv,
where µ 1 and µ 2 are given in (2.16).

Regarding p α and p as functions of (τ, u, E e , E i ), we have

p α = 1 τ (γ α -1) E α - 1 2 c α u 2 , α = e, i, p = p e + p i .
Hence, system (2.22) can be written as (2.24)

∂ t U + A(U)∂ y U = G(U), t > 0, y ∈ R, U = (τ, u, E e , E i ) T ,
which is supplemented by an initial condition (2.25)

t = 0 : U = U 0 (y) def = τ 0 (y), u 0 (y), E e0 (y), E i0 (y) , y ∈ R.
Here,

A(U) =      0 -1 0 0 ∂ τ p ∂ u p ∂ Ee p ∂ E i p c e u∂ τ p p e + c e u∂ u p c e u∂ Ee p c e u∂ E i p c i u∂ τ p p i + c i u∂ u p c i u∂ Ee p c i u∂ E i p      , G(U) =      0 0 ντ v -ντ v      , with ∂ τ p = - p τ , ∂ u p = - [c e (γ e -1) + c i (γ i -1)]u τ , ∂ Ee p = γ e -1 τ , ∂ E i p = γ i -1 τ . Let Ū = (1, 0, Ēe , Ēi ) T ,
which is an equilibrium state of (2.24) with Ēe > 0 and Ēi > 0. The result of global existence of solutions to (2.24)-(2.25) can be stated as follows.

-Ū ∈ C(R + ; H m (R)) ∩ C 1 (R + ; H m-1 (R)). Moreover, (2.26) sup t∈R
+ U(t, •) -Ū 2 m + +∞ 0 ∂ y u 2 m-1 + ∂ y p 2 m-1 + v 2 m dt ≤ c U 0 -Ū 2 m .

Energy estimates in Lagrangian coordinates

We study energy estimates for the Cauchy problem (2.24)-(2.25). Let m ≥ 2 be an integer and T > 0 such that the local smooth solution U is defined on time interval [0, T ]. We denote by • , • ∞ and • l the usual norms of L 2 (R), L ∞ (R) and H l (R) for l ∈ N, respectively. We also denote

U T = max t∈[0,T ] U(t, •) -Ū m .
We consider a smooth solution U near Ū, namely, U T is small. In the proof below, we denote by C > 0 and c 0 > 0 generic constants independent of t and T .

The global existence of smooth solutions to (2.24)-(2.25) will be proved in the three steps shown in Introduction.

3.1. An L 2 estimate. We first look at the entropy equality (1.9) in Lagrangian coordinates. From (2.6), we have

       ε e = (m i p -c i k B v)τ (γ e -1)(m e c i + m i c e ) , ε i = (m e p + c e k B v)τ (γ i -1)(m e c i + m i c e ) ,
which are strictly positive in a neighborhood of v = 0 when τ > 0 and p > 0. It follows from the definition of b α , T α and v that

- νρ T e T i (T i -T e ) 2 = -ν 1 ρv 2 ,
where ν 1 = ν 1 (τ, p, v) given by (3.1)

ν 1 = k 2 B (m e c i + m i c e ) 2 ν m e m i (m i p -c i k B v)(m e p + c e k B v) .
It is clear that, for all τ > 0 and p > 0, ν 1 > 0 in a neighborhood of v = 0. We introduce a new variable

s = τ η = - α=e,i c α b α ln (γ α -1)τ γα-1 c γα α E α - c α 2 u 2 ,
which is a function of variable U. According to the equivalence of equations (2.20) and (2.21) in two coordinates, the entropy equality (1.9) in variables (t, y) becomes

(3.2) ∂ t s = -ν 1 v 2 ,
which means that s is an entropy of system (2.24) with 0 as its entropy-flux.

Recall that an equilibrium state with zero velocity is of the form

V = (1, 0, εe , εi ) T .
By definition, V is an equilibrium state for (2.7) if H( V) = 0. Since ν > 0, by the definition of H and v, we have where

F(U) = 1 Ē * up -k B c e m e + c i m i u.
Proof. A straightforward calculation gives

-∂ τ s(U) = k B τ c e m e + c i m i , -∂ u s(U) = - c e b e ε e + c i b i ε i u, -∂ Ee s(U) = 1 b e ε e , -∂ E i s(U) = 1 b i ε i , where ε α = 1 c α E α - 1 2 u 2 , α = e, i.
Therefore,

-∂ τ s( Ū) = k B c e m e + c i m i , -∂ u s( Ū) = 0, -∂ Ee s( Ū) = -∂ E i s( Ū) = 1 Ē * .
Hence, it is easy to check that (3.4)-(3.5) are satisfied. 2

Lemma 3.2. In a neighborhood of Ū, it holds

(3.6) U(t, •) -Ū 2 + t 0 v(t , •) 2 dt ≤ C U 0 -Ū 2 , ∀ t ∈ [0, T ].
Proof. We introduce S(U) = s(U) -s( Ū) + ∇s( Ū)(U -Ū). Since η is a strictly convex entropy for (2.1), by a result in [START_REF] Wagner | Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions[END_REF], s is a strictly convex entropy for (2.24). Hence, by Taylor formula, in a neighborhood of Ū, these exist two constants

c 2 ≥ c 1 > 0 such that c 1 |U -Ū| 2 ≤ S(U) ≤ c 2 |U -Ū| 2 .
Using (2.24) and (3.2), we have

∂ t S -∇s( Ū)A(U)∂ y U = -ν 1 v 2 -∇s( Ū)G(U).
It follows from Lemma 3.1 that

∂ t S -∂ y F(U) = -ν 1 v 2 .
In a neighborhood of Ū, there is a constant ν1 > 0 such that ν 1 ≥ ν1 . Thus, integrating this equality over [0, t] × R with t ∈ [0, T ], we obtain (3.6). 2

3.2. Higher-order energy estimates. Let U = (u, p, v, s) T . We use variable U in higherorder energy estimates. From (2.22)-(2.23) and (3.2), we have

(3.7)            ∂ t u + ∂ y p = 0, ∂ t p + τ -1 µ 1 ∂ y u = -(γ i -γ e )νv, ∂ t v + τ -1 µ 2 ∂ y u = -bνv, ∂ t s = -ν 1 v 2 , t > 0, x ∈ R,
where ν 1 is defined in (3.1), µ 1 and µ 2 are defined in (2.16)-(2.17), and

b = b i c i + b e c e > 0.
In particular, µ 1 and µ 2 are linear functions of p and v. This system can be written as

(3.8) ∂ t U + A(U )∂ y U = G(U ), t > 0, x ∈ R,
where

A(U ) =      0 1 0 0 τ -1 µ 1 0 0 0 τ -1 µ 2 0 0 0 0 0 0 0      , G(U ) =      0 -(γ i -γ e )νv -bνv -ν 1 v 2     
, and τ is regarded as a function of U . By the definition in (2.6), the equilibrium state for U is Ū = (0, p, v, s) with p = (γ e -1) Ēe + (γ i -1) Ēi > 0, v = 0, s = s( Ū).

We first prove the following useful property.

Lemma 3.3. Let δ(p, v) be defined by

(3.9) δ(p, v) = bµ 1 (p, v) -(γ i -γ e )µ 2 (p, v).
There is a constant δ > 0 such that δ(p, v) ≥ δ in a neighborhood of (p, 0).

Proof. By continuity, it is sufficient to prove that δ(p, 0) > 0. From (1.2), we have

c e c i = Zm e m i .
It follows from the definition of b α in (1.8) that b i c e (γ e -1)

c i = Zb e (γ i -1), b e c i (γ i -1) c e = b i Z (γ e -1).
Since ρ = 1, from (2.15) we obtain

δ(p, 0) = γ i -1 + γ e -1 Z (Zγ e b e εe + γ i b i εi ) -(γ i -γ e ) (γ i b i εi -γ e b e εe ) .
Using the fact that b e εe = b i εi > 0 (see (3.3)), δ(p, 0) > 0 if and only if

γ i -1 + γ e -1 Z (Zγ e + γ i ) > (γ i -γ e ) 2 ,
or equivalently,

γ e Z(γ i -1) + γ i (γ e -1) > 0. Lemma 3.3 is proved since Z > 0, γ i > 1 and γ e > 1. 2
Lemma 3.4. Let the conditions of Theorem 2.1 hold. If U -Ū m is sufficiently small, for all t ∈ [0, T ], we have

U(t, •) -Ū 2 m + t 0 v(t , •) 2 m dt (3.10) ≤ C U 0 -Ū 2 m + C t 0 ∂ y u 2 m-1 + ∂ y p 2 m-1 + v 2 m U -Ū m dt . Proof. Let A 0 (U ) =      µ 0 0 0 0 0 bµ 2 γ i -γe -µ 2 0 0 -µ 2 µ 1 0 0 0 0 1      , where µ 0 (τ, p, v) = 1 (γ i -γ e )τ µ 2 (p, v)δ(p, v),
and δ(p, v) is defined in (3.9). By lemma 3.3, in a neighborhood of Ū, there are positive constants μ1 , μ2 and μ0 such that

µ 1 (p, v) ≥ μ1 , (γ i -γ e )µ 2 (p, v) ≥ μ2 , µ 0 (τ, p, v) ≥ μ0 .
Then it is easy to check that, in a neighborhood of Ū, A 0 (U ) is a symmetrizer of system (3.8), namely, A 0 (U ) is symmetric positive definite and A 0 (U )A(U ) is symmetric. In particular,

A 0 (U )A(U ) =      0 µ 0 0 0 µ 0 0 0 0 0 0 0 0 0 0 0 0      , A 0 (U )G(U ) =      0 0 -δνv -ν 1 v 2      . Let 1 ≤ k ≤ m be an integer. We denote U k = ∂ k y U . From (3.8), we have (3.11) ∂ t U k + A(U )∂ y U k = ∂ k y G(U ) + J k , where J k = A(U )∂ y U k -∂ k y (A(U )∂ y U ).
Taking the inner product of (3.11) with A 0 (U )U k in L 2 (R), we obtain the Friedrichs energy equality (3.12)

d dt A 0 (U )U k , U k = 2 A 0 (U )∂ k y G(U ), U k + 2 A 0 (U )J k , U k + div A(U )U k , U k ,
where •, • is the inner product of L 2 (R) and

div A(U ) = ∂ t A 0 (U ) + ∂ y Ã(U ), Ã = A 0 A.
By the definition of A 0 and Ã, we have div

A(U ) =      ∂ t µ 0 ∂ y µ 0 0 0 ∂ y µ 0 b γ i -γe ∂ t µ 2 -∂ t µ 2 0 0 -∂ t µ 2 ∂ t µ 1 0 0 0 0 0      , with ∂ y µ 0 (U ) = µ 0 (U )∂ y U, ∂ t µ i (U ) = µ i (U )∂ t U = µ i (U ) G(U ) -A(U )∂ y U , i = 0, 1, 2.
Since G(U ) = O(v) and the imbedding from

H m (R) to W 1,∞ (R) is continuous, we obtain (3.13) div A(U )U k , U k ≤ C ∂ y u 2 m-1 + ∂ y p 2 m-1 + ∂ y v 2 m-1 U -Ū m . Next, a direct calculation yields A 0 (U )J k =      0 τ -1 b γ i -γe µ 1 -µ 2 ∂ k+1 y u -∂ k y (τ -1 b γ i -γe µ 1 -µ 2 ∂ y u) µ 2 µ 2 ∂ k y (τ -1 µ 1 ∂ y u) -µ 1 ∂ k y (τ -1 µ 2 ∂ y u) 0      .
By the Moser-type inequalities [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF], we have

(3.14) 2 A 0 (U )J k , U k ≤ C ∂ y u 2 m-1 + ∂ y p 2 m-1 + ∂ y v 2 m-1 U -Ū m .
Moreover,

∂ k y G(U ) =      0 -(γ i -γ e )ν∂ k y v -bν∂ k y v 0      +      0 (γ i -γ e ) ν∂ k y v -∂ k y (νv) bν∂ k y v -b∂ k y (νv) -∂ k y (ν 1 v 2 )      def = G 1 + G 2 , with A 0 (U )G 1 =      0 0 -δν∂ k y v 0      and A 0 (U )G 2 =      0 a 22 (γ i -γ e ) ν∂ k y v -∂ k y (νv) -µ 2 bν∂ k y v -b∂ k y (νv) µ 1 bν∂ k y v -b∂ k y (νv) -µ 2 (γ i -γ e ) ν∂ k y v -∂ k y (νv) -∂ k y (ν 1 v 2 )     
, where

a 22 = bµ 2 γ i -γ e .
These equalities imply that

A 0 (U )G 1 • U k = -δν|∂ k y v| 2 and A 0 (U )G 2 • U k = a 22 (γ i -γ e ) ν∂ k y v -∂ k y (νv) -µ 2 bν∂ k y v -b∂ k y (νv) ∂ k y u + µ 1 bν∂ k y v -b∂ k y (νv) -µ 2 (γ i -γ e ) ν∂ k y v -∂ k y (νv) ∂ k y p -∂ k y (ν 1 v 2 )∂ k y s. (3.15)
Observe that each of three terms on the right-hand side of (3.15) is quadratic in variables (u, p, v) with coefficients depending on derivatives of U -Ū up to order m. Moreover, using Lemma 3.3, we have δν ≥ δν in a neighborhood of Ū, where ν > 0 is a constant. Thus, the Moser-type inequalities imply that

(3.16) A 0 (U )∂ k y G(U ), U k + δν ∂ k y v 2 ≤ C ∂ y u 2 m-1 + ∂ y p 2 m-1 + v 2 m U -Ū m . Since A 0 (U ) is positive definite, A 0 (U )U k , U k is equivalent to U k 2 .
Combining (3.12)-(3.16) and integrating (3.12) over [0, t] with t ∈ [0, T ], we have

U k 2 + t 0 ∂ k y v(t , •) 2 dt ≤ C U 0 -Ū 2 m + C t 0 ∂ y u 2 m-1 + ∂ y p 2 m-1 + v 2 m U -Ū m dt ,
where U 0 is the initial data of U . Finally, the change of variables U -→ U is a C ∞ -diffeomorphism in a neighborhood of Ū . Then, U -Ū l is equivalent to U -Ū l for all l ∈ N. Summing up this inequality for all k = 1, 2, • • • , m, and using Lemma 3.2, we obtain (3.10). 2

Finally, since 0 ≤ k ≤ m -1, we have

∂ k y v, ∂ k+1 y u + ∂ k y u, ∂ k+1 y p ≤ C U -Ū 2 m . Integrating (3.21) over [0, t] with t ∈ [0, T ], we obtain t 0 ∂ k+1 y u 2 + ∂ k+1 y p 2 dt ≤ C U -Ū 2 m + C U 0 -Ū 2 m + C t 0 v 2 m + ∂ y u 2 m-1 U -Ū m dt .
Summing this inequality for all k = 0, 1, • • • , m -1 and using Lemma 3.4, we obtain (3.17). 2

3.4. Proof of Theorem 2.1. From (3.10) and (3.17), we have

U(t, •) -Ū 2 m + t 0 ∂ y u 2 m-1 + ∂ y p 2 m-1 + v 2 m dt ≤ C U 0 -Ū 2 m + CU T t 0 ∂ y u 2 m-1 + ∂ y p 2 m-1 + v 2 m dt , ∀ t ∈ [0, T ].
Since U T is sufficiently small, we further obtain

U(t, •) -Ū 2 m + t 0 ∂ y u 2 m-1 + ∂ y p 2 m-1 + v 2 m dt ≤ C U 0 -Ū 2 m , ∀ t ∈ [0, T ].
This estimate together with a bootstrap argument implies (2.26) and the global existence of a solution U to (2.24)-(2.25), provided that U 0 -Ū m is sufficiently small. 2

4. Proof of Theorem 1.1

For the Cauchy problem for (2.12) with initial data given in (1.11), we first define

Y 0 (x) = x 0 ρ 0 (ξ)dξ.
Then Y 0 = ρ 0 . By the condition in Theorem 1.1, we have inf

x∈R ρ 0 (x) > 0 and ρ 0 -1 ∈ H m (R). Therefore, the continuous imbedding from H m (R) to C m-1 (R) implies that Y 0 is a C m -diffeomorphism from R to R. We denote by X 0 the inverse C m -diffeomorphism of Y 0 and define U 0 (y) = 1 ρ 0 , u 0 , 1 2 c e u 2 0 + c e ε e0 , 1 2 c i u 2 0 + c i ε i0 (X 0 (y)). Then condition V 0 -V ∈ H m (R) implies that U 0 -Ū ∈ H m (R) and condition V 0 -V m ≤ κ 0
with κ 0 being sufficiently small implies that U 0 -Ū m is sufficiently small. According to Theorem 2.1, there exists a global smooth solution U(t, y) = (τ (t, y), u(t, y), E e (t, y), E i (t, y)) T to the Cauchy problem (2.24)-(2.25). Then, we define

ρ(t, y) = 1 τ (t, y) , ε α (t, y) = 1 c α E α (t, y) - 1 2 u 2 (t, y), α = e, i.
On the other hand, the result in Theorem 2.1 also implies that U ∈ C 1 (R + × R) and U is globally Lipschitzian on R with respect to y (in particular for τ and u). Then the Cauchy problem to the following ordinary differential equation

Y 1 (t) = u(t, Y 1 (t)), Y 1 (0) = 0, admits a unique global solution Y 1 ∈ C 2 (R + ). Let us further define a function X by X(t, y) = y Y 1 (t) τ (t, η)dη. Then, X ∈ C 1 (R + × R). Similarly to Y 0 , for all t ∈ R + , X(t, •) is a C m -diffeomorphism from R to R. Let us denote by Y (t, •) the inverse C m -diffeomorphism of X(t, •). It is easy to see that X(0, y) = X 0 (y), Y (0, x) = Y 0 (x). Finally, we define V(t, x) = (ρ, u, ε e , ε i ) T t, Y (t, x) .
It is proved in [START_REF] Peng | Euler-Lagrange change of variables in conservation laws[END_REF] (see also [START_REF] Wagner | Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions[END_REF]) that entropy solutions of the Cauchy problem for a hyperbolic system of conservation laws are equivalent in Euler and Lagrangian coordinates. Moreover, there are explicit formulations of the solutions between two coordinates. Since the solutions studied here are smooth, it is obvious that this equivalence result holds for non-conservative systems. Applying this result, we see that V is a smooth solution to the Cauchy problem (2.13) and (1.11). Estimate (1.12) follows from (2.26) together with Moser-type inequalities. 2

Appendix : Strictly convex entropy and symmetrizer

There is a well-known result showing that the second-order derivative of a strictly convex entropy is a symmetrizer for the hyperbolic system of conservation laws [START_REF] Godunov | An interesting class of quasi-linear systems (Russian)[END_REF][START_REF] Boillat | Sur l'existence et la recherche d'équations de conservation supplémentaires pour les systèmes hyperboliques[END_REF]. In general, this result does not hold for a non-conservative system. In this Appendix, we want to show that the bitemperature Euler model, which is a non-conservative system, provides a good example on this topic.

More precisely, we consider the system in the form (1.5) or equivalently (1.6). Denote W = (ρ, ρu T , E e , E i ). Since η defined in (1.7) is a strictly convex entropy, η (W) is a symmetric positive definite matrix, denoted by A 0 (W). The result below implies that η (W)C j (W) is not symmetric in one space dimension.

Proposition. Consider the one-dimensional system (2.12) and denote by C 1 (W) = C(W) the related matrix. Then η (W)C(W) is symmetric if and only if T e = T i .

Proof. A straightforward calculation using (1.5) gives

C(W) =          
η i (W) = η i ρ - k B c 2 i q 2 2m i ρp i + γ i c i b i , k B c 2 i q m i p i , 0, - k B c i ρ m i p i , η (
W) = η e (W) + η i (W).

Since ∂ 2 EeE i η = 0, the hessian matrix of η is of the following form : Therefore, (η C) 31 = (η C) 13 if and only if b e ε e = b i ε i , that is to say T e = T i . This proves that η is not a symmetrizer of the system. In a same way, we can show that η C is symmetric if and only if T i = T e . 2

η (W) =     ∂ 2

2. 1 .

 1 Symmetrization of the system. System (1.5) can be written in the form(2.1)

µ 1 =

 1 ρ c e γ e (γ e -1)

Theorem 2 . 1 .

 21 Let m ≥ 2 and U 0 -Ū ∈ H m (R). Assume γ e = γ i . There are two positive constants c and κ 1 such that if U 0 -Ū m ≤ κ 1 , then the Cauchy problem (2.24)-(2.25) admits a unique global solution U satisfying U

(3. 3 )Lemma 3 . 1 .

 331 b e εe = b i εi . For all U in the domain under consideration, it holds (3.4) ∇s( Ū)G(U) = 0 and (3.5) ∇s( Ū)A(U) = ∇F(U),

1 2 3 γepe(γe- 1 )ρ + c e 3 2 3 γ i p i (γ i - 1 )ρ + c i 3 2.

 1313313 (c e γ e + c i γ i -3)u 2 -(c e γ e + c i γ i -3)u γ e -1 γ i -1 -γeupe (γe-1)ρ + 1 2 c e (c e γ e + c i γ i -2)u -c e γ e -c i γ i u 2 (γ e c e + c i )u c e (γ i -1)u γ i up i (γ i -1)ρ + 1 2 c i (c e γ e + c i γ i -2)u -c e γ e -c i γ i u 2 c i (γ e -1)u (γ i c i + c e )uLet q = ρu. From (1.3) and (1.4), we may write p α in variable W asp α = (γ α -1) E α -c α q 2 2ρ .Then η defined in (1.7) can be expressed asη = η e + η i , η α = -c α ρ b α ln p α c γα α ρ γα , α = e, i.

B c 2 e q 2 2m e ρp e + γ e c e b e , k B c 2 e q m e p e , - k B c e ρ m e p e , 0

 0 ,

2 2 qq η α = c α b α ρε α + c α u 2 b α ρε 2 αγ α c α b α ρ + c α u 4 4b α ρε 2 α 2 2 (c e γ e + c i γ i - 3 )u 2 + ∂ 2 η e 1 2 c

 222222322 ρρ η α = γ α c α b α ρ + c α u 4 4b α ρε 2 α , ∂ 2 ρq η α = -c α u 3 2b α ρε 2 α , ∂ 2 ρEα η α = -1 b α ρε α + u 2 2b α ρε 2 α , ∂ EαEα η = 1 b α c α ρε 2 α , α = e, i.The entry in the 3-th row and 1-th column of η C is(η C) 31 = ∂ 2 qEe η e ×1EeEe e (c e γ e + c i γ i -2)u 3 -c e γ e uε e e + c i b e ε e + c i (γ e -1) b i ε i u ρ .

  ρρ (η e + η i ) ∂ 2 ρq (η e + η i ) ∂ 2 ρEe η e ∂ 2 ρE i η i ∂ 2 ρq (η e + η i ) ∂ 2 qq (η e + η i ) ∂ 2 qEe η e ∂ 2 qE i η i

	∂ 2 ρEe η e ∂ 2 ρE i η i	∂ 2 qEe η e ∂ 2 qE i η i	∂ 2 EeEe η e 0	0 E i E i η i ∂ 2

Dissipation estimates.

Lemma 3.5. Let the conditions of Theorem 2.1 hold. If U -Ū m is sufficiently small, for all t ∈ [0, T ], we have

Proof. Let k be an integer with 0 ≤ k ≤ m -1. Applying ∂ k y to the first three equations in (3.7) yields

We multiply the third equation in (3.18) by (γ i -γ e ) and take the inner product with

By the Young inequality and the Moser-type inequalities, the last term above is bounded by

Moreover, by the first equation in (3.18) and an integration by parts, we have

, where β > 0 is a small constant to be chosen. This implies that

Similarly, taking the inner product of the first equation in (3.18) with ∂ k+1 y p in L 2 (R) and using an integration by parts, we have

By the second equation in (3.18), we obtain as above