Roberta Bianchini

Relative entropy in diffusive relaxation for a class of discrete velocities BGK models

We provide a general framework to extend the relative entropy method to a class of diffusive relaxation systems with discrete velocities. The methodology is detailed in the toy case of the 1D Jin-Xin model under the diffusive scaling, and provides a direct proof of convergence to the limit parabolic equation in any interval of time, in the regime where the solutions are smooth. Recently, the same approach has been successfully used to show the strong convergence of a vector-BGK model to the 2D incompressible Navier-Stokes equations.

Introduction

The model under investigation in this paper is the so-called Jin-Xin relaxation system, first introduced in [START_REF]The relaxation schemes for system of conservation laws in arbitrary space dimensions[END_REF]. This model is likely the simplest example of hyperbolic relaxation systems, i.e. semilinear approximations to hyperbolic systems of balance laws, see for instance [START_REF]Twenty-eight years with "Hyperbolic conservation laws with relaxation[END_REF] for a review on the topic. We are interested in the diffusivescaled version of this model, which reads as follows:

∂ t u + ∂ x v = 0, ε 2 ∂ t v + λ 2 ∂ x u = f (u) -v, (1)
where λ > 0 is a positive constant, u, v : (t, x) ∈ R + × R → R, and f (u) : R → R is a Lipschitz function satisfying the hypotheses below.

Assumptions (H) f (u) : R → R is a Lipschitz function such that:

• f (0) = 0;

Roberta Bianchini Sorbonne Université, LJLL, UMR CNRS-UPMC 7598, 4 place Jussieu, 75252-Paris Cedex 05 (France) & Consiglio Nazionale delle Ricerche, IAC, via dei Taurini 19 -00185 Rome (Italy), e-mail: r.bianchini@iac.cnr.it

• f (0) = a, which is a constant value independent of ε, λ;

• f (u) = au + h(u), where h(u) is a polynomial function of order higher than or equal to 2.

After a quick glance, it is clear that the equilibrium of system [START_REF] Bianchini | Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications[END_REF] has to satisfy in a suitable sense the limit parabolic equation

∂ t ū + ∂ x f ( ū) = λ 2 ∂ xx ū. ( 2
)
We will use the relative entropy method for diffusive relaxation to establish this convergence in a rigorous way. Despite its simplicity, the Jin-Xin system [START_REF] Bianchini | Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications[END_REF], and in particular its hyperbolicscaled version, has been intensively studied as a very good test case for relaxation models, first introduced by Whitham, [START_REF]Linear and Non-Linear Waves[END_REF] and then generalized by Chen, Levermore and Tai-Ping Liu [START_REF]Hyperbolic conservation laws with stiff relaxation terms and entropy[END_REF]. Far from being complete, we provide here some comments on a very shortened list of previous works. For the hyperbolic Jin-Xin system approximating scalar equations, we mention [START_REF]Zero relaxation and dissipation limits for hyperbolic conservation laws[END_REF][START_REF]The relaxation schemes for system of conservation laws in arbitrary space dimensions[END_REF][START_REF]Convergence to equilibrium for the relaxation approximations of conservation laws[END_REF] and references within [START_REF]Twenty-eight years with "Hyperbolic conservation laws with relaxation[END_REF], while the case of systems was developed in [START_REF]Relaxation limit of the Jin-Xin relaxation model[END_REF]. For the diffusive scaling considered in [START_REF] Bianchini | Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications[END_REF], the first investigations started with the papers [START_REF]Convergence of Sequences of Semigroups of Nonlinear Operators with an Application to Gas Kinetics[END_REF][START_REF]The central limit theorem for Carleman's equation[END_REF] on the Carleman equation, it was futher applied by Marcati and collaborators for the analysis of hyperbolic-parabolic relaxation limits for weak solutions of hyperbolic systems of balance laws [START_REF]Hyperbolic to Parabolic Relaxation Theory for Quasilinear First Order Systems[END_REF], while a two-velocities model is studied in [START_REF]Diffusive limit for finite velocity Boltzmann kinetic models[END_REF] by Lions and Toscani. We also quote [START_REF]Diffusive BGK Approximations for Nonlinear Multidimensional Parabolic Equations[END_REF], where the diffusion limit of (4) was studied by using monotonicity properties of the solution, and [START_REF]Diffusion limit of a hyperbolic system with relaxation[END_REF], where initial data around a traveling wave were considered. The interest in studying the one-dimensional Jin-Xin model in this work is due to its equivalence to the BGK (Bhatnagar-Gross-Krook) discrete kinetic approximation of conservation laws, first established by Natalini in [START_REF]A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF]. In the rest of the paper, we will mostly consider the BGK version of system (1), which is written below and is equivalent to [START_REF] Bianchini | Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications[END_REF] in the smooth regime. We point out that this equivalence between relaxation and discrete kinetic approximations of hyperbolic balance laws only holds in one space dimension. The BGK formulation of system (1) is obtained by applying the change of variables

u = f ε 1 + f ε 2 , v = λ ε ( f ε 1 -f ε 2 ), (3)
to system (1), which gives

         ∂ t f ε 1 + λ ε ∂ x f ε 1 = 1 ε 2 (M 1 (u) -f ε 1 ), ∂ t f ε 2 - λ ε ∂ x f ε 2 = 1 ε 2 (M 2 (u) -f ε 2 ), (4)
where the Maxwellian functions are given by:

M 1 (u) = u 2 + ε f (u) 2λ , M 2 (u) = u 2 - ε f (u) 2λ . (5)
For later purposes, we introduce the hyperbolic-scaled version of the Jin-Xin model, which was investigated in several previous works [START_REF]Convergence to equilibrium for the relaxation approximations of conservation laws[END_REF][START_REF]Long-time effect of relaxation for hyperbolic conservation laws[END_REF][START_REF]The relaxation schemes for system of conservation laws in arbitrary space dimensions[END_REF] and reads as follows:

∂ t u + ∂ x v = 0, ε 2 (∂ t v + λ 2 ∂ x u) = f (u) -v. (6)
As proved in [START_REF]Convergence to equilibrium for the relaxation approximations of conservation laws[END_REF], the equilibrium solution solves the conservation law

∂ t ū + ∂ x f ( ū) = 0. (7)
A key point for proving convergence of ( 6) to the equilibrium solution is the subcharacteristic condition, which is widely discussed in [START_REF]Twenty-eight years with "Hyperbolic conservation laws with relaxation[END_REF] and requires that the equilibrium velocities are controlled by the discrete λ, i.e.

λ ≥ | f (u)|, for any u.
The role of this condition is to ensure stability, since a straightforward Chapman-Enskog expansion of system [START_REF]Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF] shows that

(λ 2 -( f (u)) 2 ) ≥ 0
is the coefficient of the parabolic term. A very nice connection of the subcharacteristic condition with BGK approximations for hyperbolic balance laws, introduced in [START_REF]A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF] and generalized in [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF], was found in [START_REF]A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF]. In [START_REF]A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF], the author proves that a crucial hypothesis to get convergence of the BGK system to equilibrium [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF] is the monotonicity of the Maxwellians, which is actually equivalent to the subcharacteristic condition. The existence of a positively invariant region for the Jacobians of the Maxwellians and the existence of a family of convex entropies E for the hyperbolic equilibrium [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF] were the ingredients to prove convergence of the Jin-Xin system, established by Serre in [START_REF]Relaxations semi-linéaire et cinétique des systèmes de lois de conservation[END_REF]. At the same time, Bouchut in [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF] extended that result to a general class of BGK models. The main idea behind this is that the positivity of Jacobians of the the Maxwellians induces a diffeomorphims beetween the space of density functions f i and the equilibrium u. This roughly tells us that we can construct convex entropies for the relaxation system, starting from a family of convex entropies for the hyperbolic equilibrium [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF], as proved in [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF].

In the explicit case of system (4), the Jacobians of the Maxwellians are given by

M 1 (u) = 1 2 + ε f (u) 2λ , M 2 (u) = 1 2 - ε f (u) 2λ , (8)
therefore there exists a fixed ε 0 > 0 such that they are strictly positive for every

ε ≤ ε 0 , if | f (u)| is bounded.
This satisfies the assumptions of Theorem 2.1 in [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF], which provides the existence of a kinetic convex entropy

H (f ε ) = H 1 ( f ε 1 ) + H 2 ( f ε 2 )
for the diffusive Jin-Xin system in BGK formulation [START_REF]Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations[END_REF], where

f ε = ( f ε 1 , f ε 2 ) has been defined in (3). Denoting by M(u) = (M 1 (u), M 2 (u)) with M i (u), i = 1, 2 in (5), Theorem 2.1 in [7] also states that H(f ε ) ≥ H(M(u)), (9)
which, together with the convexity property, provides the following entropy estimate

∂ t H (f ε ) + Λ ε ∂ x H (f ε ) = 1 ε 2 H (f ε )(M(u) -f ε ) ≤ 1 ε 2 (H (M(u)) -H (f ε )) ≤ 0. ( 10
) This entropy inequality can be seen as a discrete version of the Boltzmann H-Theorem and tells us that the kinetic entropy H (f ε ) is dissipative, see for instance [START_REF]Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy[END_REF] for a rigorous definition. More precisely, properties (E1)-(E2) and Theorem 2.1 in [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF] assure that, for any η(u) ∈ E, defining the projector

P = 1 1 0 0 , such that Pf ε = f ε 1 + f ε 2 = u, (11)
according with (3),

η(u) = min Pf ε =u H (f ε ) = H (M(u)). (12)
In this context, the Gibbs principle for relaxation implies that

H (M(u)) ⊥ Ker(P). (13)
Since f ε -M(u) ∈ Ker(P), the convexity property of H (f ε ) together with condition (13) allow us to get the following inequality:

H (f ε )(f ε -M(u)) ≤ -c|f ε -M(u)| 2 , c = c(f ε ), (14)
namely the kinetic entropy H (M(u)) is strictly dissipative, according to the definition given for instance in [START_REF]Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy[END_REF]. This way, the entropy inequality for system (1) reads

∂ t H (f ε ) + Λ ε ∂ x H (f ε ) ≤ - c ε 2 |f ε -M(u)| 2 . ( 15
)
Together with the ones already mentioned before, other two inspiring papers for the present work are [START_REF]Relative Entropy in Hyperbolic Relaxation[END_REF][START_REF]Relative entropy in diffusive relaxation[END_REF]. In [START_REF]Relative Entropy in Hyperbolic Relaxation[END_REF], Tzavaras proved convergence of singular hyperbolic relaxation systems, endowed with a family of convex entropies, under the hyperbolic scaling. The proof relies on the relative entropy method and holds in the context of smooth solutions. We recall that the more general method of modulated energy was used by Brenier in [START_REF]Convergence of the Vlasov-Poisson system to the incompressible Euler equations[END_REF] for the Vlasov-Poisson system, and see also [START_REF] S -R | Hydrodynamic limits: some improvements of the relative entropy method[END_REF] for an interesting application to fluid-dynamics. Moreover, in [START_REF]Relative entropy in diffusive relaxation[END_REF], Lattanzio and Tzavaras extended this method to diffusive relaxation, in the particular case of 1D compressible gas dynamics with friction. This paper can be seen as an extension of [START_REF]Relative Entropy in Hyperbolic Relaxation[END_REF] to the diffusive scaling, for a class of discrete velocities relaxation models, whose hyperbolic equilibrium admits a family of convex entropies. Here we choose to present the method through the 1D diffusive Jin-Xin system (4), since this basic model admits a simple and explicit convex entropy, which is a relaxation of the entropy of the hyperbolic equilibrium satisfying [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF]. We point out that in the case of more relevant BGKs, whose cone of the positive Maxwellians is invariant, the kinetic convex entropy provided by Theorem 2.1 in [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF] is not explicit, since it comes from the application of the Inverse Function Theorem. This is a major difficulty in applying the relative entropy method in that case, since the expression of the kinetic entropy is in general not known and depends on the vanishing diffusive parameter. This obstacle has been overcome in [START_REF]Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method[END_REF], where the relative entropy method was applied to a vector-BGK model approximating the twodimensional Navier-Stokes equations. The guidelines of the method developed in [START_REF]Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method[END_REF] are contained in the present work on the diffusive Jin-Xin model. We stress that although the approach is presented here for a scalar equation ( 16), it can be easily extended to systems, as done in [START_REF]Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method[END_REF]. The result of this paper obtained by the relative entropy method is stated here.

Theorem 1 Let ū be a smooth solution to

∂ t ū + ∂ x f ( ū) = λ 2 ∂ xx ū, with initial data u 0 ∈ H s+1 (R) s > 1 2 + 1,
and consider a family of smooth solutions f ε to the discrete velocities diffusive BGK model ( 4), emanating from well-prepared initial data

f ε 0 = (M 1 (u 0 ), M 2 (u 0 )) = ( u 0 2 + ε f (u 0 ) 2λ - ελ 2 ∂ x u 0 , u 0 2 - ε f (u 0 ) 2λ + ελ 2 ∂ x u 0 ).
For any interval of time [0,T], the following strong convergence result holds:

sup t ∈[0,T ] (u -ū)(t) s ≤ cε 1 2 -s 2s ,
with s ∈ (0, s).

We end this introduction with some comments on the long-time behavior of the Jin-Xin system, which still provides convergence to a parabolic equation as [START_REF]Convergence of Sequences of Semigroups of Nonlinear Operators with an Application to Gas Kinetics[END_REF], with λ 2a 2 instead of λ 2 as the diffusion coefficient. In this regard, the paper [START_REF]Long-time behavior of solutions to a non-linear hyperbolic relaxation system[END_REF] is a complete study of the long-time behavior of the Jin-Xin system for a class of functions f (u) = |u| q-1 u, with q ≥ 2. In [START_REF]Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF], the authors extended this investigation to dissipative hyperbolic systems with convex entropies. Finally, the convergence of the Jin-Xin model (1) to the limit parabolic equation, both for large times and for the vanishing singular parameter, is proved in [START_REF]Uniform asymptotic and convergence estimates for the Jin-Xin model under the diffusion scaling[END_REF] for a general class of functions f (u) = au + h(u), with h(u) quadratic.

The relative entropy

Let ( ū, v) be a smooth solution to the limit diffusive Jin-Xin system in (1). It is well-known, see for instance [START_REF]Diffusive BGK Approximations for Nonlinear Multidimensional Parabolic Equations[END_REF][START_REF]Uniform asymptotic and convergence estimates for the Jin-Xin model under the diffusion scaling[END_REF], that ( ū, v) solves the parabolic equation

∂ t ū + ∂ x f ( ū) = λ 2 ∂ xx ū, and v = f ( ū) -λ 2 ∂ x ū. ( 16
)
The equilibria ū, v can be translated in terms of the densities,

ū = f 1 + f 2 , v = λ ε ( f 1 -f 2 ), (17)
where the densities at (diffusive) equilibrium are given by suitable corrections of the Maxwellians, as discussed for instance in [START_REF] S -R | La méthode de l'entropie relative pour les limites hydrodynamiques de modèles cinétiques[END_REF] in the case of the Boltzmann equation and in [START_REF]Uniform asymptotic and convergence estimates for the Jin-Xin model under the diffusion scaling[END_REF] for our model,

f 1 = M 1 ( ū) - ελ∂ x ū 2 =: M 1 ( ū), f 2 = M 2 ( ū) + ελ∂ x ū 2 =: M 2 ( ū), (18)
and the expressions of M 1 ( ū), M 2 ( ū) are provided in [START_REF]Relaxation limit of the Jin-Xin relaxation model[END_REF]. We recall from the previous section that under the hypothesis of positivity of the Jacobians of the Maxwellians in (8), system (4) admits a dissipative convex kinetic entropy. Under such a framework, taking a smooth solution

f ε = ( f ε 1 , f ε 2 )
to system (4) and a smooth solution ū to (16), a relative entropy is defined as follows,

H (f ε | f) = H (f ε ) -H (M( ū)) -H (M( ū)) • (f ε -M( ū)). ( 19
)
We also define the relative entropy flux,

Q(f ε | f) = λ ε (H 1 ( f ε 1 ) -H 2 ( f ε 2 )) - λ ε (H 1 (M 1 ( ū)) -H 2 (M 2 ( ū))) - λ ε H 1 (M 1 ( ū)) • ( f ε 1 -M 1 ( ū)) + λ ε H 2 (M 2 ( ū)) • ( f ε 2 -M 2 ( ū)). (20)
The aim of this section is to provide the existence of a kinetic convex dissipative entropy for system (4).

Overview on the linear case

In the linear case, it is particularly easy to find an expression of a dissipative kinetic convex entropy for the Jin-Xin system and all the computations are explicit. Consider the diffusive Jin-Xin model with linear source term

         ∂ t u + ∂ x (ε 2 v) ε 2 = 0, ∂ t (ε 2 v) + λ 2 ∂ x u = au - ε 2 v ε 2 , (21)
approximating the linear convection-diffusion equation

∂ t ū + a∂ x ū = λ 2 ∂ xx ū. ( 22
)
Introducing u = (u, ε 2 v), system (21) rewrites as

∂ t u + A∂ x u = -Bu, (23)
where

A = 0 1 ε 2 λ 2 0 , -B = 0 0 a - 1 ε 2 . ( 24
)
One can symmetrize system [START_REF]A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF], by using a positive definite symmetric matrix

Σ = 1 aε 2 aε 2 λ 2 ε 2 , with AΣ = (AΣ) T , (25)
and such that the dissipation of B in ( 24) is enhanced, see [START_REF]Uniform asymptotic and convergence estimates for the Jin-Xin model under the diffusion scaling[END_REF][START_REF]Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations[END_REF] for a detailed discussion,

-B 1 = -BΣ = 0 0 0 a 2 ε 2 -λ 2 .
Now, as usual, see for instance [START_REF] Bianchini | Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications[END_REF],

Σ -1 = 1 λ 2 -a 2 ε 2 λ 2 -a -a 1 ε 2 (26)
is a classical left symmetrizer for system [START_REF]A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF], and so it can be viewed as the Hessian matrix of an entropy function for system [START_REF]A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF], whose expression is given by

E(u) = η(u, ε 2 v) = λ 2 2 u 2 + ε 2 2 v 2 -auε 2 v. (27)
In terms of the kinetic variables,

E(u, ε 2 v) = h( f ε 1 + f ε 2 , λ( f ε 1 -f ε 2 )/ε) = ( f ε 1 ) 2 (λ 2 -aλε) + ( f ε 2 ) 2 (λ 2 + aλε) =: H 1 ( f ε 1 ) + H 2 ( f ε 2 ) = H (f ε ), (28)
i.e. H (f ε ) is an explicit convex and dissipative kinetic entropy for the linear BGK model (4), with Maxwellian functions

M 1 (u) = u 2 + aεu 2λ , M 2 (u) = u 2 - aεu 2λ . ( 29
)
We recall that the existence of such an entropy H (f ε ) is provided by the theory developed in [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF]. According to Theorem 2.1 in [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF], which tells us that the kinetic entropy is a relaxation of the entropy of the conservation law and that they coincide at equilibrium,

       H (M(u)) = H 1 (M 1 (u)) + H 2 (M 2 (u)) = η(u) = 1 2 (λ 2 -a 2 ε 2 )u 2 , ∂ f ε i H i (M i (u)) = η (u) = (λ 2 -a 2 ε 2 )u, i = 1, 2. (30)
However, a perturbed version of the original Maxwellians, M 1 (u), M 2 (u) in ( 18), naturally arises as equilibria of the diffusive-scaled system. In the linear case,

M 1 (u) = u 2 + aεu 2 - ελ∂ x u 2 , M 1 (u) = u 2 - aεu 2 + ελ∂ x u 2 , (31)
which implies that

H 1 (M 1 (u)) = η (u) -ελ 3 ∂ x u + O(ε 2 ), H 2 (M 2 (u)) = η (u) + ελ 3 ∂ x u + O(ε 2 ), ( 32
) where η (u) = λ 2 + O(ε).

The nonlinear case

We go back to the nonlinear one-dimensional Jin-Xin model ( 1) and its BGK formulation (4). In the nonlinear case, the existence of a kinetic, convex and dissipative entropy for the BGK system (4), in the vicinity of the local equilibrium f ε = M(u) is due to Theorem 2.1 in [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF]. However, Theorem 2.1 in [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF] only provides the existence of such an entropy, while its explicit expression is not known in the general case, being indeed a consequence of the Inverse Function Theorem, see [START_REF]Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method[END_REF] as an example with non-explicit kinetic entropies. In the particular case of the 1D Jin-Xin system, an explicit entropy can be shown. Nevertheless, in the following we will not rely on the expression of the kinetic entropy, in order to present the method in full generality.

The results below will be applied later on.

Proposition 1 Consider the BGK system (4) under Assumptions (H). Let η( ū) be a quadratic entropy for the hyperbolic limit of the Jin-Xin model [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF]. Then there exists a convex kinetic entropy H (f ε ) for system [START_REF]Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations[END_REF], such that ( 12)-( 14) hold, as long as T], for any fixed constant T. Proof We apply Theorem 2.1 in [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF], which yields the result provided that the derivatives of the Maxwellians M i (u), i = 1, 2 in (5) are positive,

u = f ε 1 + f ε 2 is bounded in [0,
M 1 (u) = 1 2 + ε f (u) 2λ > 0, M 2 (u) = 1 2 - ε f (u) 2λ > 0, which hold as long as |u(t)| ∞ = | f ε 1 (t) + f ε 2 (t)| ∞ is bounded. Lemma 1 Let η( ū)
be a quadratic entropy for the conservation law [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF]. Under Assumptions (H), there exists an interval of time [0,T * ], where T * depends on u 0 s+1 , s > 1 2 + 1, such that system (4) admits a kinetic convex dissipative entropy

H (f ε ) = H 1 ( f ε 1 ) + H 2 ( f ε 2 ), with H (M(u)) = η(u), H 1 (M 1 (u)) = H 2 (M 2 (u)) = η (u). Moreover, |H i (M i (u))| ∞ = 2|η (u)| ∞ + O(ε), i = 1, 2. ( 33
)
Proof The idea is to prove that the assumptions of Proposition 1 are satisfied. Therefore we need to show that there exists a fixed interval of time [0,T * ], with T * independent of ε, such that |u(t)| ∞ is bounded for t ≤ T * . For the diffusive Jin-Xin model, this is actually already proved (for global times) in [START_REF]Uniform asymptotic and convergence estimates for the Jin-Xin model under the diffusion scaling[END_REF], using a detailed description of the Green function of the linearized system. However, we perform here a classical local in time proof in order to be self-contained. Defining w such that u = Σw, the compact system is symmetric and reads

∂ t w + Ã∂ x w = -Bw + 0 h((Σw) 1 ), where à = AΣ = ÃT = a λ 2 λ 2 aε 2 λ 2 , B = BΣ = 0 0 0 λ 2 -a 2 ε 2 .
Moreover, simple computations show that Σ is positive definite and, more precisely,

1 2 w 1 2 0 + ε 2 (λ 2 -2a 2 ε 2 ) w 2 2 0 ≤ (Σw, w) 0 ≤ (1 + aε 2 ) w 1 2 0 + ε 2 (a + λ 2 ) w 2 2 0 . (34)
Then one gets the following energy estimates on the symmetric compact system:

1 2 w 1 (t) 2 s + ε 2 (λ 2 -2a 2 ε 2 ) w 2 (t) 2 s + (λ 2 -a 2 ε 2 ) ∫ t 0 w 2 (τ) 2 s dτ ≤ u 0 2 s + ε 2 c( u 0 s+1 ) + c( w 1 + aε 2 w 2 2 L ∞ t , x ) ∫ t 0 w 1 (τ) + aε 2 w 2 (τ) 2 s dτ; (35) 1 2 ∂ t w 1 (t) 2 s-1 + ε 2 (λ 2 -2a 2 ε 2 ) ∂ t w 2 (t) 2 s-1 + (λ 2 -a 2 ε 2 ) ∫ t 0 ∂ τ w 2 (τ) 2 s-1 dτ ≤ ∂ t u| t=0 2 s-1 + c( w 1 + aε 2 w 2 2 L ∞ t , x ) ∫ t 0 ∂ τ w 1 (τ) + aε 2 ∂ τ w 2 (τ) 2 s-1 dτ ≤ ∂ x f (u 0 ) -λ 2 ∂ xx u 0 2 s-1 + c( w 1 + aε 2 w 2 2 L ∞ t , x ) ∫ t 0 ∂ τ w 1 (τ) + aε 2 ∂ τ w 2 (τ) 2 s-1 dτ ≤ c u 0 2 s+1 + c( w 1 + aε 2 w 2 2 L ∞ t , x ) ∫ t 0 ∂ τ w 1 (τ) + aε 2 ∂ τ w 2 (τ) 2 s-1 dτ. ( 36
)
This provides

1 2 w 1 (t) 2 s + ε 2 (λ 2 -2a 2 ε 2 ) w 2 (t) 2 s ≤ ( u 0 2 s + ε 2 c( u 0 s+1 ))e c( w 1 +aε 2 w 2 2 L ∞ t , x )t , 1 2 ∂ t w 1 (t) 2 s-1 + ε 2 (λ 2 -2a 2 ε 2 ) ∂ t w 2 (t) 2 s-1 ≤ c u 0 2 s+1 e c( w 1 +aε 2 w 2 2 L ∞ t , x )t
, as an application of the Gronwall inequality. Now set u 0 s+1 =: M 0 and, for a fixed constant M > M 0 , define

T * := sup t ∈[0,T ε ) 1 2 w 1 (t) ∞ + ε 2 (λ 2 -2a 2 ε 2 ) w 2 (t) ∞ ≤ c S M , (37)
being T ε the maximum existence time of f ε and c S the Sobolev embedding constant. The Sobolev Embedding Theorem provides the following estimates for t ∈ [0,T * ],

1 2 w 1 (t) 2 ∞ + ε 2 (λ 2 -2a 2 ε 2 ) w 2 (t) 2 ∞ ≤ (M 2 0 + ε 2 c(M 0 ))e c(M 2 )t ≤ c 2 S M 2 , (38) 1 2 ∂ x w 1 (t) 2 ∞ + ε 2 (λ 2 -2a 2 ε 2 ) ∂ x w 2 (t) 2 ∞ ≤ cM 2 0 e c(M 2 )t ≤ c 2 S M 2 , ( 39
) 1 2 ∂ t w 1 (t) 2 ∞ + ε 2 (λ 2 -2a 2 ε 2 ) ∂ t w 2 (t) 2 ∞ ≤ cM 2 0 e c(M 2 )t ≤ c 2 S M 2 , (40)
where

T * ≤ 1 c(M 2 ) log c 2 S M 2 cM 2 0 .
Now, using the changes of variables (3) and u = (u, ε 2 v) = Σw, one gets

f ε 1 = w 1 2λ (λ + aε) + εw 2 2 (λ + aε), f ε 1 = w 1 2λ (λ -aε) - εw 2 2 (λ -aε),
and so, thanks to (38),

sup t ∈[0,T * ] max{ f ε i (t) ∞ , i = 1, 2} ≤ c(M),
which allows us to apply Proposition 1 to prove the first of part of this lemma. It remains to show the last expansion. Notice from (39)-(40

) that |ε∂ t f ε i + λ∂ x f ε i )| is bounded in [0,T * ], then f ε 1 = M 1 (u) + ε(ε∂ t f ε 1 + λ∂ x f ε 1 ) = M 1 (u) + O(ε) = u 2 + ε f (u) 2λ + O(ε), f ε 2 = M 2 (u) + ε(ε∂ t f ε 2 -λ∂ x f ε 2 ) = M 2 (u) + O(ε) = u 2 - ε f (u) 2λ + O(ε),
in the sense of the L ∞ x norm. From Proposition 1,

∂ f ε i H i ( f ε i )| f ε i =M i (u) = ∂ f ε i H i (M i (u)) = ∂ f ε i H i ( u 2 ± ε f (u) 2λ ) = g( u 2 ± ε f (u)) 2λ ) = η (u).
Recalling that

u = f ε 1 + f ε 2 , ∂ 2
f ε i H i ( f ε i ) = 1 2 g ( u 2 + O(ε)) = η (u), i.e. ∂ 2
f ε i H i ( f ε i ) = 2η (u) + O(ε). Lemma 2 Let H (f ε ) = H 1 ( f ε 1 ) + H 2 ( f ε
2 ) be a kinetic convex dissipative entropy associated with system (4). Let ū be a smooth solution to system [START_REF]Convergence of Sequences of Semigroups of Nonlinear Operators with an Application to Gas Kinetics[END_REF] and η( ū) a convex entropy for equation [START_REF]Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF]. Then, for i = 1, 2,

H i (M i ( ū)) = H i (M i ( ū) ∓ ελ 2 ∂ x ū) = H i (M i ( ū)) ∓ η ( ū) ελ 2 ∂ x ū + O(ε 2 ), H i (M i ( ū)) = η ( ū) ∓ ελη ( ū)∂ x ū + O(ε 2 ),
in the sense of the L ∞ norm.

Proof It comes directly from a Taylor expansion and the use of

H i (M i (u)) = η (u)
for any u.

Relative entropy estimate

The global in time convergence is achieved by means of the relative entropy estimate, which is stated and proved below.

Theorem 2 Consider the discrete velocities BGK model (4) under Assumptions (H), endowed with a kinetic convex dissipative entropy

H (f ε ) for t ≤ T * fixed. Let ū be a smooth solution to ∂ t ū + ∂ x f ( ū) = λ 2 ∂ xx ū, with initial data u 0 ∈ H s+1 (R), s > 1 2 + 1, and let f ε (t) = ( f ε 1 , f ε 2 )
, t ≤ T * be a sequence of solutions to the discrete velocities BGK model (4), emanating from smooth well-prepared initial data

f ε 0 = (M 1 (u 0 ), M 2 (u 0 )),
and such that

|f ε (t)| ∞ |∂ t f ε (t)| ∞ ≤ M, t ≤ T * , for any constant M > 0 independent of ε.
Then, the following stability estimate holds:

sup t ∈[0,T * ] u -ū 0 ≤ c √ εe c(| ū | ∞ , |∂ x ū | ∞ )t . ( 41
)
Proof In the following, we drop the apex ε for simplicity.

∂ t H (f ε |f) + ∂ x Q(f ε | f) = ∂ t (H 1 ( f 1 ) + H 2 ( f 2 )) + λ ε ∂ x (H 1 ( f 1 ) -H 2 ( f 2 )) -∂ t (H 1 (M 1 ) + H 2 (M 2 )) - λ ε ∂ x (H 1 (M 1 ) -H 2 (M 2 )) -∂ t (H 1 (M 1 )( f 1 -M 1 )) -∂ t (H 2 (M 2 )( f 2 -M 2 )) - λ ε ∂ x (H 1 (M 1 )( f 1 -M 1 )) + λ ε ∂ x (H 2 (M 1 )( f 2 -M 2 )) =: I 1 + I 2 + I 3 + I 4 .
We start with I 1 , whose estimate is based on Lemma 1.

I 1 = ∂ t (H 1 ( f 1 ) + H 2 ( f 2 )) + λ ε ∂ x (H 1 ( f 1 ) -H 2 ( f 2 )) = H 1 ( f 1 ) ε 2 (M 1 -f 1 ) + H 2 ( f 2 ) ε 2 (M 2 -f 2 ) = H 1 (M 1 ) + H 1 (M 1 )( f 1 -M 1 ) ε 2 (M 1 -f 1 ) + H 2 (M 2 ) + H (M 2 )( f 2 -M 2 ) ε 2 (M 2 -f 2 ) + O(ε) ≤ - (2η (u) + O(ε)) ε 2 (| f 1 -M 1 | 2 + | f 2 -M 2 | 2 ) + O(ε) = - 2η (u) 2λ 2 |v -f (u)| 2 + O(ε) = - η (u) λ 2 |ε 2 ∂ t v + λ 2 ∂ x u| 2 + O(ε) = - η ( ū) λ 2 |ε 2 ∂ t v + λ 2 ∂ x u| 2 + O(ε), (42)
where the last equalities follow from

ε 2 ∂ t v + λ 2 ∂ x u = v -f (u) in (1) and O(ε -1 f i - M i 2 ∞ ) = O(ε)
, which is due to the L ∞ bounds on f ε , ∂ t f ε . Notice that equality η (u) = η ( ū) holds since we chose a quadratic entropy η(u) for system [START_REF]Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy[END_REF]. For scalar conservation laws, there are indeed infinite entropy-entropy flux pairs (η( ū), Q( ū)) satisfying η ( ū) f ( ū) = Q ( ū). We remind to [START_REF]Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method[END_REF] for an application of the method in a more constrained case. Now we deal with I 2 , by using the expansions of Lemma 2.

I 2 = -∂ t (H 1 (M 1 ) + H 2 (M 2 )) - λ ε ∂ x (H 1 (M 1 ) -H 2 (M 2 )) = -[H 1 (M 1 )(∂ t M 1 + λ ε ∂ x M 1 ) -H 2 (M 2 )(∂ t M 2 - λ ε ∂ x M 2 )] = -[η ( ū) -ελη ( ū)∂ x ū](∂ t M 1 + λ ε ∂ x M 1 ) -[η ( ū) + ελη ( ū)∂ x ū](∂ t M 2 - λ ε ∂ x M 2 ) + O(ε) = -η ( ū)[∂ t (M 1 + M 2 ) + λ ε ∂ x (M 1 -M 2 )] -ελη ( ū)∂ x ū[-∂ t M 1 + ∂ t M 2 - λ ε ∂ x M 1 - λ ε ∂ x M 2 ] + O(ε) = -η ( ū)[∂ t ū + ∂ x f ( ū) -λ 2 ∂ xx ū] -ελη ( ū)∂ x ū[- λ ε ∂ x ū + O(ε)] + O(ε) = η ( ū)λ 2 (∂ x ū) 2 + O(ε), (43)
where the last equalities are provided by the explicit expressions of M 1 , M 2 in [START_REF]Relative entropy in diffusive relaxation[END_REF].

I 3 = -∂ t (H 1 (M 1 )( f 1 -M 1 )) -∂ t (H 2 (M 2 )( f 2 -M 2 )) = -∂ t [η ( ū)(u -ū) -ελη ( ū)∂ x ū • ( f 1 -f 2 -(M 1 -M 2 ))] + O(ε 2 ) = -∂ t [η ( ū)(u -ū)] + ε 2 η ( ū)∂ x ū • ∂ t [v -f ( ū) + λ 2 ∂ x ū] + O(ε) = -∂ t [η ( ū)(u -ū)] + η ( ū)∂ x ū • (ε 2 ∂ t v + λ 2 ∂ x u) -η ( ū)λ 2 ∂ x ū • ∂ x u + O(ε) = η ( ū)[∂ x f ( ū) -λ 2 ∂ xx ū] • (u -ū) + η ( ū) • (∂ x v -∂ x f ( ū) + λ 2 ∂ xx ū) + η ( ū)∂ x ū • (ε 2 ∂ t v + λ 2 ∂ x u) -η ( ū)λ 2 ∂ x ū • ∂ x u + O(ε).
(44) We are finally left with I 4 .

I 4 = - λ ε ∂ x (H 1 (M 1 )( f 1 -M 1 )) + λ ε ∂ x (H 2 (M 1 )( f 2 -M 2 )) = λ ε ∂ x [(-η ( ū) + ελη ( ū)∂ x ū) • ( f 1 -M 1 ) + (η ( ū) + ελη ( ū)∂ x ū) • ( f 2 -M 2 )] + O(ε) = - λ ε ∂ x [η ( ū) • ( f 1 -f 2 -(M 1 -M 2 ))] + λ 2 η ( ū)∂ x [∂ x ū • ( f 1 + f 2 -(M 1 + M 2 ))] = -∂ x [η ( ū) • (v -f ( ū) + λ 2 ∂ x ū)] + λ 2 η ( ū)∂ x [∂ x ū • (u -ū)] = -η ( ū) • (∂ x v -∂ x f ( ū) + λ 2 ∂ xx ū) -η ( ū)∂ x ū • (v -f ( ū) + λ 2 ∂ x ū) + λ 2 η ( ū)∂ xx ū • (u -ū) + λ 2 η ( ū)∂ x ū • ∂ x (u -ū) = -η ( ū) • (∂ x v -∂ x f ( ū) + λ 2 ∂ xx ū) -η ( ū)∂ x ū • (v -f (u)) -η ( ū)∂ x ū • ( f (u) -f ( ū)) -λ 2 η ( ū)(∂ x ū) 2 + λ 2 η ( ū)∂ xx ū • (u -ū) + λ 2 η ( ū)∂ x ū • ∂ x u -λ 2 η ( ū)(∂ x ū) 2 = -η ( ū) • (∂ x v -∂ x f ( ū) + λ 2 ∂ xx ū) + η ( ū)∂ x ū • (ε 2 ∂ t v + λ 2 ∂ x u) -η ( ū)∂ x ū • ( f (u) -f ( ū)) -2λ 2 η ( ū)(∂ x ū) 2 + λ 2 η ( ū)∂ xx ū • (u -ū) + λ 2 η ( ū)∂ x ū • ∂ x u. (45)
As an intermediate step, let us look at the sum

I 3 + I 4 = 2η ( ū)∂ x ū • (ε 2 ∂ t v + λ 2 ∂ x u) -η ( ū)∂ x ū • [ f (u) -f ( ū) -f ( ū) • (u -ū)] -2λ 2 η ( ū)(∂ x ū) 2 . ( 46
)
The total sum reads:

I 1 + I 2 + I 3 + I 4 ≤ - η ( ū) λ 2 |ε 2 ∂ t v + λ 2 ∂ x u| 2 + η ( ū)λ 2 (∂ x ū) 2 -2λ 2 η ( ū)(∂ x ū) 2 + 2η ( ū)∂ x ū • (ε 2 ∂ t v + λ 2 ∂ x u) -η ( ū)∂ x ū • [ f (u) -f ( ū) -f ( ū) • (u -ū)] = - η ( ū) λ 2 |ε 2 ∂ t v + λ 2 ∂ x u| 2 -λ 2 η ( ū)(∂ x ū) 2 + 2η ( ū)∂ x ū • (ε 2 ∂ t v + λ 2 ∂ x u) -η ( ū)∂ x ū • [ f (u) -f ( ū) -f ( ū) • (u -ū)] = - η ( ū) λ 2 |ε 2 ∂ t v + λ 2 ∂ x u| 2 - η ( ū) λ 2 |λ 2 ∂ x ū + ε 2 ∂ t v| 2 + O(ε) + 2 λ 2 η ( ū)λ 2 ∂ x ū • (ε 2 ∂ t v + λ 2 ∂ x u) + 2 λ 2 η ( ū)ε 2 ∂ t v • (ε 2 ∂ t v + λ 2 ∂ x u) + O(ε) -η ( ū)∂ x ū • [ f (u) -f ( ū) -f ( ū) • (u -ū)] ≤ c(| ū|, |∂ x ū|)|u -ū| 2 - η ( ū) λ 2 (ε 2 ∂ t v + λ 2 ∂ x u -(ε 2 ∂ t v + λ 2 ∂ x ū)) 2 + O(ε). (47) We end up with the following estimate ∫ R H (f ε |f)(t) dx + 1 λ 2 ∫ t 0 ∫ R η ( ū)(ε 2 ∂ τ v + λ 2 ∂ x u -(ε 2 ∂ τ v + λ 2 ∂ x ū)) 2 dτ dx ≤ ∫ t 0 ∫ R c(| ū|, |∂ x ū|)|u -ū| 2 dx ds + ∫ R H (f ε 0 |f 0 ) dx + O(ε) = ∫ t 0 ∫ R c(| ū|, |∂ x ū|)|u -ū| 2 dx ds + O(ε),
where H (f ε 0 |f 0 ) = 0 thanks to the well-prepared initial data. At this point, the definition of the relative entropy [START_REF]Diffusive limit for finite velocity Boltzmann kinetic models[END_REF] and the expansion of Lemma 2 imply that

H (f ε | f) = η ( ū)|u -ū| 2 + O(ε) in L ∞ x ,
which ends the proof after an application of the Gronwall inequality.

We are now ready to prove our result.

Proof (of Theorem 1)

As a direct consequence of Lemma 1, the assumptions of Theorem 2 are satisfied for t ∈ [0,T * ], with T * = c( u 0 s+1 ) defined in (37). Therefore, Theorem 2 provides the following estimate:

sup t ∈[0,T * ] (u -ū)(t) 0 ≤ c(M) √ ε, (48)
where T * is defined in (37) and c(M) depends on M in (37).

Conclusive remark

We point out that the main difficulty in applying this method to general diffusive vector-BGK models as the ones introduced in [START_REF]Diffusive BGK Approximations for Nonlinear Multidimensional Parabolic Equations[END_REF] is to find a symmetrizer as Σ in Section 2.2 to get uniform bounds of the densities f ε i , as done in [START_REF]Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations[END_REF] for 2D Navier-Stokes. The rest, up to heavy computations, is straightforward.

1 2 1 2 -δ ≤ c S M 2 .

 112 From (38), we also recall that u(t) s ≤ (M 0 + O(ε))e c(M)t ≤ c S M, t ∈ [0,T * ], i.e. |u(t)| ∞ ≤ c S (M 0 + O(ε))e c(M)t ≤ c S M, t ∈ [0,T * ].(49)The Interpolation Theorem for Sobolev spaces provides the following bound for any s ∈ (0, s):(uū)(t) s ≤ (uū)(t) 1-s /s 0 (uū)(t) s /s s ≤ c(M)ε s-s 2s (M 0 + (M 0 + O(ε))e C(M)t ) s /s ,where the last inequality follows from (48) and (49). By using embedding properties,|(uū)(t)| ∞ ≤ c S (uū)(t) s ≤ c(M)ε s-s 2s (M 0 + (M 0 + O(ε))e C(M)t ) s /s , i.e. |u(t)| ∞ ≤ | ū| ∞ + c(M)ε s-s 2s (M 0 + (M 0 + O(ε))e C(M)t ) s /s ≤ |u 0 | ∞ + c(M)ε s-s 2s (M 0 + (M 0 + O(ε))e C(M)t ) s /s ≤ c S M 0 + c(M)ε s-s 2s (M 0 + (M 0 + O(ε))e C(M)t ) s /s .(50)Recalling now the definition of T * in (37) and choosing M = 4M 0 , estimate (50) implies that, for t ≤ T * ,|u(t)| ∞ ≤ c S M 0 + cε -δ < 2c S M 0 = c S us assume that T * < T ε .Therefore, by definition of T * in (37), one gets|u(T * )| ∞ = 4c S M 0 = c S M.On the other hand, from (51), there exists an arbitrarily small fixed ε 0 such that, for all ε ≤ ε 0 ,|u(T * )| ∞ ≤ c S M 0 + cε Therefore T * ≥T ε by contradiction and sup t ∈[0,T ] |u(t)| ∞ is bounded for any fixed interval of time [0,T] by employing a classical continuation argument. The same reasoning applies to the relative entropy estimate in Theorem 2, which holds as far as the derivatives of the Maxwellians are positive (as far as u is bounded) as in Lemma 1, and then the strong convergence result is valid for any fixed interval of time [0,T].

Acknowledgements

The author thanks Roberto Natalini for useful comments on the Introduction of this work. This paper was partially funded by the GNAMPA (INdAM) project Partially dissipative hyperbolic systems with applications to biological models 2019. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program Grant agreement No 637653, project BLOC "Mathematical Study of Boundary Layers in Oceanic Motion". This work was supported by the SingFlows project, grant ANR-18-CE40-0027 of the French National Research Agency (ANR).


Roberta Bianchini
email: r.bianchini@iac.cnr.it

Gigliola Staffilani
email: gigliola@math.mit.edu

Revisitation of a Tartar's result on a semilinear hyperbolic system with null condition

We revisit a method introduced by Tartar for proving global well-posedness of a semilinear hyperbolic system with null quadratic source in one space dimension. A remarkable point is that, since no dispersion effect is available for 1D hyperbolic systems, Tartar's approach is entirely based on spatial localization and finite speed of propagation.

Introduction

We consider the semilinear hyperbolic system with quadratic source term:

∂ t u i + c i ∂ x u i + j,k A i jk u j u k = 0, u i (x, 0) = φ i (x), (1)
x ∈ R, t ∈ I, i = 1, • • • , p, where the coefficients A i j = A ji are symmetric, and satisfy the following condition:

(A) A i jk = 0 if c j = c k for all i = 1, • • • , p.
We now make a connection of Assumption (A) with the null condition for the semilinear wave equation, which is presented in [START_REF]The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics[END_REF] and will be discussed later on. To this end, looking at the simplest 2 × 2 case,

∂ t u 1 + c 1 ∂ x u 1 = αu 1 u 2 , ∂ t u 2 + c 2 ∂ x u 2 = βu 1 u 2 , (2)
we establish a change of variables by defining w such that

u 1 = ∂ t w -c 1 ∂ x w, u 2 = ∂ t w -c 2 ∂ x w, (3)
where u 1 , u 2 are solutions to [START_REF]Global solutions of nonlinear hyperbolic equations for small initial data[END_REF]. In terms of the new variable w, system (2) rewrites as follows:

∂ tt w -c 2 1 ∂ xx w = α(∂ t w -c 1 ∂ x w)(∂ t w -c 2 ∂ x w), ∂ tt w -c 2 2 ∂ xx w = β(∂ t w -c 1 ∂ x w)(∂ t w -c 2 ∂ x w).
Discarding the trivial solution, we end up with the compatibility condition

(C) c 2 1 = c 2 2 , α = β.
Combining (C) with (A), we are left with c 1 = -c 2 and α = β. By a simple rescaling, this yields a classical example of semilinear wave equation with null condition

∂ tt w -∂ xx w = (∂ t w) 2 -(∂ x w) 2 , (4)
first introduced by John in [START_REF]Blow-up for quasi-linear wave equations in three space dimensions[END_REF]. As an important research program in nonlinear partial differential equations, the investigation on the long-time behavior of smooth solutions to dispersive equations started in the Eighties with the seminal papers by Klainerman [START_REF]The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics[END_REF] and Christodoulou [START_REF]Global solutions of nonlinear hyperbolic equations for small initial data[END_REF]. A deep historical and mathematical survey on the topic can be found in [START_REF] Séminaire | ème année[END_REF]. A general feature is that the linear dispersive terms of the equation tend to force the solution to spread and to decay, but the contribution of the nonlinear terms is very different. Since dispersion increases with space dimension, a first class of global existence results has been obtained in dimension d = 4 by Klainerman [START_REF]Global Existence for Nonlinear Wave Equations[END_REF]. As showed by John, [START_REF]Blow-up for quasi-linear wave equations in three space dimensions[END_REF], in lower space dimensions the nonlinearity can lead to blow up in finite time for arbitrarily small data. In this case, a precise structure of the nonlinearity, the so-called null form, introduced by Klainermann [START_REF]The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics[END_REF] and Christodoulou [START_REF]Global solutions of nonlinear hyperbolic equations for small initial data[END_REF], prevents the formation of singularities.

Later, an important contribution to extend the notion of null forms was given by Germain, Masmoudi, Shatah, see [START_REF]Global existence for the Euler-Maxwell system[END_REF][START_REF]Global solutions for the gravity water waves equation in dimension 3[END_REF]. The main idea of this approach is to couple spatial localization via the space-time resonance method, using space-weighted estimates, with time oscillations, using normal forms. We refer to a recent paper by Pusateri and Shatah, [START_REF]Space-Time Resonances and the Null Condition for First-Order Systems of Wave Equations[END_REF], for a result on the semilinear wave equation with nonresonant bilinear forms.

In the more general case of systems, quadratic source terms satisfying the null condition for the wave equation are actually equivalent to the compatible forms for hyperbolic systems, [START_REF]Applications bilinéaires compatibles avec un opérateur hyperbolique[END_REF], which are the ones having the weakly sequential continuity described by compensated compactness, [START_REF]Compensated compactness and applications to partial differential equations[END_REF]. An analogous result for hyperbolic systems is indeed proved in [START_REF]Decay estimates for hyperbolic systems[END_REF] in the linear case. Actually, the first contribution in the general case of hyperbolic systems of semilinear equations is due to an unpublished paper by Tartar [START_REF]Some existence theorems for semilinear hyperbolic systems in one space variable[END_REF]. In [START_REF]Some existence theorems for semilinear hyperbolic systems in one space variable[END_REF], the author provides results on well-posedness and long-time behavior for a semilinear hyperbolic system with quadratic source term, satisfying a non-crossing condition for the charactericts (Assumption (A)), which is actually equivalent to the null condition for the semilinear wave equation, as showed in [START_REF]Applications bilinéaires compatibles avec un opérateur hyperbolique[END_REF]. The approach developed by Tartar is completely different from both the vector-field method [START_REF]The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics[END_REF] and normal forms techniques [START_REF]Global existence for the Euler-Maxwell system[END_REF]. In one space dimension, there is indeed no dispersion effect neither for the wave equation, nor for the semilinear hyperbolic system in (1). Tartar's idea is then based on spatial localization and finite speed of propagation: since the characteristic velocities of the interacting waves are different (Assumption (A)), then these waves interact only for a finite time before the separation of the cones of dependency. A similar approach was used by Bianchini and Bressan in [START_REF] Staffilani | BV Solutions for a Class of Viscous Hyperbolic Systems[END_REF].

We point out that even in the simplest 2 × 2 case, system (2) reduces to the semilinear wave equation ( 4) only in the special case of c 1 = -c 2 = 1, α = β. Therefore, besides the lack of dispersion, techiques which are built ad hoc for the wave equation do not apply to systems (1)-( 2) for generic constants c i , A i jk , and new ideas are needed in that case. The approach developed in [START_REF]Some existence theorems for semilinear hyperbolic systems in one space variable[END_REF] will be presented here. This work is indeed a revisitation of the arguments in [START_REF]Some existence theorems for semilinear hyperbolic systems in one space variable[END_REF], where we attempted to fix the notation, prove some intermediate results, as well as provide an explicit description of the objects and tools. The main result, which is due to Tartar, is stated here.

Theorem 1 (Global existence for small L 1 data) Assume condition (A) on system [START_REF] Staffilani | BV Solutions for a Class of Viscous Hyperbolic Systems[END_REF]. Then, there exists

E 0 > 0, k 1 > 0, k 2 > 1 such that, if the initial data φ i , i = 1, • • • , p satisfy φ i ∈ L 1 (R), i φ i L 1 (R) = i ε i ≤ E 0 , then system (1) admits a unique solution, which, ∀i = 1, • • • , p, satisfies i ∂ t u i + c i ∂ x u i L 1 (R + ×R) ≤ k 1 i φ i L 1 (R) . Moreover, let φi ∈ L 1 (R) with i φi L 1 (R) < E 0 ,
and let ūi be the solution to [START_REF] Staffilani | BV Solutions for a Class of Viscous Hyperbolic Systems[END_REF] with initial data φi . Then

sup t ∈R + i u i -ūi L 1 (R) ≤ k 2 i φ i -φi L 1 (R) .

Proof of the theorem

In this section, we revisit step by step the argument developed by Tartar in [START_REF]Some existence theorems for semilinear hyperbolic systems in one space variable[END_REF] for proving Theorem 1. For the sake of clarity, we consider the case of compactly supported initial data φ i (x), i = 1, • • • , p, whose support is contained in an interval J = [a, b] of the real line. However, the same proof applies to the case of more general initial conditions satisfying the assumptions of Theorem 1, as showed at the end of this work. The proof will be split in different steps.

Step

-Definition of the main tools

We start by defining

D := {(x, t) | x -c i t ∈ J = [a, b], for i = 1, • • • , p}. (5)
Consider now the transport equation

∂ t v i + c i ∂ x v i = f i (x, t), (x, t) ∈ D, v i (x, 0) = φ i (x), x ∈ J, (6)
for i = 1, • • • , p, and f i smooth enough functions on a space-time domain D. The explicit solution is given by

v i (x, t) = φ i (x -c i t) + ∫ t 0 f i (x -c i (t -s), s) ds (x, t) ∈ D. (7)
For later purposes, we define the following space

V i := {v i solution to (6) defined on D, with f i ∈ L 1 (D), φ i ∈ L 1 (J)}, (8)
equipped with norm

|||v i ||| V i = f i L 1 (D) + φ i L 1 (J) . (9)
Without loss of generality, assume now that all the speeds are positive constant values

c k > 0, k = 1, • • • , p.
According to the definition in (5), the effective domain D corresponds to the region of the space-time between the lines

x = a + ct, x = b + ct, intersecting at time T * = b -a c -c
, where

c := max k c k , c := min k c k .
More precisely, as in Figure 1. Now, for any y ∈ J = [a, b] fixed, consider the following set,

D = {(x, t) | t ∈ [0,T * ], b + ct ≤ x ≤ a + ct}, (10)
K y i := {τ ∈ I | (y + c i τ, τ) ∈ D}, (11)
as in Figure 2. For t < T * , we can also define the subset (Figure 3)

K y i t := {τ ∈ I | (y + c i τ, τ) ∈ D} ∩ {0 ≤ τ ≤ t}. (12)
Defining the new variable y = xc i t, formula (7) exactly reads

v i (y) = φ i (y) + ∫ K y i t f i (y + c i s, s) ds, y ∈ J,
and then One has the following.

|||v i ||| V i = ∫ b a |φ i (y)| dy + ∫ b a ∫ K y i | f i (y + c i s, s)| ds dy = ∫ b a |φ i (y)| dy + ∫ b a ∫ min{ y-a c-c i , b-y c i -c } 0 | f i (y + c i s, s)| ds dy. (13
Lemma 1 Consider equation ( 6),whose solution is [START_REF]Applications bilinéaires compatibles avec un opérateur hyperbolique[END_REF]. Then:

v i L 1 (D) ≤ c(T * )|||v i ||| V i , where T * = b -a c -c . ( 14
) Proof ∫ D |v i (x, t)| dx dt ≤ ∫ D |φ i (x -c i t)| dx dt + ∫ D ∫ t 0 | f i (x -c i (t -s), s)| ds dt dx ≤ ∫ b-a c-c 0 ∫ b+ct a+ct |φ i (x -c i t)| dx dt + ∫ b-a c-c 0 ∫ b+ct a+ct ∫ t 0 | f i (x -c i t + c i s, s)| ds dx dt = ∫ b-a c-c 0 ∫ b+(c-c i )t a+(c-c i )t |φ i (y)| dy dt + ∫ b-a c-c 0 ∫ b+(c-c i )t a+(c-c i )t ∫ K y i t | f i (y + c i s, s)| ds dy dt ≤ T * ∫ J |φ i (y)| dy + ∫ b-a c-c 0 dt ∫ b a ∫ K y i | f i (y + c i s, s)| ds dy ≤ T * |||v i ||| V i ,
where the last inequality follows from (13).

Step

-A Fubini-type theorem

We now perform the same computation for

∫ D |v i v j | dx dt.
We will see that space and time play somehow the same role when considering the interactions between waves with different velocities c i c j , and there will be no dependency on T * as in inequality [START_REF]Some existence theorems for semilinear hyperbolic systems in one space variable[END_REF].

Proposition 1 If v j ∈ V j and v k ∈ V k with c j c k then v j v k ∈ L 1 (D) and v j v k L 1 (D) ≤ 1 |c k -c j | |||v j ||| V j |||v k ||| V k . ( 15
)
Proof Notice that the maximal time t j such that z + c j t ∈ D for t ≤ t j is determined by the system

z + c i t = a + ct, i.e. t = z -a c -c j , z + c i t = b + ct, i.e. t = b -z c j -c .
This way, using formula [START_REF]Compensated compactness and applications to partial differential equations[END_REF], one gets

∫ D |v i (x, t)v j (x, t)| dx dt ≤ 1 |c i -c j | ∫ J×J |φ i (y)φ j (z)| dy dz + 1 |c i -c j | ∫ J |φ i (y)| dy ∫ J ∫ min{ z-a c-c j , b-z c j -c } 0 | f j (z + c j s, s)| ds dz + 1 |c i -c j | ∫ J |φ j (z)| dz ∫ J ∫ min{ y-a c-c i , b-y c i -c } 0 | f i (y + c i s, s)| ds dy + 1 |c i -c j | ∫ J×J dy dz ∫ min{ y-a c-c i , b-y c i -c } 0 ∫ min{ z-a c-c j , b-z c j -c } 0 | f i (y + c i s, s)|| f j (z + c j τ, τ)| ds dτ ≤ 1 |c i -c j | ∫ J×J |φ i (y)φ j (z)| dy dz + 1 |c i -c j | ∫ J |φ i (y)| dy ∫ J ∫ K z j | f j (z + c j s, s)| ds dz + 1 |c i -c j | ∫ J |φ j (z)| dz ∫ J ∫ K y i | f i (y + c i s, s)| ds dy + 1 |c i -c j | ∫ J dy ∫ K z j | f i (y + c i s, s)| ds ∫ J dz ∫ K y i | f j (z + c j τ, τ)| dτ,
which ends the proof again from [START_REF]Compensated compactness and applications to partial differential equations[END_REF]. The following lemma comes directly from ( 6) and ( 9). Lemma 2 Let v i be the solution to [START_REF]Global solutions for the gravity water waves equation in dimension 3[END_REF] and recall the definition of the norm in [START_REF]Global Existence for Nonlinear Wave Equations[END_REF]. If φ i L 1 (J) ≤ ε, for some positive constant ε small enough, then the following holds:

|||v i ||| V i -ε ≤ ∂ t v i + c i ∂ x v i L 1 (D) ≤ |||v i ||| V i . (16)
The fixed point scheme

We come back to the proof of the theorem defining the iterative scheme

∂ t v m i + c i ∂ x v m i + j,k A i jk v m-1 j v m-1 k = 0, v m i (x, 0) = φ i (x), (17)
for m ≥ 1 and i = 1, • • • , p, where:

• φ i ∈ L 1 (J)
are the initial data associated with the Cauchy problem in (1);

• at the first iteration m = 1,

v m-1 i = u 0 i := φ i (x -c i t),
i.e. the solution to the linear Cauchy problem:

∂ t u 0 i + c i ∂ x u 0 i = 0, u 0 i (x, 0) = φ i (x).
Notice that, by definition, v i ∈ V i in (8) at each iteration. We now prove that this iteration model has a fixed point. Denoting by

α m i := ∂ t v m i + c i ∂ x v m i L 1 (D) ,
by applying Proposition 1 to system (17), one gets:

α m i ≤ j,k | A i jk | |c j -c k | |||v m-1 j ||| V j |||v m-1 k ||| V k .
On the other hand, from Lemma 2,

α m i ≤ j,k | A i jk | |c j -c k | (α m-1 j + ε j )(α m-1 k + ε k ),
where

ε i := φ i L 1 (J) .
Define γ := max j,k i

| A i jk | |c j -c k | . Therefore, summing up i = 1, • • • , p, i α m i ≤ γ i ε i + i α m-1 i 2 ,
and if we denote

r m := i α m i , E 0 = i ε i , one gets r m ≤ γ(E 0 + r m-1 ) 2 ,
i.e. the iterative scheme maps the closed set

B r m-1 := {u ∈ V i , i = 1, • • • , p, | u(x, 0) = φ i (x), ∂ t u + c i ∂ x u L 1 (D) ≤ r m-1 } into B r m , with r m = γ(E 0 + r m-1 ) 2 .
In order to apply the fixed point theorem, we then require that

r m = γ(E 0 + r m-1 ) 2 ≤ r m-1 .

This yields

r := r m-1 ≤ 1 -2γE 0 + 1 -4γE 0 2γ , 4γE 0 < 1. (18)
We need to check that the iterative scheme defines a strict contraction. We compute

∂ t (v m i -v m-1 i ) + c i ∂ x (v m i -v m-1 i ) L 1 (D) ≤ j,k | A i jk | v m-1 j v m-1 k -v m-2 j v m-2 k L 1 (D) ≤ j,k | A i jk | v m-1 j (v m-1 k -v m-2 k ) L 1 (D) + (v m-1 j -v m-2 j )v m-2 k L 1 (D) ≤ j,k | A i jk | |c i -c j | |||v m-1 j ||| V j |||v m-1 k -v m-2 k ||| V k + |||v m-2 k ||| V k |||v m-1 j -v m-2 j ||| V j ≤ k | A i jk | |c i -c j | (r + E 0 )|||v m-1 k -v m-2 k ||| V k + j | A i jk | |c i -c j | (r + E 0 )|||v m-1 j -v m-2 j ||| V j = 2(r + E 0 ) j | A i jk | |c i -c j | |||v m-1 j -v m-2 j ||| V j . Therefore Lemma 2 yields i |||v m i -v m-1 i ||| V i ≤ 2γ(r + E 0 ) j |||v m-1 j -v m-2 j ||| V j .
Choosing r = E 0 (in accordance with (18)), the Lipschitz constant is 4γE 0 and the iterative scheme defines a strict contraction on B r provided that 4γE 0 < 1.

Uniqueness

Now let u m i , ūm i be two outputs of the iterative scheme, with initial data φ i , φ i respectively. We use the following notation:

φ i L 1 (J) = ε i , i ε i = E 0 ; φ i L 1 (J) = ε i , i ε i = Ē0 < E 0 .
Taking the difference, one has

i |||u m i -ūm i ||| V i ≤ γ j,k (α j + ε j )|||u m-1 k -ūm-1 k ||| V k + (α k + ε k )|||u m-1 j -ūm-1 j ||| V j ≤ γ(2r + E 0 + Ē0 ) k |||u m-1 k -ūm-1 k ||| V k ≤ 4γE 0 k |||u m-1 k -ūm-1 k ||| V k ≤ • • • ≤ (4γE 0 ) m k φ i -φ i L 1 (J) ,
which yields the proof of uniqueness, since 4γE 0 < 1.

L 1 initial data

In the case of more general initial data φ i ∈ L 1 (J), we consider a partition of the real line R = ∪ h J h , where J h are closed connected intervals with h φ h L 1 (J h ) ≤ E 0 . Let D h , defined as ( 5), be the domain of the solution u h i related to the interval J h . When J h 1 ∩ J h 2 ∅, the two solutions u h 1 i , u h 2 i coincide thanks to uniqueness proved above. Therefore we can glue the u h i together to get our solution.

xFig. 1

 1 Fig. 1 Representation of the domain D.

Fig. 2 Fig. 3

 23 Fig.2Representation of K y i . In this picture, the triangle D = ∪ y∈J K y i .

Acknowledgement

Roberta Bianchini thanks Roberto Natalini for introducing her this line of research and the beautiful paper by Tartar [14]. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program Grant agreement No 637653, project BLOC "Mathematical Study of Boundary Layers in Oceanic Motion". This work was supported by the SingFlows project, grant ANR-18-CE40-0027 of the French National Research Agency (ANR).