Xavier Blanc

Marc Josien

From the Newton equation to the wave equation : the case of shock waves

We study the macroscopic limit of a chain of atoms governed by the Newton equation. It is known from the work of Blanc, Le Bris, Lions, that this limit is the solution of a nonlinear wave equation, as long as this solution remains smooth. We show, numerically and mathematically that, if the distances between particles remain bounded, it is not the case any more when there are shocks -at least for a convex nearest-neighbour interaction potential with convex derivative.

Introduction

Motivation We investigate here the macroscopic limit of the time-dependent Newton equation ruling the evolution of a set of particles at the microscopic scale. We perform our study in a simplified context: the particles form a one-dimension chain and we suppose that the interactions between the particles are nearest-neighbour interactions. It has been proven in [START_REF] Blanc | From the Newton equation to the wave equation in some simple cases[END_REF] that, when the potential is convex, this system tends to a wave equation, provided that the solution of this wave equation is regular. However, non-linear wave equations are known to develop shocks in finite time. Our aim is to examine how this phenomenon impacts the convergence of Newton equations to wave equation. Consider 2N particles, indexed by j ∈ [[-N , N -1]] and with positions X j which interact through the Newton equation, for j ∈ [[-N + 1, N -2]]:

d 2 d t X j (t ) = W X j +1 (t ) -X j (t ) -W X j (t ) -X j -1 (t ) , ( 1
)
where W is the interaction potential. Throughout the article, we assume that W is even. The initial and boundary conditions are:

X j +1 -X j (0) = φ x 0 j N and d d t X j (0) = φ τ 0 j N , (2)
X -N (t ) = N φ l and X N (t ) = N φ r .

(3)

We introduce the following rescaling:

t = N τ, j = N x.
The time t is the microscopic time while τ is the macroscopic time. Then, the semidiscrete equation ( 1) is consistent with the wave equation:

∂ 2 τ φ(τ, x) = ∂ x W (∂ x φ(τ, x)) , (4)
with initial and boundary conditions:

∂ x φ(τ = 0, x) = φ x 0 (x) and ∂ τ φ(τ = 0, x) = φ τ 0 (x), (5)
φ(τ, -1) = φ l and φ(τ, 1) = φ r .

Remark 1. It is worth pointing out that the natural variables in the hyperbolic system (4) are ∂ τ φ and ∂ x φ, in the sense that:

∂ τ ∂ τ φ ∂ x φ = ∂ x W ∂ x φ ∂ τ φ ,
which is a p-system (see [START_REF] Serre | Systems of Conservation Laws 1: Hyperbolicity, entropies, shock waves[END_REF], p 127-131). We therefore introduce their discrete analogues:

U j = X j +1 -X j and V j = d X j d t .
Remark 2 (About inversion). One could a priori think that (1) may lead to some inversions of atom positions, especially when shocks occur (see [START_REF] Brenier | Approximation of a simple navier-stokes model by monotonic rearrangement[END_REF]). Put differently, one could have X j +1 (t ) < X j (t ) for certain t and j , even if X j (t = 0) was increasing. This would question the physical relevance of (1), for the j -th particle is supposed to interact with its nearest neighbours (which are the j -1-th and the j + 1-th particles if and only if X j is monotone). However, numerical simulations show that, for many interesting initial conditions (including many of those that lead to shocks), such inversions never occur. We therefore assume throughout the article that condition X j (t ) < X j +1 (t ) holds for all t , j .

In the regular case and if W is convex, it has been proven in [START_REF] Blanc | From the Newton equation to the wave equation in some simple cases[END_REF] that (1) converges to (4) in the following sense: Theorem 1.1. Assume that W ∈ C 4 (R), and that W ≥ α > 0. Suppose φ l = -1 and φ r = 1. Assume that φ ∈ C τ C 4

x is a solution to (4) for the initial and boundary conditions [START_REF] Berezhnyy | Continuum limit for three-dimensional massspring networks and discrete Korn's inequality[END_REF] and [START_REF] Blanc | From the Newton equation to the wave equation in some simple cases[END_REF]. Let X j (t ) be the unique solution to [START_REF] Adams | Sobolev spaces[END_REF] for the initial and boundary conditions [START_REF] Allaire | Analyse numérique et optimisation: Une introduction à la modélisation mathématique et à la simulation numérique[END_REF] and [START_REF] Arnold | Mathematical methods of classical mechanics[END_REF]. Then we have the following convergences:

∀τ ∈ [0, T [, sup -N ≤i ≤N -1 1 N X j (N τ) -φ τ, i N → N →∞ 0, (7)
∀τ ∈ [0, T [, sup

-N ≤i ≤N -1 d X j d t (N τ) -∂ τ φ τ, i N → N →∞ 0. ( 8
)
Remark 3. Theorem 1.1 is not stated in [START_REF] Blanc | From the Newton equation to the wave equation in some simple cases[END_REF] for the initial condition (2), but with:

X j (t = 0) = N j /N -1 φ x 0 (x)d x and d X j d t (t = 0) = φ τ 0 j N .
As easily seen, its proof however also applies to the initial condition [START_REF] Allaire | Analyse numérique et optimisation: Une introduction à la modélisation mathématique et à la simulation numérique[END_REF].

When W is convex but not quadratic, even if φ x 0 and φ τ 0 are smooth, shocks generally occur in finite time for solutions of [START_REF] Benzoni-Gavage | Semi-discrete shock profiles for hyperbolic systems of conservation laws[END_REF]. By shock, we mean that the solution φ of (4) becomes irregular (see [START_REF] Serre | Systems of Conservation Laws 1: Hyperbolicity, entropies, shock waves[END_REF] for examples). An interesting question is what happens after such shocks for the discrete system [START_REF] Adams | Sobolev spaces[END_REF], and in particular if there is still a link between (1) and ( 4). To answer this question, we will consider Riemann-like initial conditions, as is customary in the study of hyperbolic systems. Let us underline that [START_REF] Adams | Sobolev spaces[END_REF], which can be seen as a semi-discrete numerical scheme, is taken for granted, as it comes from a physical model. Some authors take the opposite way, and modify given schemes (adding viscosity for example) in order to go from the discrete system to the continuous one (see [START_REF] Mielke | From discrete visco-elasticity to continuum rateindependent plasticity: rigorous results[END_REF]), or to help the numerical computation of hyperbolic systems ( [START_REF] Tadmor | The numerical viscosity of entropy stable schemes for systems of conservation laws[END_REF]). Let us also mention that their exists a quite detailed study on discrete systems ruled by [START_REF] Adams | Sobolev spaces[END_REF] in the particular case where:

W (u) = exp(-u). (9)
In that case, called the Toda lattice (see [START_REF] Deift | A continuum limit of the Toda lattice[END_REF], [START_REF] Holian | Shock waves in the Toda lattice: Analysis[END_REF], [START_REF] Toda | Theory of nonlinear lattices[END_REF], [START_REF] Venakides | The Toda shock problem[END_REF]), the discrete Hamiltonian system is completely integrable: this allows for a detailed description of the solutions. It is well-known that (4) does not describe well the limiting system and that the solutions are dispersive waves. This is linked with Lax pairs, and helps to make the connection with the Korteweg-de Vries equation (see [START_REF] Lax | The small dispersion limit of the Korteweg-de Vries equation[END_REF]). We will not investigate in this article this particular case, which is, in our understanding, closely linked with the special structure induced by the potential [START_REF] Brenner | Poincaré-Friedrichs inequalities for piecewise H 1 functions[END_REF]. We shall however demonstrate that the solutions associated with more general potentials globally display the same features as the dispersive waves of the Toda lattice (see Section 5).

Numerics In order to have a better understanding of (1), we perform some numerical experiments. To do so, we use a Verlet scheme (see [START_REF] Bris | Systèmes multi-échelles: modélisation et simulation[END_REF], p 111) on the variables U j and Z j := dU j d t . More explicitly, we simulate:

                 U n+1/2 j = U n j + δt 2 Z n j , Z n+1/2 j = Z n j + δt 2 W U n+1/2 j +1 -2W U n+1/2 j + W U n+1/2 j -1 , U n+1 j = U n j + δt Z n+1/2 j , Z n+1 j = Z n j + δt W U n+1/2 j +1 -2W U n+1/2 j + W U n+1/2 j -1 , ( 10
)
where X n j is an approximation for X j (nδt ). We take an initial condition corresponding to a Riemann problem or a smooth initial condition that develops shocks in finite time (for the sake of simplicity, we only use Riemann problems for illustrations in this article). The crucial feature of [START_REF] Bressan | A uniqueness condition for hyperbolic systems of conservation laws[END_REF] is that it preserves the Hamiltonian properties of (1) (for ( 10) is symplectic). The error we make on U j in L 2 norm is of order O(N T δt ) (see [START_REF] Hairer | Geometric Numerical Integration Structure-Preserving Algorithms for Ordinary Differential Equations[END_REF] p13), where T is the final macroscopic time of simulation, which allows to simulate (1) for a reasonably large number 2N of particles (N 10 4 ), and thus to have a fair experimental knowledge of the system (1).

Outline of the article

In Section 2, we introduce the notations and collect some classical facts about (1) and ( 4). In particular, we focus on the initial and boundary conditions, that are supposed to mimic the Riemann problem. We also focus on the natural energy of these systems. In Section 3, we state and next illustrate our main results. We focus first on the simple quadratic potential W (u) = u 2 /2 and claim that the convergence of (1) to ( 4) is true for a large class of initial conditions. This is proved in Section 4. Then we examine the case where both W and W are strongly convex. We show that, if the distances between neighbouring particles remain bounded and if the energy of the continuous system (4) is not preserved, solutions of (1) do not converge to solutions of (4). It is based on the fact that the system (1) displays the property of light cone: the perturbations propagate with a finite speed at macroscopic level. This is proved in Section 5. We state next a conjecture about a uniform bound on the distances between particles of the system (1), that we justify with numerics and that we question through a study of the linear case. This conjecture is motivated by the fact that the assumption of boundedness of the distances between particle is a major assumption in every result of Section 5. We discuss it in Section 6. Finally, we state that discrete shock waves do not exist, either when W (u) = u 2 2 or when W and W are strictly convex. It is proved in Section 7.

Preliminaries

General notations

Let q ∈ [1, ∞]. For Y j , with j ∈ [[-N , N -1]],
we denote by:

Y j l q j = N -1 i =-N Y q j 1/q if q < ∞, max j ∈[[-N ,N -1]] Y j if q = ∞.
We denote by C p the set of piecewise continuous functions on [-1, 1], and C 1 p the set of piecewise continuous functions on [-1, 1] that have piecewise continuous derivatives. We use the subscript per for functional spaces to indicate that we intersect these spaces with the space of 2-periodic functions. We use the subscripts x, τ, t for functional spaces to indicate that these spaces have their variables x in [-1, 1], τ in [0, T ], respectively t in [0, N T ]. For example:

H 1 τ (C x ) := H 1 ([0, T ], C ([-1, 1])) .

Initial data and boundary conditions

In the present article, we mainly use Dirichlet boundary conditions. They have the advantage of being consistent with Riemann problems. In Section 6, we will also use periodic boundary conditions for technical reasons; more specifically, when the potential W is quadratic, it allows for an explicit resolution of (1). We say that (1) (respectively (4)) is set with Dirichlet boundary conditions if (2) and

(3) (respectively ( 5) and ( 6)) are satisfied, with φ x 0 , φ τ 0 ∈ C p and φ l , φ r ∈ R being compatible in the following sense:

1 -1 φ x 0 (x)d x = φ r -φ l , φ τ 0 (-1) = 0, φ τ 0 (1) = 0. ( 11
)
We say that the system (1) is set with periodic boundary conditions if [START_REF] Adams | Sobolev spaces[END_REF] is satisfied for all j with the convention that X N + j = X -N + j . The associated initial conditions are (3) with φ τ 0 , φ x 0 ∈ C p such that the compatibility condition:

1

-1 φ x 1 (x)d t = 0 is satisfied.

Hypotheses on W

We suppose that W is C 2 and strongly convex:

W ≥ α > 0. ( 12
)
Indeed, this assumption implies that (4) is a strictly hyperbolic system (if not, the theory for (4) is far more complex). A very particular case is when W is quadratic:

W (u) = u 2 2 . ( 13
)
When we consider a non-quadratic potential, we also assume that W is C 3 and that W is strictly convex:

W > 0. ( 14
)
Let us emphasize that ( 4) is genuinely non-linear when W > 0 (see [START_REF] Serre | Systems of Conservation Laws 1: Hyperbolicity, entropies, shock waves[END_REF] p 113 and p 127). We speak about the linear case (respectively the nonlinear case) when ( 13) is satisfied (respectively when [START_REF] Dafermos | Estimates for conservation laws with little viscosity[END_REF] and ( 14) are satisfied). The terminology may seem ambiguous, but it is justified by (4), which involves W and not W . The convexity [START_REF] Dafermos | Estimates for conservation laws with little viscosity[END_REF], and a fortiori [START_REF] Deift | A continuum limit of the Toda lattice[END_REF], is obviously a strong and non-physical simplification, as a physical potential should be even (and non-constant even potentials with other minima than 0 cannot satisfy (12) on R). For example, our results do not directly cover this "quadratic" potential:

W (u) = (|u| -1) 2 . ( 15
)
Our numerical experiments suggest that for given initial conditions, the distances between particles is bounded from below and from above (see Remark 2 and Section 6). Hence one can require [START_REF] Dafermos | Estimates for conservation laws with little viscosity[END_REF] or ( 14) to be true only on the corresponding intervals. For example, if we know a priori that the order of the particles is preserved, one can apply our results with the potential (15).

The discrete system

Notations For the discrete system (1), we denote:

V j (t ) = d X j d t (t ), U j (t ) = X j +1 (t ) -X j (t ), Z j (t ) = dU j d t (t ).
Remark 4 (Dependence on N ). X j and the other discrete quantities implicitly depend on N . When necessary, we write X N j , U N j , et cetera.

The correspondence between the discrete system and the continuous system in encoded in the following notations:

k N (x) := N x , θ N (x) := N x -k N (x), φ N (τ, x) := 1 -θ N (x) N X k N (x) (t ) + θ N (x) N X k N (x)+1 (t ), ζ N (τ, x) := ∂ τ φ N (τ, x) = 1 -θ N (x) d d t X k N (x) + θ N (x) d d t X k N (x)+1 (t ), ξ N (τ, x) := d d t X k N (x) (t ).
Remark that ∂ τ φ N is the linear interpolation of V j , and is therefore not equal to ξ N (τ, x), which corresponds to V j (t ). In any case, we extend the functions φ N , ξ N by continuity with constant branches on ] -∞, -1] ∪ [1, +∞[. For example, we have

φ(x) = φ l if x < -1.
Properties of the discrete system The discrete system (1) is an Hamiltonian system, with the energy:

E D (t ) := 1 2N N -1 j =-N V 2 j (t ) + 1 N N -1 j =-N W (U j (t )). (16)
The energy ( 16) is the total mechanical energy of the system. The kinetic energy is the first term and the potential energy is the second term. Either in Dirichlet or in periodic setting, an elementary calculation shows that the discrete energy E D is preserved:

d d t E D (t ) = 0. ( 17
)
As a consequence, the energy (16) being convex, a direct application of the Cauchy-Lipschitz theorem implies that (1) has a unique solution for every time t ∈ [0, +∞[, provided W satisfies [START_REF] Dafermos | Estimates for conservation laws with little viscosity[END_REF]. For later purpose, we define the notion of discrete compatibility.

Definition 2.1 (D-compatibility).

We say that T > 0 is D-compatible with φ x 0 and φ τ 0 if there exist δ > 0 and C > 0 such that:

∂ x φ N (τ, x) -φ x 0 (-1) ≤ C N -1 ∀(τ, x) ∈ [0, T ] × [-1, -1 + δ], ∂ x φ N (τ, x) -φ x 0 (1) ≤ C N -1 ∀(τ, x) ∈ [0, T ] × [1 -δ, 1],
and

∂ τ φ N converges uniformly to 0 on [0, T ] × {[-1, -1 + δ] ∪ [1 -δ, 1]}, as N goes to infinity.
D-compatibility means that the solution of (1) is almost not perturbed near the boundary x = -1 and x = 1, until time T .

The continuous system

Let T > 0. Following [START_REF] Serre | Systems of Conservation Laws 1: Hyperbolicity, entropies, shock waves[END_REF], p 28, we say that φ ∈ W 1,∞ τ,x is a weak solution of (4) with initial conditions [START_REF] Berezhnyy | Continuum limit for three-dimensional massspring networks and discrete Korn's inequality[END_REF]

if, for all g ∈ C ∞ c (] -∞, T [×] -1, 1[): T 0 1 -1 ∂ x g W ∂ x φ -∂ τ g ∂ τ φ (τ, x)d xd τ = 1 -1 g (0, x)φ τ 0 (x)d x, ( 18
) T 0 1 -1 ∂ x g ∂ τ φ -∂ τ g ∂ x φ (τ, x)d xd τ = 1 -1 g (0, x)φ x 0 (x)d x. ( 19
)
We say that a weak solution φ of ( 4) is an entropy solution if it also satisfies in the weak sense (see [START_REF] Serre | Systems of Conservation Laws 1: Hyperbolicity, entropies, shock waves[END_REF], p 82):

d d τ E C (τ) ≤ 0, ( 20
)
where E C is the continuous energy associated with φ:

E C (τ) = 1 -1 1 2 ∂ τ φ 2 + W ∂ x φ (τ, x)d x. ( 21
)
We are interested in weak entropy solutions φ of ( 4) satisfying

E C (T ) < E C (0). (22)
Shocks satisfy [START_REF] Mielke | From discrete visco-elasticity to continuum rateindependent plasticity: rigorous results[END_REF]. We recall now the definition of the Riemann problems:

Definition 2.2 (Riemann problem). Let u l , u r , v l , v r ∈ R
, and:

φ x 0 (x) := u l if x < 0, u r if x ≥ 0, φ τ 0 (x) := v l if x < 0, v r if x ≥ 0. ( 23
)
Solving the Riemann problem associated with (u l , u r , v l , v r ) consists in finding φ an entropy solution of (4) on [0, T ] × R with initial conditions (5). This is the classical Riemann problem. However, it is possible to use weaker assumptions on φ x 0 and φ τ 0 , that simulate what we call a boundary Riemann problem. This second definition is more flexible and allows to work with a very large class of initial conditions (for example, smooth initial data that develop discontinuities in finite times, in system (4)). Namely: Definition 2.3 (Boundary Riemann problem). Let u l , u r ∈ R 2 , and:

φ x 0 (x) := u l if x < -1/2, u r if x > 1/2, φ τ 0 (x) := 0 if x < -1/2, 0 if x > 1/2, ( 24
)
without further requirement on φ x 0 and φ τ 0 between -1/2 and 1/2. Solving this boundary Riemann problem consists in finding φ(τ, x), entropy solution of (4) with initial and boundary conditions (5) and (6), and X j (t ), solution of (1) with initial and boundary conditions (2) and (3), with φ l and φ r being constant so that (11) is satisfied.

We impose φ τ 0 to vanish near the boundary in the boundary Riemann problem (24) so that φ l and φ r are constant; this is useful to avoid some technicalities about boundary conditions. The solutions of the Riemann problem [START_REF] Müller | Variational models for microstructure and phase transitions[END_REF] are combinations of rarefaction waves and shock waves. One does not change the solution of (4) (for T sufficiently small) if one restricts φ to x ∈ [-1, 1] and solves (4) with Dirichlet boundary conditions [START_REF] Blanc | From the Newton equation to the wave equation in some simple cases[END_REF]. For example (see [START_REF] Serre | Systems of Conservation Laws 1: Hyperbolicity, entropies, shock waves[END_REF], p 127-131), if u r > u l and if the following Rankine-Hugoniot condition is satisfied:

v r -v l = σ (u r -u l )
and

W (u r ) -W (u l ) = σ(v r -v l ), (25)
then the entropy solution of the Riemann problem reads as:

∂ x φ(τ, x) = u l if x < -στ u r if -στ ≤ x , ∂ τ φ(0, x) = v l if x < -στ v r if -στ ≤ x ,
and satisfies [START_REF] Mielke | From discrete visco-elasticity to continuum rateindependent plasticity: rigorous results[END_REF]. We are interested in boundary Riemann problems. As a consequence, we focus on weak solutions φ ∈ W 1,∞ τ,x of (4) in the Dirichlet setting that can be continued by a constant on the right and on the left: Definition 2.4 (C -compatibility). Let φ x 0 and φ τ 0 ∈ C p , φ l and φ r ∈ R satisfying [START_REF] Brillouin | Wave propagation and group velocity[END_REF]. Assume that φ is an entropy solution of (4) on [0, T ]×[-1, 1] with initial and boundary conditions [START_REF] Berezhnyy | Continuum limit for three-dimensional massspring networks and discrete Korn's inequality[END_REF] and [START_REF] Blanc | From the Newton equation to the wave equation in some simple cases[END_REF]. We say that T is C -compatible with φ x 0 and φ τ 0 if there exists δ > 0 such that:

φ([0, T ] × [-1, -1 + δ]) = φ l and φ([0, T ] × [1 -δ, 1]) = φ r . ( 26
)
If T is D-compatible and C -compatible with φ x 0 and φ τ 0 , we say that it is DCcompatible. Basically, DC -compatibility provides a strong control on the solutions of ( 1) and (4) near the boundary x = -1 and x = 1, until time T .

For the linear system, we have the following theorem of existence and uniqueness (see Theorem 3 p 384 and Theorem 4 p 385 of [START_REF] Evans | Partial Differential Equations[END_REF]): Theorem 2.1 (Existence and uniqueness in the linear case). Let

φ x 0 , φ τ x ∈ L 2 x . Suppose that a, b ∈ C 1
x , and a ≥ a 0 > 0. Then, there exists one and only one solution φ ∈ H 1 τ,x to:

∂ 2 τ φ(τ, x) = ∂ x a(x)∂ x φ(τ, x) + b(x) ,
with initial and boundary conditions (5) and [START_REF] Blanc | From the Newton equation to the wave equation in some simple cases[END_REF]. In addition, the energy ẼC of the system is preserved:

ẼC = 1 -1 1 2 ∂ τ φ(τ, x) 2 + a(x) 2 ∂ x φ(τ, x) 2 + b(x)∂ x φ(τ, x) d x.
It is clear that this energy extends the above definition (21).

Discrete shock waves

Definition 2.5. We say that X j (t ), j ∈ Z, t ∈ R + , is a discrete shock wave of (1) associated with (u l , u r ) ∈ R 2 , u l = u r , if X j (t ) satisfies (1) and if there exist φ ∈ C 2 (R) and c ∈ R such that:

X j (t ) = φ( j -c t ) ∀t ∈ R + , ∀ j ∈ Z, lim x→-∞ φ (x) = u l , lim x→+∞ φ (x) = u r .
The definition implies that:

c 2 φ (x) = W (φ(x + 1) -φ(x)) -W (φ(x) -φ(x -1)). ( 27
)

Results

We state here our main results and illustrate them with some numerical results.

The linear case

When W (u) = u 2 /2, one observes that φ N converges in H 1 τ,x to φ. One can even see that for regular initial conditions, this convergence seems to hold in every W 1,p τ,x . This is illustrated by Figure 1: We prove this convergence in a generalized framework, where the quadratic potential W depends not only on u but also on x:

Theorem 3.1. Let T > 0, φ l , φ r ∈ R, φ x
0 and φ τ 0 ∈ C p satisfy [START_REF] Brillouin | Wave propagation and group velocity[END_REF]. Assume that:

W (x, u) = 1 2 A(x)u 2 + B (x)u.
with A, B ∈ C 1 x and A satisfying:

A ≥ α > 0, ( 28
)
for α a given positive constant. Consider the solution X N j (t ) to:

d 2 d t 2 X N j (t ) = ∂ u W j N ,U N j (t ) -∂ u W j -1 N ,U N j -1 (t ) ( 29
)
for all j ∈ [[-N +1, N -2]
] for the initial and boundary conditions (2) and (3). Then the associated φ N converges:

φ N → N →+∞ φ in H 1 τ,x ,
where φ is the unique solution of:

∂ 2 τ φ(τ, x) = ∂ x ∂ u W (x, ∂ x φ(τ, x) , ( 30
)
for the initial and boundary conditions [START_REF] Berezhnyy | Continuum limit for three-dimensional massspring networks and discrete Korn's inequality[END_REF] and [START_REF] Blanc | From the Newton equation to the wave equation in some simple cases[END_REF].

It has a direct corollary: [START_REF] Brillouin | Wave propagation and group velocity[END_REF]. Assume that W satisfies [START_REF] Dafermos | Energy methods for quasilinear hyperbolic initial-boundary value problems[END_REF]. Let φ be the solution of (4) for the initial and boundary conditions [START_REF] Berezhnyy | Continuum limit for three-dimensional massspring networks and discrete Korn's inequality[END_REF] and [START_REF] Blanc | From the Newton equation to the wave equation in some simple cases[END_REF], and X N j be the solution of (1) for the initial and boundary conditions (2) and (3). We have the following convergence:

Corollary 3.2. Let T > 0, φ l , φ r ∈ R, φ x 0 and φ τ 0 ∈ C p satisfying
φ N → N →+∞ φ in H 1 τ,x .
Remark 5 (Less restrictive assumptions). For both Theorem 3.1 and Corollary 3.2, it is sufficient to assume that φ x 0 and φ τ 0 ∈ L 2 x as long as X N j satisfies the initial condition:

X j +1 (0) -X j (0) = ∂ x φ N τ = 0, j N and d d t X j (0) = ∂ τ φ N τ = 0, j N ,
such that:

∂ x φ N (τ = 0, .) → φ x 0 in L 2 x , ξ N (τ = 0, .) → φ τ 0 in L 2 x .

The non-linear case

If the potential W is convex but not quadratic, when there is a shock, we observe on numerical simulations that ∂ x φ N does not converge strongly to the associated ∂ x φ.

It does not even converge weakly. Actually ∂ x φ N oscillates with a high frequency and an amplitude that does not decrease when N grows. We believe that this situation is generic for basically any potential such that W is not affine on the zone where U j evolves. We can to prove this non-convergence, under the extra-hypothesis that W is strictly convex, and under the assumption that the distance between particles U N j is bounded uniformly in N , j and t ∈ [0, N T ] (the latter assumption is discussed in Section 6).

We illustrate this non-convergence with Figure 2 comparing ∂ x φ N and ∂ x φ. Remarkably enough, even if there are large oscillations, let us remark that distances between particles remain bounded. We check numerically that ∂ x φ N coincides with ∂ x φ outside a region of space away from the shock that grows linearly in macroscopic time; this property is known for Toda lattice [START_REF] Holian | Shock waves in the Toda lattice: Analysis[END_REF]. It is also known that the Korteweg-de-Vries equation [START_REF] Lax | The small dispersion limit of the Korteweg-de Vries equation[END_REF] has a similar behaviour. 12) and [START_REF] Deift | A continuum limit of the Toda lattice[END_REF]. Assume that φ x 0 , φ τ 0 ∈ C 1 p,x and φ l , φ r ∈ R satisfy [START_REF] Brillouin | Wave propagation and group velocity[END_REF] and [START_REF] Niculescu | Convex functions and their applications: a contemporary approach[END_REF], for u l , u r ∈ R. Let T 0 > 0. Let X N j (t ) be the solution of (1) for the initial and boundary conditions (2) and (3). Suppose that:

U N j (t ) ∈ [a, b] ∀ j ∈ [[-N , N -1]], ∀N > 0, ∀t ∈ [0, N T 0 ]. ( 31
)
Then there exists a D-compatible T ≤ T 0 . Assume that φ ∈ W 1,∞ τ,x , is an entropy solution of (4) for the initial and boundary conditions [START_REF] Berezhnyy | Continuum limit for three-dimensional massspring networks and discrete Korn's inequality[END_REF] and [START_REF] Blanc | From the Newton equation to the wave equation in some simple cases[END_REF], that T is C -compatible and that there exists T 1 < T satisfying:

E C (T 1 ) < E C (0). ( 32
)
Then φ N does not converge to φ in the sense of distribution in space and time D τ,x .

Remark 6 (Entropy solution). In Theorem 3.3, we only compare φ N to the entropy solution φ of (4). We think that φ N cannot converge to any weak solution of (4). Indeed, if φ N converges to φ, which is a solution of (4), Lemma 5.2 below implies that ∂ x φ N converges strongly to ∂ x φ. But numerical experiments show that ∂ x φ N oscillates too much so that it cannot converge strongly to anything: this justifies our conclusion.

Remark 7 (Reversibility). ( 1) is a reversible system, but (4) is not when shocks occur, whereas both systems are reversible as long as the solution φ of (4) remains smooth enough. In the first case, the discrete system does not converge to the continuous one (Theorem 3.3), but it does in the second case (Theorem 1.1).

Remark 8 (Convergence breakdown). Suppose that W satisfies ( 12) and [START_REF] Deift | A continuum limit of the Toda lattice[END_REF]. Let φ x 0 and φ τ 0 be smooth functions. Define X N j and φ as in Theorem 3.3. Suppose that there exists a DC -compatible T > 0 such that E C (T ) < E C (0) -in other words, a shock occurs. If Conjecture 3.5 above holds for initial data φ x 0 and φ τ 0 , it leads to the following paradoxical situation:

1. until a certain time T 0 < T , φ(τ, .) is sufficiently smooth, so that Theorem 1.1 applies. Thus:

φ N → φ in C ([0, T 0 ] × [-1, 1]) , 2. applying Theorem 3.3, we get that φ N does not converge to φ in D ([0, T 1 ] × [-1, 1]) as soon as E C (T 1 ) < E C (0).
Therefore, shocks break the discrete-to-continuum convergence of (1) to (4).

It is immediate to see that (1) instantly propagates perturbations in the discrete system. It can be proven by linearizing [START_REF] Adams | Sobolev spaces[END_REF] and assuming a small perturbation on a fixed j 0 -th particle:

X j (0) = X j (0) + δ j j 0 , d X j d t (0) = d X j d t (0).
We assume that both X j and X j satisfy (1). Integrating iteratively (1) for small time ∆t , we get (for j > 0) at leading order in (the proof of this formal expansion is in the spirit of the proof of Proposition 5.1 below):

X j 0 + j (∆t ) -X j 0 + j (∆t ) (∆t ) 2 j (2 j )! j -1 k=0 W U j 0 +k (0) .
However, on the macroscopic level, this propagation has a finite speed. This paradox is due to the fact that the influence of perturbation on x 0 at t 0 decays exponentially outside a cone |x -x 0 | ≤ c |tt 0 |. It is noticeable that this light cone property is an important feature of hyperbolic systems. It is however a key ingredient to prove that (1) does not converge to (4).

We formalize the fact that perturbations of the discrete system propagate with a finite speed on the macroscopic level by the following theorem:

Theorem 3.4. Let W ∈ C 2 (R) satisfy (12). Let T > 0. Assume that X N j (t ) and X N j (t ) satisfy (1), for j ∈ [[0, N -1]], t ∈ [0, N T ], with right boundary condition X N N = X N N = N φ r .
We denote by:

Ṽ N j (t ) = d d t X N j (t ), Ũ N j (t ) := X N j +1 (t ) -X N j (t ).
Suppose that

X N j (t = 0) = X N j (t = 0),∀ j ∈ [[1, N -1]], V N j (t = 0) = Ṽ N j (t = 0),∀ j ∈ [[1, N -1]].
Assume that there exists C ∈ R + such that, ∀N > 0:

sup t ∈R + N -1 j =0 W U N j (t ) + 1 2 N -1 j =0 V N j (t ) 2 ≤ C N , (33)
and:

K = sup u∈]u 1 ,u 2 [ W (u) < +∞, (34)
where:

u 1 := inf N > 0, j ∈ [[-N , N -1]], t ∈ [0, N T ] min U N j (t ), Ũ N j (t ) , u 2 := sup N > 0, j ∈ [[-N , N -1]], t ∈ [0, N T ] max U N j (t ), Ũ N j (t ) .
Let c = exp(2) K . Let x ∈]0, 1[. Then, for all τ < x c , we have:

lim N →+∞        N sup 0 ≤ t < N τ, j > N x U N j (t ) -Ũ N j (t )        = 0, ( 35
) lim N →+∞ sup 0 ≤ t < N τ, j > N x V N j (t ) -Ṽ N j (t ) = 0. ( 36
)
Figure 3: Light cone on the surface t ,U N j (t ) , for Riemann initial conditions corresponding to a shock wave, for a non-linear potential W (u) = u 6 6 j U j t A few remarks are in order: Remark 9. The speed c in Theorem 3.4 is not optimal -we see it from numerical experiments-but it has the same order as the natural speed of (4), given by [START_REF] Pachpatte | On discrete inequalities of the Poincaré type[END_REF]. Formally:

exp(2) K ∝ sup u∈[u l ,u r ] |W (u)| ∝ W (u r ) -W (u l ) u r -u l .
The above formal calculation is exact when W is quadratic.

Remark 10. Assumption (34) of Theorem 3.4 is automatically fulfilled if there exists α, β ∈ R such that α ≤ W (u) ≤ β, ∀u ∈ R. This is the case when the potential W is quadratic. However, such a bound cannot hold if W is strongly convex: one needs to know a priori that the distances U N j between particles is bounded uniformly in N .

Uniform

L ∞ t l ∞ j bound
Most of the results we are able to prove in the non-linear case require the assumption that, for given initial data, the distance between particles remains bounded uniformly in N (that is, (31) is satisfied). We have not been able to prove that this assumption is fulfilled. We formulate the following conjecture:

Conjecture 3.5. Suppose that W ∈ C 2 (R) satisfies [START_REF] Dafermos | Estimates for conservation laws with little viscosity[END_REF]. Assume that φ x 0 and φ τ 0 ∈ C 1 p,x . Then, there exist T > 0 and a < b ∈ R depending only on W , φ x 0 and φ τ 0 such that, for X j (t ) satisfying (1) for the initial and boundary conditions (2) and (3):

a ≤ U N j (N τ) ≤ b, ∀ j ∈ [[-N , N -1]], ∀N ∈ N, ∀τ ∈ [0, T ]. ( 37
)
Note that in the case of Riemann problem [START_REF] Müller | Variational models for microstructure and phase transitions[END_REF], the initial conditions satisfy the hypotheses of Conjecture 3.5. We checked Conjecture 3.5 numerically for a large set of piecewise smooth initial data, with potentials of the form W (u) = Au γ +Bu 2 , γ > 2, A, B ∈ R + . When φ is sufficiently smooth, Conjecture 3.5 can be proven (by Theorem 1.1).

Let us point out the fact that it seems necessary to require some smoothness on the initial conditions in Conjecture 3.5. In other words, one cannot hope that, for X N j solution of (1), for given

T > 0, U N j l ∞ j ,t is controlled by U N j (t = 0) l ∞ j and V N j (t = 0) l ∞ j
uniformly in N . Indeed, we can prove the following proposition:

Proposition 3.6. Let W (u) = u 2 2
, and τ 0 > 0. There exists a sequence of initial conditions U N j (t = 0), V N j (t = 0) such that, for X N j (t ) the corresponding solutions of (1) with periodic boundary conditions, we have:

U N j (t = 0) l ∞ j ≤ 1, V N j (t = 0) l ∞ j ≤ 1, U N j (N τ 0 ) l ∞ j → N →∞ +∞.
The following plot shows the explosion:

Non-existence of discrete shock waves

A natural question is whether or not there exist non-trivial discrete shock waves for the Newton equation ( 1). Should such discrete progressive waves exist, one could expect that they would describe an important feature of the limit of ( 1) system when N → +∞. Unfortunately, we prove that discrete shock waves do not exist, even when the potential is quadratic. More specifically, we prove the following propositions: Proposition 3.7. Suppose W satisfies [START_REF] Dafermos | Energy methods for quasilinear hyperbolic initial-boundary value problems[END_REF]. Then there exist no discrete shock wave for [START_REF] Adams | Sobolev spaces[END_REF], in the sense of Definition 2.5.

Proposition 3.8. Suppose W satisfies [START_REF] Dafermos | Estimates for conservation laws with little viscosity[END_REF] and [START_REF] Deift | A continuum limit of the Toda lattice[END_REF]. Then there exist no discrete shock wave for [START_REF] Adams | Sobolev spaces[END_REF], in the sense of Definition 2.5.

Remark 11. It is straightforward from the proof that there does not exist any other discrete wave than the constant ones in the linear case. In the non-linear case, we do not know if there exists solitons, that is X j (t ) satisfying Definition 2.5, with the slight modification that u l = u r .

The linear case

When the potential is quadratic, the corresponding wave equation ( 4) is linear. Its characteristic lines do not cross, therefore, when the initial conditions are regular, shocks never occur. Furthermore, the energy is preserved: the continuous system ( 4) is thus conservative, as the discrete one (1). This is the reason why the discrete system naturally tends to the continuous one, and we show it with simple arguments, essentially using weak compactness of H 1 τ,x . This extends the result of [START_REF] Blanc | From the Newton equation to the wave equation in some simple cases[END_REF].

Let us prove Theorem 3.1. We first prove that the discrete energy is preserved:

Lemma 4.1. Under the hypotheses of Theorem 3.1, the following generalized discrete energy is preserved:

Ẽ N D (t ) := N -1 k=-N W k N ,U k (t ) + 1 2 N -1 k=-N (V k (t )) 2 . ( 38
)
Proof. Using (1), we get that:

d d t Ẽ N D (t ) = N -1 k=-N ∂ u W k N ,U k (V k+1 -V k ) (t ) + N -1 k=-N +1 V k d 2 d t 2 X k (t ), = N -1 k=-N ∂ u W k N ,U k (V k+1 -V k ) (t ) + N -1 k=-N +1 V k ∂ u W k N ,U k -∂ u W k -1 N ,U k-1 (t ).
Reorganizing the sum, we obtain:

d d t Ẽ N D (t ) = ∂ u W N -1 N ,U N -1 V N -∂ u W (-1,U -N )V -N (t ).
Yet, as (3) is satisfied, we have V N = V -N = 0. Therefore:

d d t Ẽ N D (t ) = 0,
which implies the desired result.

Next we prove that ( 29) is consistent with (30):

Lemma 4.2. Under the hypotheses of Theorem 3.1, we have the following convergences for all g

∈ C ∞ c (] -∞, T [×] -1, 1[): T 0 1 -1 ∂ x g ∂ u W x, ∂ x φ N -∂ τ g ∂ τ φ N (τ, x)d xd τ - 1 -1 g (0, x)φ τ 0 (x)d x → N →+∞ 0, ( 39
) T 0 1 -1 ∂ x g ∂ τ φ N -∂ τ g ∂ x φ N (τ, x)d xd τ - 1 -1 g (0, x)φ x 0 (x)d x → N →+∞ 0. ( 40
)
Proof. It is easy to prove (40) by an integration by parts:

T 0 1 -1 ∂ x g ∂ τ φ N -∂ τ g ∂ x φ N (τ, x)d xd τ = - T 0 1 -1 g ∂ x ∂ τ φ N -g ∂ x ∂ τ φ N (τ, x)d xd τ + 1 -1 g (0, x)∂ x φ N (0, x)d x → N →+∞ 1 -1 g (0, x)φ x 0 (x)d x.
Before proving (39), let us introduce the operators:

D - N : f (x) → N f (x) -f x - 1 N D + N : f (x) → N f (x) -f x + 1 N ,
which are adjoint of each other. We integrate by parts:

T 0 1 -1 ∂ x g ∂ u W x, ∂ x φ N -∂ τ g ∂ τ φ N (τ, x)d xd τ = T 0 1 -1 ∂ x g ∂ u W x, ∂ x φ N + g ∂ 2 τ φ N (τ, x)d xd τ + 1 -1 g (0, x)∂ τ φ N (0, x)d x. ( 41
)
It is clear that:

1

-1 g (0, x)∂ τ φ N (0, x)d x → N →+∞ 1 -1 g (0, x)φ τ 0 (x)d x. ( 42
)
We focus on the other integrals. By definition, if x ∈ [-1 + 1/N , 1 -1/N ]:

∂ 2 τ φ N (τ, x) = N d 2 d t 2 1 -θ N (x) X k N (x) + θ N (x)X k N (x)+1 (t ) = 1 -θ N (x) D - N A k N (x) N U k N (x) (t ) + B k N (x) N + θ N (x)D - N A k N (x) + 1 N U k N (x)+1 (t ) + B k N (x) + 1 N = 1 -θ N (x) D - N A k N (x) N ∂ x φ N (τ, x) + B k N (x) N + θ N (x)D - N A k N (x) + 1 N ∂ x φ N (τ, x + 1/N ) + B k N (x) + 1 N .
Remark that D ± N and θ N commute, in the sense that:

D ± N θ N (x) f (x) = θ N (x)D ± N f (x) .
Hence:

∂ 2 τ φ N (τ, x) = D - N 1 -θ N (x) A k N (x) N ∂ x φ N (τ, x) + B k N (x) N + θ N (x) A k N (x) + 1 N ∂ x φ N (τ, x + 1/N ) + B k N (x) + 1 N , if |x| < 1-1/N .
We assume that N is sufficiently large, so that Supp(g ) ⊂ [-1+2/N , 1-2/N ]. Then:

T 0 1 -1 ∂ 2 τ φ N (τ, x)g (τ, x)d xd τ = T 0 1 -1 g (τ, x)D - N 1 -θ N (x) A k N (x) N ∂ x φ N (τ, x) + B k N (x) N + θ N (x) A k N (x) + 1 N ∂ x φ N (τ, x + 1/N ) + B k N (x) + 1 N d xd τ.
As D - N and D + N are adjoint of each other:

T 0 1 -1 ∂ 2 τ φ N (τ, x)g (τ, x)d xd τ = T 0 1 -1 D + N g (τ, x) 1 -θ N (x) A k N (x) N ∂ x φ N (τ, x) + B k N (x) N + θ N (x) A k N (x + 1/N ) N ∂ x φ N (τ, x + 1/N ) + B k N (x + 1/N ) N d xd τ = T 0 1 -1 1 -θ N (x) D + N g (τ, x) + θ N (x)D + N g (τ, x -1/N ) A k N (x) N ∂ x φ N (τ, x) + B k N (x) N d xd τ.
Now, since A, B and g are regular:

D + N g (τ, x) = -∂ x g (τ, x) + h N g (τ, x) N , A k N (x) N = A(x) + h N A (x) N , B k N (x) N = B (x) + h N B (x) N ,
where: sup

N sup τ,x h N g (τ, x) + h N A (τ, x) + h N B (τ, x) < +∞.
Therefore:

T 0 1 -1 ∂ 2 τ φ N (τ, x)g (τ, x)d xd τ = - T 0 1 -1 ∂ x g (τ, x) A(x)∂ x φ N (τ, x) + B (x) d xd τ +C N , ( 43
)
where, by the Cauchy-Schwarz inequality:

C N ≤ C N   1 + T 0 1 -1 ∂ x φ N (τ, x) 2 d xd τ   . ( 44
)
Using Lemma 4.1 and (28), we get that:

1

-1 ∂ x φ N (τ, x) 2 d x ≤ C . (45) 20
Whence, from (43), ( 44) and (45) we deduce:

T 0 1 -1 ∂ 2 τ φ N g + ∂ x g ∂ u W (x, ∂ x φ N (τ, x)d xd τ → N →+∞ 0. ( 46
)
From ( 41), ( 42) and ( 46), we obtain (39).

We are now able to prove Theorem 3.1.

Proof. Using Lemma 4.1 and ( 28), we estimate:

N -1 k=-N A k N U 2 k + V 2 k (t ) (28) ≤ C +C N -1 k=-N A k N (U k ) 2 + B k N U k + (V k ) 2 (t ) ≤C 1 + Ẽ N D (t ) ≤C 1 + Ẽ N D (0) .
Since φ x 0 and φ τ 0 are sufficiently regular, we have:

E D (0) → N →∞ 1 -1 W x, φ x 0 (x) d x + 1 2 1 -1 φ τ 0 (x) 2 d x.
We therefore obtain that, for all τ > 0:

1

-1 A k N (x) N ∂ x φ N (τ, x) 2 + ∂ τ φ N (τ, x) 2 d x ≤ C .
And by smoothness of A and thanks to the fact that A ≥ α > 0, we obtain:

1 -1 A(x) ∂ x φ N (τ, x) 2 + ∂ τ φ N (τ, x) 2 d x ≤ C . ( 47
)
We take the following scalar product on Ḣ 1 τ,x :

〈g , h〉 H 1 τ,x := T 0 1 -1 A(x)∂ x g ∂ x h + ∂ τ g ∂ τ h (τ, x)d xd τ.
This scalar product induces a norm which is equivalent to the classical one on Ḣ 1 τ,x , as A ≥ α > 0 and A is bounded. We denote H 1 τ,x for Ḣ 1 τ,x endowed with this scalar product, respectively H 1 τ,x for H 1 τ,x endowed with the scalar product:

〈g , h〉 H 1 τ,x = 〈g , h〉 H 1 τ,x + T 0 1 -1 g (τ, x)h(τ, x)d xd τ.
From (47) and the fact that φ N (-1) = φ l , we get by the Poincaré inequality that φ N is bounded in H 1 τ,x . By weak compactness of this space, we extract:

φ N φ ∞ in H 1 τ,x . ( 48
) Lemma 4.2 implies that, for all g ∈ C ∞ c ([0, T [, ] -1, 1[): T 0 1 -1 ∂ x g ∂ u W x, ∂ x φ ∞ -∂ τ g ∂ τ φ ∞ (τ, x)d xd τ = 1 -1 g (0, x)φ τ 0 (x)d x, T 0 1 -1 ∂ x g ∂ τ φ ∞ -∂ τ g ∂ x φ ∞ (τ, x)d xd τ = 1 -1 g (0, x)φ x 0 (x)d x.
Therefore, φ ∞ is φ, the unique solution of (4) for the initial and boundary conditions ( 5) and ( 6) given by Theorem 2.1.

We now prove that this convergence is strong. As Ẽ N D is preserved, we have:

φ N 2 H 1 τ,x =2T Ẽ N D (0) 2 + 2 T 0 1 -1 A k N (x) N -A(x) ∂ x φ N 2 -B k N (x) N ∂ x φ N (τ, x)d x
Yet, as A and B are C 1 , we have:

lim N →+∞ sup x∈[-1,1] A k N (x) N -A(x) = 0, B k N (x) N → B (x) in L 2 x .
Moreover:

Ẽ N D (0) → N →∞ 1 -1 A(x) 2 φ x 0 (x) 2 + B (x)φ x 0 (x) + 1 2 φ τ 0 (x) 2 d x.
Therefore, we have the following convergence:

φ N 2 H 1 τ,x → N →∞ 2T 1 -1 A(x) 2 φ x 0 (x) 2 + B (x)φ x 0 (x) + 1 2 φ τ 0 (x) 2 d x -2 T 0 1 -1 B (x) ∂ x φ (τ, x)d x. ( 49
)
But, from Theorem 2.1, the continuous energy E C is also preserved. This implies:

2T 1 -1 A(x) 2 φ x 0 (x) 2 + B (x)φ x 0 (x) + 1 2 φ τ 0 (x) 2 d x -2 T 0 1 -1 B (x) ∂ x φ (τ, x)d x = T 0 1 -1 A(x) 2 ∂ x φ 2 + 1 2 ∂ τ φ 2 (τ, x)d x. ( 50
)
From ( 49) and (50), we obtain:

φ N H 1 τ,x → N →∞ φ H 1 τ,x
.

Whence φ N strongly converges to φ in H 1 τ,x (and as a consequence, in H 1 τ,x ).

The non-linear case

This section is devoted to the proof of Theorem 3.3.

Remark 12 (Boundedness of ∂ x φ). Under the hypotheses of Theorem 3.3, if ∂ x φ does not belong to [a, b] on a non-zero measure set, then ∂ x φ N cannot converge weakly to ∂ x φ. Therefore, we henceforth assume that

∂ x φ(τ, x) ∈ [a, b], ∀x ∈ [-1, 1], ∀τ < T 0 .
The latter assumption holds for some a and b related to the initial conditions φ 0 x , φ 0 τ ∈ C 1 p,x .

Light cone

The system (1) has the property that perturbations propagate at a finite speed on the macroscopic level. This is stated in Theorem 3.4, but before proving it, we have to derive a Grönwall-type estimate:

Proposition 5.1. Under the hypotheses of Theorem 3.4, if, for fixed N :

M = U 0 (.) -Ũ0 (.) L ∞ t < +∞, (51)
then we have, for all j ∈ [[1, N -1]], t ∈ R + : U j (t ) -Ũ j (t ) ≤ M 2t K 2 j (2 j )! exp 2t K , ( 52
) V j (t ) -Ṽj (t ) ≤ M 2K 2t K 2 j +1 (2 j + 1)! exp 2t K + 2t K 2 j -1 (2 j -1)! exp 2t K . ( 53
)
Proof. Remark first that it is straightforward to get (53) from (52) ((52) also holds for j = 0) by integrating (1). Indeed, using (1), we have, for j ∈ [[1, N -2]]:

V j (t ) -Ṽj (t ) ≤ t 0 W U j (s) -W Ũ j (s) + W Ũ j -1 (s) -W U j -1 (s) d s (34) ≤ K t 0 Ũ j (s) -U j (s) + Ũ j -1 (s) -U j -1 (s) d s (52) ≤ 2M K t 2t K 2 j (2 j + 1)! exp 2t K + t 2t K 2( j -1)
(2 j -1)! exp 2t K .

We now prove the estimate (52). We do it by induction on j in the expression:

S j (t ) := max k∈[[ j ,N -1]] U k (t ) -Ũk (t ) .
Using (1), we get, for j ≥ 1:

U j (t ) -Ũ j (t ) ≤ t 0 V j +1 (s) -Ṽj+1 (s) + Ṽj (s) -V j (s) d s ≤K t 0 s 0 U j +1 -Ũ j +1 + 2 U j -Ũ j + U j -1 -Ũ j -1 (r )d r d s, ≤4K t 0 s 0 S j -1 (r )d r d s.
Thus:

S j (t ) ≤ 4K t 0 s 0 S j -1 (r )d r d s. ( 54
)
From hypothesis (34), we have that S 0 (0) ≤ M . Using the same argument as above, we get for j = 0:

S 0 (t ) ≤ M + 4K t 0 s 0 S 0 (r )d r d s.
Using the Grönwall Lemma, we obtain:

S 0 (t ) ≤ M exp 2t K . (55)
Therefore, we obtain from (54) that:

S j (t ) ≤(4K ) j t =t j 0 s j 0 t j -1 0 s j -1 0 ... t 1 0 s 1 0 S 0 (r )d r d s 1 d t 1 ...d s j -1 d t j -1 d s j d t j ≤(4K ) j t 0 (t -r ) 2 j -1 (2 j -1)! S 0 (r )d r ≤M (4K ) j t 2 j (2 j )! exp 2t K ,
which concludes the proof.

We are now able to prove Theorem 3.4:

Proof. Let j > N x, t ∈ [0, N τ[.
By strong convexity of W and by the Cauchy-Schwarz inequality, (33) implies:

M N := sup t ∈R + U N 0 (t ) ≤ C N .
By Proposition 5.1, we have:

U j (t ) -Ũ j (t ) ≤ M N 2t K 2 j (2 j )! exp 2t K .
Using logarithm and Stirling formula, we obtain:

ln N U j (t ) -Ũ j (t ) = ln(M N ) + 2 j ln 2t K -ln((2 j )!) + 2t K + ln(N ) ≤C (1 + ln(N )) + 2 j ln 2et K 2 j + 2t K , ≤C (1 + ln(N )) + 2 j ln exp(2)t K j + (2t K -2 j ) ≤C (1 + ln(N )) + 2 j ln c N τ N x + 2 c exp(2) N τ -N x ≤C (1 + ln(N )) + 2N x ln cτ x + c exp(2) τ -x .
As cτ < x, we obtain that:

lim N →+∞ sup 0≤t <N τ max j >N x ln N U N j (t ) -Ũ N j (t ) = -∞,
which gives (35). The same method applies for (36).

Strengthened convergence

In this section, we prove that if φ N converges weakly to φ in H 1 τ,x , then convergence holds in a strong sense for

∂ x φ N to ∂ x φ in L 2 τ,x .
Lemma 5.2. Under the hypotheses of Theorem 3.3, if T is DC -compatible, then the following implication is true:

φ N φ in H 1 τ,x =⇒ ∂ x φ N → ∂ x φ in L 2 τ,x .
To get it, we first prove two parallel integral identities, for the continuous system (4) and the discrete one (1). Lemma 5.3. Assume that W ∈ C 2 (R) satisfies [START_REF] Dafermos | Estimates for conservation laws with little viscosity[END_REF]. Let φ ∈ W 1,∞ τ,x be a weak entropy solution of (4) the initial and boundary conditions (5) and [START_REF] Blanc | From the Newton equation to the wave equation in some simple cases[END_REF]. Then, if T is C -compatible, we have:

T 0 1 -1 (1 -x) ∂ τ φ(τ, x) -φ τ 0 (x) d xd τ = T 0 1 -1 (T -τ) W ∂ x φ(τ, x) -W φ x 0 (-1) d xd τ. ( 56
)
A similar identity can be derived for the discrete system: Lemma 5.4. Let W ∈ C 1 (R) satisfy [START_REF] Dafermos | Estimates for conservation laws with little viscosity[END_REF] and [START_REF] Deift | A continuum limit of the Toda lattice[END_REF]. Let X j (t ) be the solution of (1) for the initial and boundary conditions (2) and (3). Suppose that T is D-compatible. Assume that:

∂ τ φ N ξ ∞ in L 2 τ,x , (57)
∂ x φ N ψ ∞ in L 2 τ,x . (58)
Assume that ∂ x φ N is uniformly bounded in τ, x, N . Let ν τ,x be the Young measure associated with the convergence of ∂ x φ N . Then, we have:

T 0 1 -1 (1 -x) ξ ∞ (τ, x) -φ τ 0 (x) d xd τ = T 0 1 -1 (T -τ) R W (λ)d ν τ,x (λ) -W (φ x 0 (-1)) d τd x. ( 59
)
See Theorem 3.1 p 31 of [START_REF] Müller | Variational models for microstructure and phase transitions[END_REF] for the definition of Young measures.

We temporarily admit Lemma 5.3 and 5.4 and proceed with the proof of Lemma 5.2.

Proof. Using Lemmata 5.3 and 5.4, we get that, if ν τ,x characterizes the weak convergence of ∂ x φ N to ∂ x φ:

T 0 1 -1 (T -τ) W ∂ x φ(τ, x) - R W (λ)d ν τ,x (λ) d xd τ = 0.
But, as W is strongly convex, using Theorem 1.1.8 p 47 of [START_REF] Niculescu | Convex functions and their applications: a contemporary approach[END_REF], we obtain:

ν τ,x = δ ∂ x φ(τ,x) .
Then, Corollary 3.2 p 34 of [START_REF] Müller | Variational models for microstructure and phase transitions[END_REF] implies:

∂ x φ N → ∂ x φ in L 2 τ,x .
Next, we prove Lemma 5.3:

Proof. As T is C -compatible, we can take the following g (τ, x) = (T -τ)(1x) in [START_REF] Holian | Shock waves in the Toda lattice: Analysis[END_REF]. Thus, we have:

T 0 1 -1 -(T -τ)W ∂ x φ(τ, x) + (1 -x)∂ τ φ(τ, x) d xd τ = T 1 -1 (1 -x)φ τ 0 (x)d x -2 T 0 W φ x 0 (-1) (T -τ) d τ.
This is equivalent to (56).

We now prove Lemma 5.4:

Proof. We sum (1) and get:

k j =-N d 2 d t 2 X j (t ) = k j =-N +1 W (U j )(t ) -W (U j -1 )(t ) = W (U k )(t ) -W (U -N )(t ).
Then we multiply the above expression by (N Tt ) and integrate with respect to t :

N T 0 (N T -t ) W (U k ) -W (U -N ) (t )d t = N T 0 t 0 k i =-N d 2 d t 2 X j (s)d sd t = k j =-N N T 0 d X j d t (N τ) - d d t X j (0) d t .
If we rescale it and sum over k ∈ [[-N , N -1]], we obtain:

1 N N -1 k=-N T 0 (T -τ) W ∂ x φ N τ, k N -W (U -N (N τ)) d τ = 1 N 2 N -1 k=-N k j =-N T 0 ξ N τ, k N -ξ N 0, k N d τ = 1 N N -1 k=-N (N -k) N T 0 ξ N τ, k N -ξ N 0, k N d τ = T 0 1 -1 1 - N x N ξ N (τ, x) -ξ N (0, x) d τ.
Remark first that:

1 - N x N → 1 -x in L ∞ x .
As T is D-compatible, we have the following convergence, uniform for τ ∈ [0, T ]: 1) . From initial conditions (2), we get that:

W (U -N (N τ)) → W φ x 0 (-
ξ N (0, .) → φ τ 0 (.) in L 2
x , and from the hypotheses (57) and (58) that:

T 0 1 -1 (1 -x) ξ ∞ (τ, x) -φ τ 0 (x) d xd τ = T 0 1 -1 (T -τ) R W (λ)d ν τ,x (λ) -W (φ x 0 (-1)) d τd x.

From strong convergence of ∂ x φ N to strong convergence of ∂ τ φ N

We now prove that the strong convergence of ∂ x φ N in L 2 τ,x implies the strong convergence of ∂ τ φ N in L 2 τ,x .

Lemma 5.5. Under the hypotheses of Theorem 3.3, if T is DC -compatible, the following implication is true:

   φ N → φ in L 2 τ,x ∂ x φ N → ∂ x φ in L 2 τ,x ∂ τ φ N ∂ τ φ in L 2 τ,x =⇒ ∂ τ φ N → ∂ τ φ in L 2 τ,x ξ N → ∂ τ φ in L 2 τ,x , for ξ N (τ, x) = V N k N (x) (N τ).
We first prove that it is enough to have

∂ τ φ N → ∂ τ φ. Lemma 5.6. Assume that ∂ τ φ N → ζ in L 2 τ,x . Then ξ N → ζ in L 2 τ,x . Proof. Recall that ζ N = ∂ τ φ N . Assume that ζ N → ζ.
We suppose first that:

ζ(τ, x) = N -1 k=-N ζ k (τ)1 k N , k+1 N (x). (60)
Then, for all j ∈ [[-N , N -1]], τ ∈ [0, T ]:

( j +1)/N j /N ζ N -ζ 2 (τ, x)d x = 1 N 1 0 θ(ξ N (τ, j /N ) -ζ(τ, j /N )) + (1 -θ)(ξ N (τ, ( j + 1)/N ) -ζ(τ, j /N )) 2 d θ = 1 N 1 3 ξ N (τ, j /N ) -ζ(τ, j /N ) 2 + 1 3 ξ N (τ, ( j + 1)/N ) -ζ(τ, j /N ) 2 + 1 3 ξ N (τ, j /N ) -ζ(τ, j /N ) ξ N (τ, ( j + 1)/N ) -ζ(τ, j /N ) ≥ 1 6N ξ N (τ, j /N ) -ζ(τ, j /N ) 2 ≥ 1 6 ( j +1)/N j /N ξ N (τ, x) -ζ(τ, x) 2 d x.
Therefore, we have:

T 0 1 -1 ζ N -ζ 2 (τ, x)d xd τ ≥ 1 6 T 0 1 -1 ξ N -ζ 2 (τ, x)d xd τ. Since ∂ x φ N → ∂ x φ in L 2 τ,
x , and since ∂ x φ N is bounded in L ∞ , we have:

W ∂ x φ N → W ∂ x φ in L 2 τ,x .
We claim that it implies:

Ψ N -W ∂ x φ L 2 τ,x → N →+∞ 0. ( 62
)
Indeed, as ∂ x φ N is bounded in L ∞ τ,x uniformly in N , it suffices to bound as follows:

1

-1 Ψ N (τ, x) -W ∂ x φ(τ, x) d x ≤ 1 -1 N -1 j =-N Ξ N j -1 (x) W U j (N τ) -W ∂ x φ(τ, x) d x ≤ 1 -1 N -1 j =-N Ξ N j -1 (x) W U j (N τ) -W ∂ x φ(τ, x) d x ≤ 1 -1 N -1 j =-N 1 j -1 N , j +2 N (x) W U j (N τ) -W ∂ x φ(τ, x) d x ≤ 1 j =-1 1 -1 W U k(x)+ j (N τ) -W ∂ x φ(τ, x) d x ≤ 1 j =-1 1 -1 W ∂ x φ N τ, x + j N -W ∂ x φ (τ, x) d x → 0.
Interpolating with L ∞ τ,x , this gives (62). We define:

α N := ζ N -ζ, β N := Ψ -W ∂ x φ , γ N := ∂ x φ N -∂ x φ.
Remark that, by definition, we have:

∂ τ α N = ∂ x β N , (63)
∂ x α N = ∂ τ γ N . ( 64
)
D-compatibility of T implies the following convergences:

T 0 α N (τ, -1) 2 → 0, T 0 α N (τ, 1) 2 d τ → 0, 1 -1 α N (0, x) 2 d x → 0. ( 65
)
From (62), and by D-compatibility, we deduce that:

lim N →+∞ β N L 2 τ,x = 0, lim N →+∞ β N (., 1) L 2 τ + β N (., -1) L 2 τ = 0. ( 66
)
The energy estimates for (1) and ( 4) give:

sup

N γ N L 2 τ,x ≤ C , sup N γ N (0, .) L 2 x + γ N (T, .) L 2 x ≤ C . ( 67
)
We now claim that (63), ( 64), (66), (67), (65) imply:

lim sup N →+∞ α N L 2 τ,x = 0, (68)
which gives the desired result, thanks to Lemma 5.6. To prove (68), we write:

α N (τ, x) (63) = α N (τ, -1) + x -1 ∂ τ γ N (τ, y)d y. α N (τ, x) (64) = α N (0, x) + τ 0 ∂ x β N (ν, x)d ν.
Therefore:

T 0 1 -1 α N (τ, x) 2 d xd τ = T 0 1 -1 α N (τ, -1) + x -1 ∂ τ γ N (τ, y)d y α N (0, x) + τ 0 ∂ x β N (ν, x)d ν d xd τ = T 0 1 -1 α N (τ, -1)α N (0, x)d xd τ + T 0 1 -1 α N (τ, -1) τ 0 ∂ x β N (ν, x)d νd xd τ + T 0 1 -1 α N (0, x) x -1 ∂ τ γ N (τ, y)d yd xd τ + T 0 1 -1 x -1 ∂ τ γ N (τ, y)d y τ 0 ∂ x β N (ν, x)d νd xd τ =: T N 1 + T N 2 + T N 3 + T N 4 .
We deal separately with T N 1 , T N 2 , T N 3 , T N 4 . By the Cauchy-Schwarz inequality:

T N 1 ≤ T 0 α N (τ, -1)d τ 1/2 1 -1 α N (0, x)d x 1/2 (65) → 0. (69)
Integrating over x in T N 2 , we obtain:

T N 2 = T 0 α N (τ, -1) τ 0 β N (ν, 1) -β N (ν, -1) d νd τ (66),( 65
) → 0. (70)
Next, integrating over τ in T N 3 , we get:

T N 3 = 1 -1 α N (0, x) x -1 γ N (T, y) -γ N (0, y) d yd x (67),( 65
) → 0. (71)
We deal with T N 4 by a double integration by parts:

T N 4 = T 0 1 -1 ∂ τ γ N (τ, y)d y τ 0 β N (ν, 1)d νd τ - T 0 1 -1 ∂ τ γ N (τ, x) τ 0 β N (ν, x)d νd xd τ = 1 -1 γ N (T, y)d y T 0 β N (τ, 1)d τ - T 0 1 -1 γ N (τ, x)β N (τ, 1)d xd τ - T 0 1 -1 β N (τ, x)γ N (T, x)d τd x T 0 1 -1 β N (τ, x)γ N (τ, x)d τd x. (66),( 67
) → 0. (72)
From ( 69), ( 70), ( 71), (72), we obtain (68), which concludes the proof.

Proof of Theorem 3.3

We are now in position to prove Theorem 3.3.

Proof. We first prove the existence of a D-compatible T > 0. φ x 0 and φ τ 0 satisfies [START_REF] Niculescu | Convex functions and their applications: a contemporary approach[END_REF], and :

W (U j (t )) L ∞ t (l ∞ j ) ≤ W L ∞ ([a,b]
) . Moreover, the discrete energy is preserved, which implies that:

E D (t ) ≤ N j =-N +1 1 2 φ τ 0 j N 2 + W φ x 0 j N ≤N φ τ 2 L ∞ x + 2N W L ∞ ([a,b]) .
Therefore, we can apply Theorem 3.4. Let X j (t ) satisfy (1) with initial conditions Ũ j (t = 0) = u r , Ṽj (t = 0) = v r and Dirichlet boundary conditions (remark that this means that U j (t ) and V j (t ) do not depend on time). We compare Ũ j (t ) = u r and Ṽj (t ) = v r with U j (t ) and V j (t ), respectively. Using Theorem 3.4, there exists c > 0 such that, if T < 1/(4c) and T < T 0 :

sup t ∈[0,N T ] max j ∈[3N /4,N ] N U j (t ) -u r → N →+∞ 0, sup t ∈[0,N T ] max j ∈[3N /4,N ] N V j (t ) -v r → N →+∞ 0.
The same argument applies for j < -3N /4. Therefore, there exists a D-compatible T > 0.

We now prove that φ N does not converge to φ. We argue by contradiction and assume that:

φ N → φ in D τ,x . (73)
Since the discrete energy E D is preserved and as W is strongly convex, we get an H 1 estimate over φ N :

1

-1 ∂ x φ(τ, x) 2 + ∂ τ φ(τ, x) 2 ≤ C .
which directly implies:

φ N φ in H 1 τ,x . (74)
By Lemma 5.2, we get that:

∂ x φ N → ∂ x φ in L 2 τ,x . (75)
Whence, by Lemme 5.5, we have:

ξ N → ∂ τ φ in L 2 τ,x . (76)
W is continuous. Therefore (75), ( 31) and (76) imply:

T 0 1 -1 1 2 ∂ τ φ N 2 + W ∂ x φ N (τ, x)d xd τ → T 0 1 -1 1 2 ∂ τ φ 2 + W ∂ x φ (τ, x)d xd τ. ( 77
)
But the left-hand term of (77) also converges, by discrete energy conservation, to:

T 0 1 -1 1 2 ∂ τ φ N 2 + W ∂ x φ N (τ, x)d xd τ → T 0 1 -1 1 2 φ τ 0 2 + W φ x 0 (τ, x)d xd τ = T E C (0),
and the right-hand term of (77) is nothing but the energy E C (τ). As φ is an entropy solution, we have:

T 0 E C (τ)d τ = T 1 0 E C (τ)d τ + T T 1 E C (τ)d τ (32)
< T E C (0).

Therefore, we reach a contradiction, and φ N cannot converge to φ.

A uniform bound on the distance between particles

Notice that it is important to assume some regularity on the initial conditions in Conjecture 3.5. It is indeed possible to build some initial conditions that are small in l ∞ j such that the associated solutions of (1) are not bounded uniformly in N at a fixed macroscopic time τ > 0. The following proof of Proposition 3.6 uses the reversibility of equation ( 1) and also a linearization of eigenvalues λ k . We first derive explicit formulae for solution of linear periodic system (1). Let I ∈ M 2N (R) be the identity matrix, and J ∈ M 2N (R) the circular permutation:

J j k := δ j k+1 .
When the potential is quadratic and satisfies [START_REF] Dafermos | Energy methods for quasilinear hyperbolic initial-boundary value problems[END_REF], system (1) with periodic boundary conditions is equivalent to:

d 2 X d t 2 + 2I -J -J -1 X = 0.
We diagonalize this system using its eigenvectors:

Ω j = 1 2N      1 ω j .. ω 2N -1 j     
, where ω j = exp i j π N . The associated eigenvalues are:

λ j = 2 1 -cos j π N = 4 sin 2 j π 2N . ( 78
)
Thus, a solution of (1) with periodic boundary conditions satisfies:

X j (t ) = 2N -1 k=1 cos t λ k (Ω k |X (0)) (Ω k ) j + 2N -1 k=1 1 λ k sin t λ k (Ω k |V (0)) (Ω k ) j + 1 2N [(Ω 0 |X (0)) + (Ω 0 |V (0)) t ] , (79)
where, for two vectors Y , Z ∈ C 2N , (|) denotes the hermitian product:

(Y |Z ) := N -1 k=-N Y k Z * k .
One easily derives such formulae for V j , U j and Z j by linearity.

We can now prove Proposition 3.6.

Proof of Proposition 3.6. Using the reversibility of (1), it is enough to prove that, if X N j is a solution of (1) with periodic boundary condition such that U N j (t = 0) = δ 0 j and V N j (t = 0) = 0, then:

U N j (N τ) l ∞ j → 0, ( 80
)
V N j (N τ) l ∞ j → 0. (81)
Indeed, let Xi be the solution of (1) with periodic boundary conditions and the following initial conditions:

Ũ N j (t = 0) := K N U N j (N τ), Ṽ N j (t = 0) := -K N V N j (N τ).

By linearity and reversibility of (1), we get:

Ũ N j (N τ) = K N U N j (0), Ṽ N j (N τ) = -K N V N j (0). Setting K N = max U N j (N τ) l ∞ j , V N j (N τ) l ∞ j -1
gives the desired result.

We only show (80), as the proof of (81) is similar. Thanks to (79):

U N k (N τ) =

2N

N -1 j =-N cos 2N τ sin j π 2N e i j kπ/N .

To simplify the proof, we suppose N = nm (it can be generalized with a few technicalities). Thus:

U N k (N τ) = 1 2N n-1 l =-n m-1 j =0
cos 2N τ sin (l m + j )π 2N e i (l m+ j )kπ/N .

Let us bound terms of the type:

Q l ,k ± := 1 N m-1 j =0
exp ±i 2N τ sin (l m + j )π 2N e i (l m+ j )kπ/N . Without loss of generality, we focus only on Q l ,k + . There exist at most two solutions s 1 and s 2 ∈ [-π/2, π/2] to the equation:

τπ cos(s) + kπ N = 0. Let 1 > δ > 0. If l mπ 2N ∉]s 1 -δ, s 1 + δ[∪]s 2 -δ, s 2 + δ[, then: 1 -exp(i γ + (k, l , m, N , τ)) > C δ 2 . ( 82
)
for some universal constant C . Whence, if l mπ 2N ∉]s 1 -δ, s 1 +δ[∪]s 2 -δ, s 2 +δ[, we have:

Q l ,k + ≤C m 3 N 2 + 1 N C δ 2 .
Moreover, it is immediate from the definition of Q l ,k ± that for all l , k:

Q l ,k ± ≤ m N .
We denote:

E := [[-n, n -1]] ∩ 2N mπ (s 1 -δ), 2N mπ (s 1 + δ) ∪ 2N mπ (s 2 -δ), 2N mπ (s 2 + δ) ,
and bound the sum:

n-1 l =-n Q l ,k + = l ∈E Q l ,k + + l ∉E Q l ,k + ≤ l ∈E m N +C l ∉E m 3 N 2 + 1 N δ 2 ≤ 8N mπ δ m N + 2C n m 3 N 2 + 1 N δ 2 ≤C δ + m 2 N + 1 mδ 2 .
We can now prove Proposition 3.7:

Proof. If c = 0, Lemma 7.1 gives the desired result. Suppose c = 0. Using Fourier transform on [START_REF] Serre | Discrete shock profiles: Existence and stability[END_REF] implies:

c 2 ξ 2 F (φ)(ξ) = exp(i ξ) -2 + exp(-i ξ) F (φ)(ξ).
The equation:

c 2 ξ 2 = 2 (1 -cos(ξ)) , (84)
has a finite number of solutions ξ j , j ∈ J . Therefore, there exist K j ∈ N, a j k ∈ R such that:

F (φ) = J j =1 K j k=0 a j k δ (k) ξ j
.

Thus:

φ(x) = J j =1 K j k=0
a j k (i x) k exp(i x ξ j ).

Since φ has a limit for x → +∞ then a j k = 0 if j = 0 or k > 1. Thus, there exists no discrete shock wave with c = 0.

Convex non-linear potential

We now prove Proposition 3.8.

Proof. Suppose u l = u r . Test now [START_REF] Serre | Discrete shock profiles: Existence and stability[END_REF] with φ :

Figure 1 :

 1 Figure 1: Comparison between ∂ x φ N (black curve) and ∂ x φ (red curve) for Riemann shock initial conditions. W (u) = u 2 2 , N = 10000, τ = 0.3.

Figure 2 : 6 . 3 . 3 .

 2633 Figure 2: Comparison between ∂ x φ N (black curve) and ∂ x φ (red curve) for Riemann shock initial conditions, and a non-linear potential. N = 5000, τ = 0.075, W (u) = u 6 6 .

Figure 4 :

 4 Figure 4: Successive pictures of U j (N τ). N = 1000, τ = 0, 0.08, ..., 0.4

2 N 2 r 1 N 1 N 2 1

 22112 (N , m, l , j ), where r (N , m, l , j ) < C , independently of N , m, l , j . As a consequence: m, l , j ) e i j kπ/N ≤ exp(i γ ± (k, l , m, N , τ)),where:γ ± (k, l , m, N , τ) = kπ N ± τπ cos l mπ 2N .

WWR R- 1 φ 1 φ 2 W

 112 (φ(x + 1) -φ(x)) -W (φ(x) -φ(x -1)) φ (x)d x (φ(x + 1) -φ(x)) φ (x + 1) -φ (x) d x + (x + 1)W (φ(x + 1) -φ(x))d x -(x + 1)W (φ(x + 1) -φ(x))d x =W (u r ) -W (u l ) + W (u r )u r -W (u l )u l .Using (83), we get:1 (u r ) -W (u l ) = W (u r ) -W (u l ) u ru l . (85)

Acknowledgement

We wish to thank Claude Le Bris and Frédéric Legoll for their help and Gabriel Stoltz and Frédéric Lagoutière, for fruitful discussions.

If ζ is now only L 2 , we approximate it by ζ that has the form (60). Thus, by the triangle inequality, and applying the former result on ζ, we have:

Hence:

We can now proceed with the proof of Lemma 5.5.

Proof. By definition:

Using (1), we obtain:

We define:

Let Ξ N j (x) := Ξ(N xj ). We have:

Newt, we define:

We have:

Let δ = (m) -1/3 , m = N 3/7 . We get:

Doing the same manipulations on Q l ,k -, we get that, for all k ∈ [[-N , N -1]]:

whence (80).

Remark 13. One can remove the technical assumption N = nm with m, n ∈ N, by fixing µ := N 1/7 , and then m := µ 3 , n := µ 4 . Remarking that N -µ 7 < C N 6/7 , we can apply the same proof as above and derive the same estimates.

Non-existence of discrete shock waves

We prove in this section that there do not exist discrete shock waves. We use some ideas from [START_REF] Benzoni-Gavage | Semi-discrete shock profiles for hyperbolic systems of conservation laws[END_REF], where an existence result is proven for upwind schemes.

Quadratic potential

In this section, we prove Proposition 3.7. We first show a lemma which is valid for a wide class of potentials W :

Then there exists no discrete shock wave to the equation (1) with zero speed. That is, there does not exist X j (t ) satisfying Definition 2.5, with associated c = 0.

Proof. Integrating [START_REF] Serre | Discrete shock profiles: Existence and stability[END_REF], we get:

If x → -∞ and y → +∞, we obtain:

which is the Rankine-Hugoniot equation [START_REF] Pachpatte | On discrete inequalities of the Poincaré type[END_REF]. It cannot hold if c = 0.

Yet:

and as W is strictly convex: u rs u ru l W (u l ) + su l u ru l -W (s) > 0, ∀s ∈]u l , u r [. This is contradictory. Therefore, as u l = u r , there does not exist any discrete shock wave of (1).


Dmitry S Chezganov

Evgenii O Vlasov

Maxim M Neradovskiy

Lubov V Gimadeeva

Elisaveta Neradovskaia

Maria A Chuvakova

H Tronche

F Doutre

P Baldi

Marc P De Micheli

E A Neradovskaya

V Ya Shur

Periodic domain patterning by electron beam of proton exchanged waveguides in lithium niobate

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The fact was attributed to ineffective screening of depolarization field in the crystals presenting a surface layer modified by the SPE process. The dependences of the domain sizes as a function of the dose and the distance between irradiated areas have been measured. Finally, we have demonstrated that electron beam patterning using surface resist layer can produce high quality periodical poling after channel waveguide fabrication. Second harmonic generation with normalized nonlinear conversion efficiency up to 48%/(Wcm 2 ) has been achieved in such waveguides.

Periodically poled (PP) nonlinear-optical ferroelectric crystals are widely used for frequency converters [START_REF] Nalwa | Handbook of Advanced Electronic and Photonic Materials and Devices[END_REF][START_REF] Byer | [END_REF] and various photonic devices. 3,4 Lithium niobate (LiNbO3, LN) is one of the most popular materials for domain engineering due to its large electro-optical and nonlineara) Author to whom correspondence should be addressed. Electronic mail: chezganov.dmitry@urfu.ru optical coefficients. 5 The ability to combine periodical poling and low-loss waveguides fabrication makes LN a very attractive material for integrated optics. 6 Applying an external electric field (e-field poling) via electrodes is a standard and well developed technique used to fabricate commercially available periodically poled crystals. [7][8][9] Its main drawback is domain broadening outside the electrode area 7 that prevents reaching short periods (< 5 µm).

It stimulates development of alternative poling techniques such as electron beam (e-beam)

writting. [START_REF] Volk | Lithium Niobate: Defects, Photorefraction and Ferroelectric 11 Switching[END_REF] The first realization of e-beam patterning was reported by P.W. Haycock et al [START_REF] Haycock | [END_REF] , who combined e-beam writing with application of an external field at high temperature.

Later, a standard e-beam lithography process was applied to LN for 1D and 2D domain patterning. [12][13][14][15][16][17][18] Nevertheless, the random growth of obtained domains resulted in components with low conversion efficiencies. 12,13,19 Formation of PPLN by e-beam has been realized in various waveguides, such as He-implanted 20 and Ti-diffused 19,[21][22][23] ones. Recently, it has been reported that covering the irradiated surface by a thin dielectric layer improves the quality of e-beam patterning. [24][25][26] Using this technique, we have produced high quality bulk PPLN in MgO doped LN (MgO:LN). 27,[START_REF] Chezganov | Simulation of spatial distribution of electric field after electron beam irradiation of MgO-doped LiNbO3 covered by resist layer[END_REF] Proton exchange (PE) is another popular technique for waveguides production in LN.

Waveguides obtained by conventional PE tend to exhibit lower electro-optical 29 and nonlinear-optical coefficients 30 as well as instability of the refractive index profile. 31 The annealed proton exchange (APE) process allows fabricating high optical quality waveguides with restored coefficients. 32,33 Green and blue light generation was demonstrated in PPLN APE waveguides with domain structure produced by e-field patterning. [34][35][36] The highest value of the normalized conversion efficiency was obtained in embedded waveguides produced by reverse proton exchange (RPE) waveguides. 37 Soft Proton Exchange (SPE) is an interesting alternative to get, with one fabrication step only, waveguides presenting a surface index increase in the range from 3×10 -2 to 5×10 -2 . 38 Electro-optical operation with surface SPE waveguide is much easily achievable than with RPE waveguides; 39 moreover, higher changes of refractive index resulted in less bending losses 40 and higher nonlinear efficiency.

Therefore, SPE could be considered as the most attractive process for fabricating nonlinear integrated optical circuits. Nevertheless, we have observed that the formation of SPE waveguides in PPLN can lead to the appearance of uncontrolled nanodomains that perturb the initial periodical structure, mainly in the Z + areas, and, therefore, the QPM effect. 41 Attempts to create PPLN by e-field application after SPE have not succeeded due to changes in domain growth resulting in formation of domain rays. [START_REF] Neradovskiy | Investigation of domain kinetics in congruent lithium niobate modified by proton exchange[END_REF] In this paper, we present the first detailed study of domain structure formation by e-beam irradiation in congruent LN (CLN) containing waveguides produced by the SPE process and covered by e-beam resist layer. Finally, we present an attempt to produce a PPLN channel SPE waveguide using this technique and give some information about the nonlinear results that will be presented in detail elsewhere.

Starting from Z-cut 0.5-mm thick CLN wafers produced by Gooch & Housego, we produced planar SPE waveguides on both polar surfaces. The SPE process 38 was carried out by dipping the crystal for 72 hours into a benzoic acid bath diluted by 3.1 weight % of lithium benzoate (BL) and melted at 300°C.

A 100-nm-thick Cu electrode was sputtered on the Z + -polar surface. It was grounded during irradiation. A 2.5-m-thick layer of negative electron-beam resist AZ nlof 2020 Microchemicals was deposited on the Z --polar surface by spin coating.

The domain patterning was made by irradiation of the Z -surface of CLN with a scanning electron microscope (Auriga Crossbeam workstation, Carl Zeiss) with Schottky field emission gun. The exposure parameters and e-beam positioning were controlled by the ebeam lithography system Elphy Multibeam from Raith GmbH. The design was specified by the Raith Nanosuite software. Dot (lattice of dots) and stripe (periodic stripe grating) exposures were used. The e-beam diameter was about 10 nm. The beam position was fixed for dot exposure. The stripes were defined using a raster-scan covering the desired surface.

The sample was irradiated at accelerating voltage of 10 kV and e-beam current (I) of 1.3 nA.

The irradiation doses (the charge in the incident beam) were defined as Dd = I×t for dot exposure and as DS = I×t/A for stripe exposure, where t is the exposure time and A is the stripe area. The domain patterns were visualized by piezoresponse force microscopy (PFM).

The surface relief of developed resist layer was measured by atomic force microscopy (AFM).

The formation of periodic stripe domains structure was carried out to fit the e-beam patterning technique utilizing surface dielectric layer developed by us for periodical poling in bulk MgO:LN crystals with SPE waveguides. 27 Test stripe domain patterns oriented along Y direction with periods (P) ranging from 3 to 10 μm and irradiated stripe width 0.5 μm were formed using doses ranging from 0.1 to 1 mC/cm 2 .

The influence of the dose on the domain patterns at the polar surface visualized by PFM after resist removal was revealed. At low doses (0.1-0.7 mC/cm 2 ) (Fig. 1(a)), submicron domains appear with a normalized switched domain area decreasing from the center to the border of the irradiated stripe (Fig. 1(c)). Increasing the dose enlarges the switched domain area due to domains growth and merging (Fig. 1(b)). The formation of continuous stripe domains was obtained for dose above 0.7 mC/cm [START_REF] Byer | [END_REF] . In all cases, the obtained domains are non-through with their maximum depth up to 150 µm for continuous stripes in the studied dose range. It is necessary to point out that after e-beam irradiation a number of small domains appear in the vicinity of the stripe domain wall at any dose (Fig. 1(c)). We have revealed that the domain stripes width varies linearly with the charge dose (Fig. 2(a)):

𝑑(𝐷 𝑠 ) = 𝑑 𝑚𝑖𝑛 + 𝐵(𝐷 𝑠 -𝐷 𝑠,𝑡ℎ ),

where dmin is the minimal width of the stripe domain, Ds,th is the threshold dose necessary for the formation of continuous stripes domain and B is a constant.

The interaction between neighboring domains in the gratings has been studied for doses larger than 2 mC/cm 2 . It has been shown that the stripe domain width increases with the period up to 7 m and remains almost constant for the larger periods (Fig. 2(b)).

FIG. 2

The stripe domain width as a function of: (a) the dose for 10-m-period grating, (b) the grating period for a dose of 2 mC/cm 2 .

The observed evolution of the domain structure with isolated submicron domains represented the discrete switching caused by ineffective screening of depolarization field in the ferroelectric with surface dielectric layer produced by SPE process. 43 The self-organized domain evolution was demonstrated in LN under highly non-equilibrium switching conditions, caused by completely ineffective screening of depolarization field, which prevented the conventional domain growth by sideways wall motion and stimulated formation of isolated nanodomains. 5,[44][45][46] Domain kinetics in this case is governed by correlated nucleation effect leading to formation of quasi-regular nanodomain structures [47][48][49] .

It was shown that the averaged period of the domain structure was about the thickness of the surface dielectric layer produced by PE process. 45 Thereby, it is possible to distinguish the main stages of the domain stripes formation:

(1) discrete switching by appearance of isolated submicron-scale domains with decreasing of the normalized switched domain area from the center to the border of the irradiated stripe;

(2) growing and merging of isolated domains; (3) formation of stripe domains; (4) further spreading of the stripe domains by sideways wall motion with step generation by merging with isolated micro-and nano-domains appeared in front of the wall. Such mechanism leads to continuous wall motion in contrast to jump-like wall motion caused by determined nucleation observed in the crystals of LN family. 50 It should be noted that presence of the resist layer is mandatory for periodical poling of CLN with SPE planar waveguides. Without this resist layer, it was impossible to create any domain structure by e-beam irradiation. This fact can be attributed to a fast redistribution of the space charges caused by the higher conductivity of the SPE layer.

The growth of isolated domains as a function of dot exposure was studied. Several patterns of 10×10 dots with a 10-m-period were irradiated, with dose increasing from one pattern to the other in the range from 10 to 100 pC. The formation of circular domains on the irradiated polar surface was revealed by PFM (Fig. 3(a)). The absence of domains on the opposite polar surface indicated that the domains were non-through. Increasing the dose did not modify the domain shape. The monotonic nonlinear dependence of domain radius as a function of the deposited charge is similar to that obtained in MgO:LN without PE layer (Fig. 3(c)). 27 Nevertheless, it is necessary to point out that in the PE crystals the domain size variations in each array are much smaller than those observed in the virgin crystals. The circular domain shape is a clear demonstration of isotropic domain growth, which was discussed for stripe domains (Fig. 3(b)). Finally, observation by PFM revealed that the e-beam patterning was nearly not affected by the presence of the waveguide in contrast to what was observed for electric field poling [START_REF] Neradovskiy | Investigation of domain kinetics in congruent lithium niobate modified by proton exchange[END_REF] (Fig. 5). Therefore, this study shows that it is possible to get state-of-the-art SHG efficiency by

FIG. 1

 1 FIG. 1 PFM images of domains appearing after e-beam stripe exposure for different doses: (a) 0.7 mC/cm 2 , (b) 1 mC/cm 2 , (c) dependences of the profile of the switched domain area as a function of the charge dose (the dashed lines indicate the area that was exposed by e-beam irradiation).

FIG. 3

 3 FIG. 3 PFM image of (a) circular domain array and (b) individual circular domain after dot exposure, (c) dose dependence of domain radius.Finally, we have demonstrated the possibility to produce stripe domain structures

Fig. 4 PFM

 4 Fig.4PFM images of (a) periodic domain stripes oriented along X direction, (b) ring-shaped domain.

FIG. 5

 5 FIG. 5 PFM images of PPLN structure (horizontal) fabricated perpendicular to channel waveguide (vertical) produced by SPE method in CLN: darkinitial domain state, lightswitched domains.We have created PPLN structure in SPE channel waveguide with aspect ratio close to

periodic domain patterning after the waveguides fabrication. This is of major interest for producing devices combining nonlinear sections with more complicated circuits. The authors would like to acknowledge V. Kvashnin for participation in the experiments, E. Vaskina, Dr. A.R. Akhmatkhanov, and Dr. D.O. Alikin for technical assistance. The equipment of the Ural Center for Shared Use "Modern nanotechnology" UrFU was used. The research was made possible in part by RFBR (15-32-21102-mol_a_ved), by Government of the Russian Federation (Act 211, Agreement 02.A03.21.0006), and by President of Russian Federation grant for young scientists (Contract 14.Y30.16.8441-МК). V.S. acknowledges financial support within the State Task from the Ministry of Education and Science of the Russian Federation (Project No. 1366.2014/236).