Hassan Ibrahim
email: hassan.ibrahim@ul.edu.lbvivian.rizik@utc.fr

Vivian Rizik

Zaynab Salloum
email: salloum-zaynab@hotmail.com

Ahmad El Hajj
email: ahmad.el-hajj@utc.fr

Global BV solution for a non-local coupled system modeling the dynamics of dislocation densities

Keywords: AMS Classication: 35K50, 35K40, 35K55 Boundary value problems, parabolic systems, comparison principle AMS Classication: 35Q53, 35Q72, 49L25, 35F25, 35L40, 35L45, 65M06, 65M12, 65M15, 74H20, 74H25 Hamilton-Jacobi system, non-local Eikonal system, non-local transport system, BV estimate, discontinuous viscosity solution, dislocation dynamics AMS Classication: 35A01, 74G25, 35F20, 35F21, 70H20, 35Q53 Hamilton-Jacobi system, non-linear Eikonal system, non-linear hyperbolic system, BV estimate, discontinuous viscosity solution, gas dynamics

In this thesis we are interested in the analysis of the dynamics of dislocation densities, where dislocations are crystalline defects appearing at the microscopic scale in metallic alloys. In particular, the study of the Groma-Czikor-Zaiser model (GCZ) and the study of the Groma-Balogh model (GB) are considered. It is actually a system of parabolic type equations for GCZ model and non-linear Hamilton-Jacobi type for GB model. Initially, we demonstrate an existence and uniqueness result of a regular solution using the comparison principle and a xed point argument for the GCZ model. Next, we establish a time-based global existence result for the GB model, based on notions of discontinuous viscosity solutions and a new estimate of total solution variation, as well as nite velocity propagation of the governing equations. This result is extended also to the case of general Hamilton-Jacobi equation systems. Résumé Dans cette thèse on s'intéresse á l'analyse de la dynamique des densités des dislocations, où les dislocations sont des défauts cristallins, apparaissant à l'échelle microscopique dans les alliages métalliques. En particulier, on considère dans un premier temps l'étude du modèle de Groma-Czikor-Zaiser (GCZ) et dans un second temps l'étude du modèle de Groma-Balogh (GB). Il s'agit en réalité d'un système d'équations de type parabolique et de type Hamilton-Jacobi non-linéaires. Au départ, nous démontrons, pour le modèle GCZ, un résultat d'existence et d'unicité xv

Acknowledgment

I am using this opportunity to express my gratitude to everyone who gave me the support during my Ph.D. and the help to bring it to fruition. Throughout these lines, I would like to express my thankfulness to each aspiring guidance, constructive criticism and friendly advice.

First, I would like to acknowledge my indebtedness and render my warmest thanks to my supervisor, Professor Ahmad EL HAJJ, who made this work possible. With no doubt, his friendly guidance and expert advice have been invaluable and carved my way to success. I have been extremely lucky to have a supervisor who cared so much about my work, and who responded to my questions and queries so promptly. Also, I would like to show my gratitude and special appreciation to my supervisor Professor Hassan IBRAHIM for sharing his pearls of wisdom with me during this work. I am so deeply grateful for his help, professionalism and valuable guidance. You have been tremendous mentors for me and this mentorship was paramount in providing a well rounded experience consistent my long-term career goals. I am sincerely grateful to Professors Olivier LEY and Ulrich RAZAFISON for having the extreme kindness of being reporters of this thesis. Their illuminating views, professional commentaries and valuable suggestions contributed greatly to the improvement of the thesis. Also, I am pleased to express my gratitude to the committee members Jean-Paul CHEHAB, Ali WEHBE, Faten JELASSI and Batoul ABDELAZIZ. I am immensely thankful for the chance, the honor and the pride to have them in my jury.

I wish to express my gratitude to Professors Florian DE VUYST and Ali WEHBE, the directors of LMAC and KALMA laboratories, for their continuous support, and to all the laboratory members whose enthusiasm reects a bright light to my work and creates a pleasant working environment. I greatly appreciate the help of Maryline SCHAEFFLEN and Wissam BERRO in every administrative issue. • L p loc (E): Set of functions whose p th power is locally integrable

• W 1,∞ loc (E): Set of locally bounded functions whose rst order derivatives are also locally bounded

• C α,α/2 (E): Set of continuous functions u(x, t) in E whose derivatives of the form D r t D s x u, for 2r + s < α, are also continuous Chapter 1

Introduction

This thesis is involved in the study of two models describing dislocation dynamics. The rst model, called Groma-Czikor-Zaiser model, is a coupling of parabolic equations.

Whereas, the second model, known as Groma-Balogh model, is a coupling of non-linear, non-local Hamilton-Jacobi equations or, in some particular cases, it is a coupling of hyperbolic equations. Upon treating these equations, we use, for Groma-Czikor-Zaiser model, the classical theory of Sobolev and Hölder parabolic equations. Whereas, for Groma-Balogh model, we deal with Hamilton-Jacobi equations and the notion of viscosity solutions arises as a suitable framework.

This introduction is a collection of ve sections. In the rst section, we present a brief physical description of dislocations and gas dynamics. Sections 1.2, 1.3 and 1.4, that will be expanded in Chapters 3, 4 and 5 respectively, are concerned with the mathematical results that we have obtained. The last section 1.5 is a separated result for 1D isentropic gas dynamics.

Focusing on elucidating the crucial ideas and emphasizing on their preservation in order not to be lost in the technic, we settle this introduction as an overview of our essential results. The chapters afterwards come as an accurate and detailed announcement of these results.

Physical motivation

The combination of mathematical science and specialized knowledge generates applied mathematics which, in turn, describes the professional specialty in working on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which became the subject of study in pure mathematics where abstract concepts are studied for their own sake. In our work, aiming to predict, explain or perhaps understand phenomena, we develop problems that are applied on some branches of mechanics as dislocation dynamics and gas dynamics.

Dislocation dynamics

The response of crystalline materials to an external force can be roughly classied as elastic, that is a recoverable change in shape involving bond stretching without any modication in the structure of atoms, or plastic which is permanent caused by the break down of atomic bonds upon exerting a sucient stress. The elastic behavior of crystalline solids has been characterized and understood since the pioneering work of R. Hooke [START_REF] Hooke | De Potentia restitutiva of spring explaining the power of springing bodies[END_REF] in 1678, that is more than three centuries ago, and its theory was developed by V. Volterra [START_REF] Volterra | Sur l'équilibre des corps élastiques multiplement connexes[END_REF] in 1907. However, only since 1934, the theoretical works of E. Orowan [START_REF] Orowan | Kristallplastizitat[END_REF], M. Polanyi [START_REF] Polanyi | Uber eine art gitterstorung die einem kristall plastisch machen konnte[END_REF] and G. I. Taylor [START_REF] Taylor | The mechanism of plastic deformation of crystals[END_REF] linked the plastic behavior of crystalline solids to the existence of linear defects within the crystals. These defects, that are called dislocations, have been directly observed for the rst time in a transmission electron microscope only in 1956 in the independent works of W. Bollmann [START_REF] Bollmann | Interference eects in the electron microscopy of thin crystal foils[END_REF] and P. B. Hirsch, R. W. Horne, M. S. Whelan [START_REF] Hirsch | Direct observations of the arrangement and motion of dislocations in aluminium[END_REF]. They are represented by the black lines appearing in Figure 1.1. From here, the denition of dislocations arises as a crystallographic disorder or irregularity within a crystal structure. A dislocation line consists of a local atomic rearrangement of the crystalline lattice, which is therefore described at the nanometer scale, while dislocation motion can be over distances that are at the micrometer scale. The movement of dislocations aects the strength and toughness of the material and constitutes the fundamental basis for understanding its plastic behavior. We refer the reader to F. R.

N. Nabarro [START_REF] Nabarro | Theory of crystal dislocations[END_REF] and J. R. Hirth and L. Lothe [START_REF] Hirth | Theory of dislocations[END_REF] for a detailed physical presentation.

Dislocations are characterized by two vectors, the line direction, represented by ⃗ ξ, and the Burgers vector ⃗ b [START_REF] Burgers | Geometrical considerations concerning the structural irregularities to be assumed in a crystal[END_REF] describing the direction of propagation of dislocations.

Before the discovery of the dislocation, no one could gure out how the properties of a metal could be greatly changed at the macroscopic level after a slight microscopic disorder. This became even bigger mystery when in the early 1900s scientists estimated that metals undergo plastic deformation at forces much smaller than the theoretical strength of the forces that are holding the metal atoms together. This physical ambiguity was the motivation to explain the created deformation based on the theory of dislocations. For a detailed view, let us consider a perfect crystal where the atoms are well organized (Figure 1.2). The deformation inside this crystal is a consequence of the elimination of a half of a plane from above or below as seen in Figure 1.2.

Hence, the interaction force between atoms causes the nearby planes to bend towards the dislocation. Therefore, the adjacent planes of atoms are not straight. The region in which the defect occurs is the dislocation area. When enough force is applied from one side of the crystal structure, the gap caused by the missing plane passes through the other planes of atoms breaking and joining bonds until reaching the crystal boundary. It was realized that that dislocations exist in many complicated forms. However, these various structures of dislocations in a crystal are generated by two basic types which are edge dislocations and screw dislocations. An edge dislocation is formed after the contraction that occurs in the elimination position. It is a type of line defect in which the direction of propagation of dislocations subjected to stress is perpendicular to the dislocation line formed. However, a screw dislocation results when displacing planes relative to each other through shear stress. It is another type of line defect in which the irregularity occurs when the planes of atoms in the crystal lattice trace a helical path around the dislocation line. In this case, the Burgers vector is parallel to the dislocation line (see Figure 1.3). Throughout this thesis, we were highly interested in two particular models for edge dislocations. The rst model is Groma-Balogh model that treats the movement of straight parallel edge dislocations regardless of the short range dislocation-dislocation interactions that represent a regime in which the dislocation separations are small (for a detailed explanation for this kind of interactions, refer to Chapter 2). Taking into account the periodicity of this phenomena along time, we shed light on what is happening inside the material away from its boundary. The other model is Groma, Czikor and Zaiser model which stands as a generalization to the previous one by focusing on the short range interactions. Here, we were interested in studying the acquisition of edge dislocations on the boundary layer of a crystal that is subjected to an exterior stress and having a nite width.

In addition to our interest in the application of these models on dislocation dynamics in unique and multi-directions of propagation, we present in this thesis a separated result on isentropic gas dynamics.

Gas dynamics

Conservation principles are axioms of mechanics and represent statements that cannot be proved. They provide predictions which are consistent with empirical observations and accurately capture the complicated surface motion that satises the global entropy condition for propagating fronts. In our work, an analysis of the coupling of this level set formulation to a system of conservation laws for compressible gas dynamics is presented. We study a particular system illustrating mass and momentum conservation laws for propagation of gas (see Figure 1.4).

Conservation of mass

This principle states that in a material volume, which is a volume that always encompasses the same uid particles, the mass is constant. The equation essentially says that the net accumulation of mass within a control volume is attributable to the net ux of mass in and out of the control volume. In Gibbs notation, and in 1-dimensional space, it is written in the following form

∂ρ ∂t + ∂ x (ρu) = 0,
where ρ is the density of the uid and u is its speed.

Conservation of linear momenta

This is really Newton's Second Law of Motion which states that the time rate of change of linear momentum of a body equals the sum of the forces acting on it. So, we get ma = ∑ F.

Under certain conditions, we neglect the body forces and surface forces. Thus, the external forces are only presented by the pressure. Consequently, in vector form the above equality is written as follows

∂ t (ρu) + ∂ x (ρu 2 ) = -∂ x P (ρ),
where P is the pressure of the uid. As with the mass equation, the time derivative can be interpreted as the accumulation of linear momenta within a control volume, and the divergence term can be interpreted as the ux of linear momenta into the control volume.

The combination of these two conservation laws gives a system of hyperbolic equations modeling isentropic gas dynamics where the pressure and density are related by Laplace's law for perfect isentropic gases presented by the following γ-relation

P (ρ) = kρ γ ,
with k is a positive constant and γ > 1. For an expanded overview to such kind of systems, we refer the reader to P. L. Lions et al [START_REF] Lions | Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates[END_REF] and R. J. Diperna [START_REF] Diperna | Convergence of approximate solutions to conservation laws[END_REF][START_REF] Diperna | Compensated compactness and general systems of conservation laws[END_REF].

Formal derivation and existence result of an approximate model on dislocation densities

Our interest here lies in studying a one dimensional model developed by Groma, Czikor and Zaiser describing the dynamics of dislocation densities in a bounded crystal. In this work, we focus on the short range dislocation-dislocation interactions neglecting the long range ones (refer to Chapter 2 for a detailed explanation). Previous results [START_REF] Hajj | Dislocation dynamics: from microscopic models to macroscopic crystal plasticity[END_REF][START_REF] Hajj | Derivation and study of dynamical models of dislocation densities[END_REF][START_REF] Ibrahim | Existence and uniqueness for a non-linear parabolic/Hamilton-Jacobi coupled system describing the dynamics of dislocation densities[END_REF][START_REF] Ibrahim | Existence and uniqueness for a dislocation model with short range interactions and varying stress eld[END_REF][START_REF] Ibrahim | Dynamics of dislocation densities in a bounded channel. Part 1. Smooth solutions to a singular coupled parabolic system[END_REF][START_REF] Ibrahim | Dynamics of dislocation densities in a bounded channel. Part 2. Existence of weak solutions to a singular parabolic/Hamilton-Jacobi strongly coupled system[END_REF] were obtained for such models by considering a constant or bounded spacetime dependent exterior stress eld. However, up to our knowledge, our result is the rst to treat the real physical stress eld scaled in [START_REF] Groma | Spatial correlations and higher-order gradient terms in a continuum description of dislocations dynamics[END_REF] as a non-linear term depending on dislocation densities. The positive and negative dislocation densities are represented, respectively, by ρ + x and ρ - x , where ρ + and ρ -are unknown scalars. Their dierence ρ = ρ + -ρ -represents the plastic deformation and their sum is denoted by κ = ρ + + ρ -. Let us dene the domains I T and I as follows I T = I × (0, T ) and I = (0, 1) T > 0.

Our main objective is to examine, under suitable boundary conditions, the short time existence and uniqueness of an approximated solution of the following coupled system of non-linear parabolic equations { κ t κ x = ρ t ρ x on I T , ρ t = ρ xx + κ x √ κ x on I T .

(

The derivation of this system is presented in details in Chapter 2. By adding a viscosity term, we regularize our system to get the following approximated model

   κ t = εκ xx + ρ x ρ xx κ x + ρ x √ κ x on I T , ρ t = (1 + ε)ρ xx + κ x √ κ x on I T , (1.2)
taking into consideration the following initial data ρ(x, 0) = ρ 0 (x) and κ(x, 0) = κ 0 (x) x ∈ I, (1.3) and the boundary data of Dirichlet type ρ(0, t) = ρ(1, t) = κ(0, t) = 0 and κ(1, t) = 1 t > 0.

(1.4) Now, we state the short time existence and uniqueness result of solutions to problem (1.2)-(1.4).

Formal derivation and existence result of an approximate model on dislocation densities

Theorem 1.2.1. (Short time existence and uniqueness)

Let ε > 0 be a xed constant. Consider ρ 0 , κ 0 ∈ C ∞ (I), such that ρ 0 (0) = ρ 0 (1) = 0 and κ 0 (0) = 0, κ 0 (1) = 1 with κ 0 x > |ρ 0

x | on I.

(1.5)

Assume

{ (1 + ε)ρ 0 xx + κ 0 x √ κ 0 x = 0
on ∂I,

(1 + ε)κ 0 xx + ρ 0 x √ κ 0 x = 0
on ∂I.

(1.6)

Then there exists a short time T > 0 and a unique solution (ρ, κ) of problem (1.2)- (1.4), satisfying

(ρ, κ) ∈ ( C 3+α, 3+α 2 (I T ) ∩ C ∞ (I × (0, T )) ) 2 ,
with 0 < α < 1 and κ x > |ρ x | on I × (0, T ).

(1.7)

Condition (1.5) is of physical origin representing the initial positivity of dislocation densities. By establishing the comparison principle on the gradient of the solution, we prove this condition up to a short time T > 0. This later result helps in avoiding the singularity presented in our system which stands as the main complication of this work. Thus, let us rst state our comparison principle that is the rst step to accomplish our existence and uniqueness result.

Proposition 1.2.2. Let (ρ, κ) be a regular solution of (1.2) on the compact I T with κ x > 0, and the initial data (ρ 0 , κ 0 ) satises κ 0

x β γ 0 (ρ 0 x ), γ 0 ∈ (0, 1), (1.8) where

β δ (x) := √ x 2 + δ 2 , x, δ ∈ R.
Then there exists a C 1 positive function γ : [0, T ] → ]0, +∞[, with γ(0) = γ 0 , such that κ x β γ (ρ x ) on I T .

(1.9)

Considering a constant or bounded exterior stress eld and lacking the real physical stress in previous articles dealing with similar systems enable to linearize the equation after getting rid of the singularity. Consequently, the long time existence can be attained.

Whereas, in our problem, the presence of the square root of the gradient term makes it limited by the short time existence and uniqueness that will be proved using a xed point argument after an articial modication of system (1.2) in order to avoid dividing by zero.

Thus, we carefully truncate the gradients ρ x and κ x conserving the same initial condition (1.3) and boundary condition (1.4). Using a xed point argument, we will prove that the truncated system, that will be presented in Chapter 3, coincides with (1.2) under a specic inequalities satised by ρ x and κ x and has a unique solution.

Proposition 1.2.3. Let p > 3 and let ρ 0 , κ 0 ∈ C ∞ (I), be two given functions such that ρ 0 (0) = ρ 0 (1) = κ 0 (0) = 0 and κ 0 (1) = 1. Suppose furthermore that

     κ 0 x γ 0 on I, ∥D s x ρ 0 ∥ L ∞ (I) M 0 , s = 1, 2, ∥D s x κ 0 ∥ L ∞ (I) M 0 , s = 1, 2, (1.10)
with 0 < γ 0 M 0 . Then there exists a unique solution (ρ, κ) ∈ Y 2 of (1.2), (1.3) and (1.4) where

T = T (M 0 , γ 0 , ε, p), 0 < T < 1,
and the parabolic Sobolev space Y is dened by

Y = W 2,1 p (I T ) = {u ∈ L p (I T ); D r t D s x u ∈ L p (I T ) for 2r + s 2},
where

D k z (u) = ∂ k u ∂z k for an integer k. Moreover, this solution satises    γ 0 2 κ x 2M 0 on I T , |ρ x | 2M 0 on I T .
(

Finally, to fulll the demonstration of our theorem, it remains to exhibit the regularity of the solution relying on a bootstrap argument which is included in the following proposition.

Proposition 1.2.4. Under the same hypothesis of Proposition 1.2.3 and if, in addition, the functions ρ 0 , κ 0 satisfy the condition

{ (1 + ε)ρ 0 xx + κ 0 x √ κ 0 x = 0 on ∂I, (1 + ε)κ 0 xx + ρ 0 x √ κ 0 x = 0
on ∂I, (1.12) then the solution (ρ, κ) obtained in Proposition 1.2.3 satises

(ρ, κ) ∈ (C 3+α, 3+α 2 (I T ) ∩ C ∞ ( Ī × (0, T ))) 2 .
(1.13)

1.3 Global BV solution for a non-local coupled system modeling the dynamics of dislocation densities

Neglecting the short range dislocation-dislocation interactions and looking for the variations inside the material, our results for Groma-Balogh model [START_REF] Groma | Link between the individual and continuum approaches of the description of the collective behavior of dislocations[END_REF][START_REF] Groma | Investigation of dislocation pattern formation in a two-dimensional self-consistent eld approximation[END_REF], that deals with the long range interactions, will be briey presented in the coming section. Periodic boundary conditions are naturally considered in the mathematical study of such models

BV (R) = { u ∈ L 1 loc (R); T V (u) < +∞ } ,
where the total variation of a function u

∈ L 1 loc (R) is dened by T V (u) = sup {∫ R u(x)Φ ′ (x)dx, Φ ∈ C 1 c (R) and ∥Φ∥ L ∞ (R) 1 } .
The current work is a simplication of the two dimensional Groma-Balogh model formed by taking the Burgers vector ⃗ b = (1, 0), assuming that our domain is 1-periodic in x 1 and x 2 , and supposing that the dislocation densities depend only on the variable x = x 1 + x 2 . Here, note that (x 1 , x 2 ) is the coordinate of a generic point in R 2 . Thus, the two dimensional model of [START_REF] Groma | Link between the individual and continuum approaches of the description of the collective behavior of dislocations[END_REF][START_REF] Groma | Investigation of dislocation pattern formation in a two-dimensional self-consistent eld approximation[END_REF] reduces to the following system of one dimensional coupled non-local Hamilton-Jacobi equations

           ∂ρ + ∂t (x, t) = - ( (ρ + -ρ -)(x, t) + α ∫ 1 0 (ρ + -ρ -)(y, t)dy + a(t) ) ∂ρ + ∂x (x, t) in R × (0, T ), ∂ρ - ∂t (x, t) = ( (ρ + -ρ -)(x, t) + α ∫ 1 0 (ρ + -ρ -)(y, t)dy + a(t) ) ∂ρ - ∂x (x, t) in R × (0, T ), (1.14)
where ρ + , ρ -are the unknown scalars, that we denote for simplicity by ρ ± . We refer the reader to El Hajj and Forcadel [START_REF] Hajj | A convergent scheme for a non-local coupled system modeling dislocations densities dynamics[END_REF]Lemme 3.1] for more modeling details. Here, the dierence (ρ + -ρ -) represents the plastic deformation, and the space derivatives ∂ρ ± ∂x represent the dislocation densities of ± dislocations respectively. The constant α depends on the elastic coecients and the material size, while the function a(t) represents the exterior shear stress eld, which is assumed to satisfy the following regularity a ∈ L ∞ (0, T ).

(1.15)

The initial conditions associated to system (1.14) are dened as

ρ ± (x, 0) = ρ ± 0 (x) = P ± 0 (x) + L 0 x on R, (1.16)
where P ± 0 are 1-periodic functions satisfying

P ± 0 ∈ L ∞ (T) ∩ BV (T), (1.17)
where T = R/Z is the [0, 1) periodic interval (we refer to Chapter 4 for a detailed explanation).

We seek to work with BV space since in reality, dislocations are randomly distributed in a crystal. This enables us to study the real mechanism of dislocations far from considering the very specic perfect case of continuity that rarely exists.

From a physical viewpoint, the dislocation density has a nonnegative sign ∂ρ ± ∂x 0. In this work, however, we do not consider this assumption but rather we allow the density to change sign.

We prove the existence of a global BV solution for system (1.14). In order to attain this result, we pass by some key steps that we now briey expose. First, we consider an associated local problem which is obtained by freezing the integral term

           ∂ρ + ∂t (x, t) = - ( (ρ + -ρ -)(x, t) + L(t) ) ∂ρ + ∂x (x, t) in R × (0, T ), ∂ρ - ∂t (x, t) = ( (ρ + -ρ -)(x, t) + L(t) ) ∂ρ - ∂x (x, t) in R × (0, T ), (1.18)
with the initial conditions (1.16) and assuming L ∈ L ∞ (0, T ).

(

Then, we regularize, by a classical convolution argument, the function L(•) and the initial conditions in (1.16). This approximation leads to the study, for every 0 < ε < 1, of the following system

                   ∂ρ + ε ∂t (x, t) = - ( (ρ + ε -ρ - ε )(x, t) + L ε (t) ) ∂ρ + ε ∂x (x, t) in R × (0, T ), ∂ρ - ε ∂t (x, t) = ( (ρ + ε -ρ - ε )(x, t) + L ε (t) ) ∂ρ - ε ∂x (x, t) in R × (0, T ), ρ ± ε (x, 0) = P ± 0,ε (x) + L 0 x in R, (1.20)
where L ε and P ± 0,ε are the regularization of the functions L and P ± 0 respectively (refer to Chapter 4 for the denitions of L ε and P ± 0,ε ). Next, we use the existence result obtained in [START_REF] Hajj | A convergent scheme for a non-local coupled system modeling dislocations densities dynamics[END_REF] to prove that this regularized system has a unique Lipschitz continuous viscosity solution in the sense of the following denition proposed by Ishii and Koike in [START_REF] Ishii | Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games[END_REF] where

0 η 1.
Denition 1.3.1. (Continuous viscosity sub-solution, super-solution and solution)

A function ρ ± ε ∈ C(R × [0, T ))
is a viscosity sub-solution of (1.20) if and only if

• ρ ± ε (x, 0) ρ ± 0,ε (x),
• for all k ∈ {+, -} and for any test-

function ϕ ∈ C 2 (R × (0, T )) such that ρ k ε -ϕ reaches a local maximum at a point (x 0 , t 0 ) ∈ R × (0, T ), we have ∂ϕ ∂t (x 0 , t 0 ) + k ( (ρ +,ε η -ρ -,ε η )(x 0 , t 0 ) + L ε (t 0 ) ) ∂ϕ ∂x (x 0 , t 0 ) -η ∂ 2 ϕ ∂x 2 (x 0 , t 0 ) 0.
In a similar way, a function

ρ ± ε ∈ C(R × [0, T ))
is a viscosity super-solution of (1.20) if and only if

• ρ ± ε (x, 0) ρ ± 0,ε (x),
• for all k ∈ {+, -} and for any test-

function ϕ ∈ C 2 (R × (0, T )) such that ρ k ε -ϕ reaches a local minimum at point (x 0 , t 0 ) ∈ R × (0, T ), we have ∂ϕ ∂t (x 0 , t 0 ) + k ( (ρ +,ε η -ρ -,ε η )(x 0 , t 0 ) + L ε (t 0 ) ) ∂ϕ ∂x (x 0 , t 0 ) -η ∂ 2 ϕ ∂x 2 (x 0 , t 0 ) 0.
Finally, a continuous function ρ ± ε is a viscosity solution of (1.20) if and only if it is a sub-solution and a super-solution of (1.20).

After proving the existence and the uniqueness of a Lipschitz continuous viscosity solution ρ ± ε satisfying a local L ∞ uniform estimate in ϵ, we show that the relaxed semilimits of Barles and Perthame [START_REF] Barles | Exit time problems in optimal control and vanishing viscosity method[END_REF][START_REF] Barles | Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations[END_REF] ρ Assume that L is bounded on (0, T ) and ρ ± 0 is locally bounded on R. Assume moreover that

± (x, t) = lim sup ⋆ ρ ± ε (x, t) = lim sup ε-→0 (y,s)-→(x,t) ρ ± ε (y, s), (1.21) and ρ ± (x, t) = lim inf ⋆ ρ ± ε (x, t) = lim inf ε-→0 (y,s)-→(x,t) ρ ± ε (y, s), ( 1
L + sub = L - super = L ⋆ and L - sub = L + super = L ⋆ . (1) (Discontinuous viscosity sub-solution) An upper semi-continuous function ρ ± on R × [0, T ) is a discontinuous viscosity sub- solution of (1.18) if it satises (i) ρ ± (x, 0) (ρ ± 0 ) ⋆ (x),
(ii) for all k ∈ {+, -} and for any test-function ϕ ∈ C 1 (R × (0, T )) such that ρ k -ϕ reaches a local maximum at a point (x 0 , t 0 ) ∈ R × (0, T ), we have

∂ϕ ∂t (x 0 , t 0 ) + k ( (ρ + -ρ -)(x 0 , t 0 ) + L k sub (t 0 )
) ∂ϕ ∂x (x 0 , t 0 ) 0.

(1.23)

(2) (Discontinuous viscosity super-solution)

A lower semi-continuous function

ρ ± on R × [0, T ) is a discontinuous viscosity super- solution of (1.18) if it satises (i) ρ ± (x, 0) (ρ ± 0 ) ⋆ (x),
(ii) for all k ∈ {+, -} and for any test-

function ϕ ∈ C 1 (R × (0, T )) such that ρ k -ϕ reaches a local minimum at a point (x 0 , t 0 ) ∈ R × (0, T ), we have ∂ϕ ∂t (x 0 , t 0 ) + k ( (ρ + -ρ -)(x 0 , t 0 ) + L k super (t 0 )
) ∂ϕ ∂x (x 0 , t 0 ) 0.

(1.24)

(3) (Discontinuous viscosity solution)

Finally, we say that a locally bounded function ρ ± dened on R × [0, T ) is a discontinuous viscosity solution of (1.18) if its upper semi-continuous (respectively lower semicontinuous) envelope is a viscosity sub-solution (respectively super-solution).

Note that f ⋆ and f ⋆ the respective upper and lower semi-continuous envelopes of a locally bounded function f dened on [0, T ) × Ω, where Ω is an open subset of R n , and given by

f ⋆ (X, t) = lim sup (Y,s)→(X,t) f (Y, s) and f ⋆ (X, t) = lim inf (Y,s)→(X,t)
f (Y, s).

(1.25)

Finally, by establishing some ε-independent a priori estimates, we will be able to prove that ρ ± (•, t) = ρ ± (•, t) almost everywhere in R, for all t > 0. This almost everywhere equality holds true due to the fact that the set of discontinuous points of a BV function is at most countable and the nite speed propagation property. Consequently, we ensure the existence of functions ρ ± , dened as a strong limit of

ρ ± ε in C([0, T ); L 1 loc (R)),
that is almost everywhere discontinuous viscosity solution of (1.18). This solution is called weak discontinuous viscosity solution.

Many results arise on similar eikonal systems. However, all these result were established in the regular case away from the discontinuous case. In [START_REF] Hajj | Global solution for a non-local eikonal equation modelling dislocation dynamics[END_REF], based on an energy estimate, the global existence and uniqueness of a solution in the class of non-decreasing W 1,2 loc (R × [0, +∞)) functions has been established. However, in [START_REF] Hajj | A convergent scheme for a non-local coupled system modeling dislocations densities dynamics[END_REF], the authors have used the notion of viscosity solutions (initially introduced by Crandall and Lions [START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF][START_REF] Crandall | On existence and uniqueness of solutions of Hamilton-Jacobi equations[END_REF] to deal with Hamilton-Jacobi equations) in order to solve (1.14). Afterwards, a local existence and uniqueness result in Hölder spaces has been shown in [START_REF] Hajj | Short time existence and uniqueness in Hölder spaces for the 2D dynamics of dislocation densities[END_REF], based on some commutator estimates.

Related to our analysis, we also obtain, as a consequence, the global existence of a discontinuous viscosity solution of (1.18) for non-decreasing initial data (see Theorem 1.3.4 afterwards). Moreover, using a xed point argument, we get a similar result for the non-local system describing the dynamics of dislocation densities which will be presented later in Theorem 1.3.5.

Therefore, we are going to present rst, in Theorem 1.3.3, a global existence result of a weak discontinuous viscosity solution of (1.18). This solution can be seen as a discontinuous viscosity solution but in some weak sense, since it only veries an almost everywhere equality in space between ρ ± and ρ ± , which is reected by (1.35) presented after. As a consequence, we show, in Theorem 1.3.4, that this solution is indeed a classical discontinuous viscosity solution in the case of non-decreasing solutions. Moreover, based on these two theorems, we present, in Theorem 1.3.5, some similar results for a model related to the dynamics of dislocation densities.

We will now present our rst main result.

ρ ± ε (x, t) = P ± ε (x, t) + L 0 x,
where P ± ε are 1-periodic functions (with respect to the space variable). Moreover, for all T > 0, we have the following uniform a priori estimates

max ± ( ∥P ± ε ∥ L ∞ (T×(0,T )) ) M 0 , (1.26) ∂ρ ± ε ∂x L ∞ ((0,T );L 1 (T)) |ρ ± 0 | BV (T) , (1.27) ∂ρ ± ε ∂t L ∞ ((0,T );L 1 (T)) [ 2M 0 + ∥L∥ L ∞ (0,T ) ] |ρ ± 0 | BV (T) , (1.28)
with

M 0 = max ± ( ∥P ± 0 ∥ L ∞ (T) ) + |L 0 |∥L∥ L ∞ (0,T ) T. (1.29)
ii) Sub-and super-solutions of the limit problem Let ρ ± ε be the solution of (1.20) constructed in (i), then the relaxed semi-limits ρ ± and ρ ± , dened by (1.21), (1.22), are respectively discontinuous viscosity sub-solution and supersolution of (1.18) in sense of the denition of discontinuous viscosity solution presented in Chapter 4.

iii) Convergence and existence of weak solution Assume that ρ ± ε satises (1.26), (1.27) and (1.28). Then, up to the extraction of a subsequence, the functions ρ ± ε converge, as ε tends to zero, to a function

ρ ± ∈ L ∞ loc (R × (0, T )) ∩ L ∞ ((0, T ); BV (T)) ∩ C([0, T ); L 1 loc (R)), (1.30)
strongly in C([0, T ); L 1 loc (R)). Moreover,

ρ ± (x, t) = P ± (x, t) + L 0 x, ( 1.31)
where P ± are 1-periodic functions (with respect to the space variable) and ρ ± satisfy, for all T > 0, the following estimates

max ± ( ∥P ± ∥ L ∞ (T×(0,T )) ) M 0 , (1.32) ∥ρ ± ∥ L ∞ ((0,T );BV (T)) |ρ ± 0 | BV (T) , (1.33) ∥ρ ± (•, t) -ρ ± (•, s)∥ L 1 (T) ([ 2M 0 + ∥L∥ L ∞ (0,T ) ] |ρ ± 0 | BV (T) ) |t -s|, for all s, t ∈ [0, T ), (1.34)
and the following equality holds

ρ ± (•, t) = ρ ± (•, t) = ρ ± (•, t), for all t ∈ [0, T ), (1.35)
except at most on a countable set in D ⊂ R.

This theorem is proved relying on the uniform BV estimate (1.27) on ρ ± ε . We rst consider the parabolic regularization of (1.18) and we show that the Lipschitz continuous viscosity solution admits the L ∞ bound (1.26), and the fundamental BV estimate (1.27).

These estimates allow us, by relying on the stability property of viscosity solutions (see Barles [10,Theorem 4.1]), to pass to the limit when the regularization vanishes, and then to show that the relaxed semi-limits ρ ± and ρ ± are, respectively, sub-and super-solutions of (1.18). Moreover, these estimates also imply that the solution ρ ± (•, t) is continuous except at most on a countable set D t ⊂ R. Taking into account the nite speed of propagation property of (1.18) and the time continuous estimate (1.29), it is then possible to construct the set D out of D t ; hence proving (1.35).

From a mathematical point of view, in the framework of non-decreasing solutions, system (1.14) is related to other similar models such as transport equations. Lax [START_REF] Lax | Hyperbolic systems of conservation laws and the mathematical theory of shock waves[END_REF] proved the existence and uniqueness of non-decreasing smooth solutions for diagonal 2×2 strictly hyperbolic systems. Let us mention that, in the case of general strictly hyperbolic systems, Bianchini and Bressan [START_REF] Bianchini | Vanishing viscosity solutions of nonlinear hyperbolic systems[END_REF] proved a striking global existence and uniqueness result, assuming that the initial data has small total variation. This approach is mainly based on a careful analysis of the vanishing viscosity approximation. Moreover, an existence result has been also obtained by LeFloch and Liu [START_REF] Lefloch | Existence theory for nonlinear hyperbolic systems in nonconservative form[END_REF], and LeFloch [START_REF] Lefloch | Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form[END_REF], in the non-conservative case. However, in this framework, the fact that our system becomes hyperbolic but not strictly hyperbolic makes it more complicated to establish the desired results. This is due to the fact that we have no sign property on the term ρ + -ρ -. Thus, we do not have ordered velocities as the case of strictly hyperbolic systems (λ 1 < ... < λ d ).

Therefore, related to our analysis in Theorem 1.3.3, it is then possible to get the following theorem as a by-product. Assume that the assumptions (1.16), (1.17) and (1.19) hold. If the initial data ρ ± 0 are non-decreasing, then the system (1.18) admits a discontinuous non-decreasing viscosity solution ρ ± , in the sense of the denition 1.3.2, satisfying (1.30), (1.31), (1.32), (1.33) and (1.34).

Applying Theorems 1.3.3, 1.3.4, and using a xed point argument, we will prove the following result for the non-local system (1.14) describing the dynamics of dislocation densities.

Theorem 1.3.5. (Global existence results of dislocation system) Under the assumptions (1.15), (1.16) and (1.17), the following points hold.

i) Global existence of weak viscosity solution

There exists a weak discontinuous viscosity solution of (1.14) in the sense of Theorem 1.3.3, satisfying (1.30), (1.31) and (1.35).

ii) Global existence of viscosity non-decreasing solution Assume ρ ± 0 are non-decreasing, then the system (1.14) has a discontinuous non-decreasing viscosity solution ρ ± satisfying (1.30) and (1.31).

The proof of Theorem 1.3.5 is an application of Theorems 1.3.3 and 1.3.4.

Remark 1.3.6. The uniqueness of the solution obtained in Theorem 1.3.5 remains an open question in the case of BV initial data. Here, the problem lies in the fact that the initial data is discontinuous, and hence we can not establish a comparison principle result. On the other hand, it is possible to have such a result if we consider uniformly continuous data (see the result of Ishii and Koiki [START_REF] Ishii | Viscosity solution for monotone systems of second-order elliptic PDEs[END_REF]).

BV solution for a non-linear Hamilton-Jacobi system

As a generalization of the work done in Section 1.3, this section presents briey the results that we have attained on a strongly coupled Hamilton-Jacobi system of d equations. Our aim lies rst in proving the global in time existence of the solution of the regularized system using a xed point argument and a comparison principle in an unbounded domain.

By passing to the limit, we come later into the existence of the solution of the main system.

More precisely, we are looking for solutions of the form u(t, x) =

( u i (t, x) ) i=1,...,d of the following one dimensional Hamilton-Jacobi system    ∂ t u i (t, x) = λ i (t, x, u) |∂ x u i (t, x)| in (0, T ) × R, u i (0, x) = u i 0 (x) in R, (1.36)
for T > 0 and i = 1, ..., d, where d ∈ N * . The function u i is real-valued, ∂ t u i and ∂ x u i stand, respectively, for its time and spatial derivatives.

Here, the velocity λ i is assumed to satisfy the following assumption

λ i ∈ L ∞ ((0, T ) × R × K) for T > 0 and for all compact K ⊂ R d .
(1.37)

We introduce also the below non-decreasing assumption of λ i with respect to the variable

u i For all u i v i , (r j ) j=1,••• ,d,j̸ =i ∈ R d-1 and (t, x) ∈ (0, T ) × R, we have λ i (t, x, r(u i )) λ i (t, x, r(v i ))
where r(

u i ) = (r 1 , • • • , u i , • • • , r d ) and r(v i ) = (r 1 , • • • , v i , • • • , r d ).
(

Our study of system (1.36) is motivated by the consideration of a model describing the dynamics of dislocation densities (see [START_REF] Hajj | Global continuous solutions for diagonal hyperbolic systems with large and monotone data[END_REF]Section 5] for more details about the modeling), which is given by

∂ t u i = ( ∑ j=1,...,d B ij u j ) |∂ x u i | for i = 1, . . . , d, (1.39)
where (B ij ) i,j=1,...,d is a real matrix. This model can be seen as a special case of system (1.36).

The goal of this work is to establish the global existence of discontinuous viscosity solutions of system (1.36) assuming condition (1.37) and the following regularity on the initial data

u i 0 ∈ L ∞ (R) ∩ BV (R).
(1.40)

Many existence and uniqueness results were brightened up on similar Eikonal systems.

Let us mention the most known results. First, motivated by dislocation dynamics, we can point out the result done by El Hajj and Boudjerada in [START_REF] Boudjerada | Global existence results for eikonal equation with BV initial data[END_REF] who were able to prove the global existence of discontinuous viscosity BV solutions for scalar one dimensional nonlinear and non-local Eikonal equations, including in particular the case d = 1 in system (1.36), where the velocity does not contain the solution. Also, considering dislocation dynamics as a motivation, this result has been extended to a more general non-linear

(2 × 2) system which is our previous result on the non-local coupled Hamilton-Jacobi system. Also, an existence and uniqueness result of a Lipschitz viscosity solution was proved by El Hajj and Forcadel in [START_REF] Hajj | A convergent scheme for a non-local coupled system modeling dislocations densities dynamics[END_REF] for the same system. In the case of general (d × d)

1.4.
BV solution for a non-linear Hamilton-Jacobi system system, it is worth mentioning the result of Ishii, Koike [START_REF] Ishii | Viscosity solution for monotone systems of second-order elliptic PDEs[END_REF] and Ishii [START_REF] Ishii | Perron's method for monotone systems of second-order elliptic partial dierential equations[END_REF], who had shown the existence and uniqueness of continuous viscosity solutions for Hamilton-Jacobi systems of the form

   ∂ t u i + H i (t, x, u, Du i ) = 0 with u = (u 1 , . . . , u d ) ∈ R d , x ∈ R N
, and t ∈ (0, +∞),

u i (0, x) = u i 0 (x) for x ∈ R N ,
where the Hamiltonian H i is quasi-monotone in u (see the denition in Ishii, Koike [START_REF] Ishii | Viscosity solution for monotone systems of second-order elliptic PDEs[END_REF]Th.4.7]).

Our result, allows to give meaning to system (1.36) in the framework of discontinuous viscosity solutions. This enables us to enlarge the area of applications to touch dislocation dynamics in multi-directions of propagation. More precisely, we present a global existence result for the strongly coupled Hamilton-Jacobi system (1.36) considering large BV initial data. This result is obtained without sign restrictions on the velocity λ i and also unconditional monotonicity of the solution. We only consider the case when the initial data and the velocity satisfy the assumptions (1.37) and (1.40) without any better regularity. However, the state of having non-decreasing initial data is presented as a particular case of our work in Theorem 1.4.3. In its full generality, the fundamental issue of uniqueness for global solution remains open. This question is related to the fact that system (1.36) is not only non-linear but it is also non-monotone which means that the comparison principle, that plays a central role in the level-set approach, does not hold and then we cannot apply directly the viscosity solutions theory. Therefore, the uniqueness of solutions cannot be proved via standard viscosity solutions methods. We refer the reader to [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF][START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF][START_REF] Crandall | User's guide to viscosity solutions of second order partial dierential equations[END_REF] for a complete overview of viscosity solutions. We also refer to Barles [11] for an interesting counter-example on the uniqueness of discontinuous viscosity solution.

First, by a classical convolution argument, we regularize the velocity and the initial data which were announced in system (1.36). This approximation brings us to consider, for every 0 < ε < 1, the following system   

∂ t u i ε (t, x) = λ i ε (t, x, ρ 1 ε ⋆ u ε (t, •)(x)) |∂ x u i ε (t, x)| in (0, T ) × R, u i ε (0, x) = u i 0,ε (x) in R, (1.41)
where λ i ε and u i 0,ε are the regularization of the functions λ i and u i 0 respectively, and they are given by

u i 0,ε (x) = u i 0 ⋆ ρ 1 ε (x) and λ i ε (t, x, w) = λi ⋆ ρ d+2 ε (t, x, w) ∀ (t, x, w) ∈ R × R × R d , (1.42)
with λi is an extension in R d+2 of λ i by 0, and ρ n ε for n = 1, d + 2 are the standard molliers dened as follows

ρ n ε (•) = 1 ε n ρ n ( • ε ) , such that ρ n ∈ C ∞ c (R n ), supp {ρ n } ⊆ B(0, 1), ρ n 0, and ∫ R n ρ n = 1.
(1.43)

Our result lies initially in proving the global in time existence of the solution of the above regularized system using a xed point argument and a comparison principle of the associated linear problem obtained by freezing u in the velocity. Afterwards, to pass from the solution of the regularized system (1.41) to that of system (1.36), we will show that the upper and lower relaxed semi-limits, which are dened as follows

u i (x, t) = lim sup ⋆ u i ε (x, t) = lim sup ε-→0 (y,s)-→(x,t) u i ε (y, s), ( 1.44)
and Assume that λ i is locally bounded on (0, T )×R×R d and

u i (x, t) = lim inf ⋆ u i ε (x, t) = lim inf ε-→0 (y,s)-→(x,t) u i ε (y, s),
u 0 = (u i 0 ) i=1,••• ,d is locally bounded on R. Let u = (u i ) i=1,••• ,d be a locally bounded function dened on [0, T ) × R.
(1) (Discontinuous viscosity sub-solution)

We call u a discontinuous viscosity sub-solution of (1.36) if it satises

(i) (u i ) ⋆ (0, x) (u i 0 ) ⋆ (x), for all i = 1, • • • , d and x ∈ R. (ii) If whenever ϕ ∈ C 1 ((0, T ) × R), i = 1, • • • , d and (u i ) ⋆ -ϕ attains its local maximum at (t 0 , x 0 ) ∈ (0, T ) × R, then we have min { ∂ t ϕ(t 0 , x 0 ) -(λ i ) ⋆ (t 0 , x 0 , r) |∂ x ϕ(t 0 , x 0 )| : r ∈ U (t 0 , x 0 ), r i = (u i ) ⋆ (t 0 , x 0 ) } 0.
(1.46)

(2) (Discontinuous viscosity super-solution)

Similarly, we call u a discontinuous viscosity super-solution of (1.36) if it satises

(i) (u i ) ⋆ (0, x) (u i 0 ) ⋆ (x), for all i = 1, • • • , d and x ∈ R. (ii) If whenever ϕ ∈ C 1 ((0, T ) × R), i = 1, • • • , d and (u i ) ⋆ -ϕ attains its local minimum at (t 0 , x 0 ) ∈ (0, T ) × R, then we have max { ∂ t ϕ(t 0 , x 0 ) -(λ i ) ⋆ (t 0 , x 0 , r) |∂ x ϕ(t 0 , x 0 )| : r ∈ U (t 0 , x 0 ), r i = (u i ) ⋆ (t 0 , x 0 ) } 0.
(1.47)

(3) (Discontinuous viscosity solution)

Finally, we call u a discontinuous viscosity solution of (1.36) if it is both a discontinuous viscosity sub-solution and super-solution of (1.36).

Here, U : (0, T ) × R → 2 R d is the graph closure of u and it is dened by

U (t, x) = {r ∈ R d : there is a sequence {(t n , x n )} ⊂ (0, T ) × R such that (t n , x n ) -→ n→+∞ (t, x) and u(t n , x n ) → r}.
Lastly, leaning on some ε-independent a priori estimates, we come to prove the almost everywhere equality between u i and u i in R, for all t > 0. This shows the existence of a function u i , dened as a strong limit of

u i ε in C([0, T ); L 1 loc (R)),
u i ε L ∞ ((0,T )×R) u i 0 L ∞ (R) , (1.48) ∂ x u i ε L ∞ ((0,T );L 1 (R)) u i 0 BV (R) , (1.49) ∂ t u i ε L ∞ ((0,T );L 1 (R)) λ i L ∞ ((0,T )×R×K 0 ) u i 0 BV (R) , (1.50)
where

K 0 = ∏ d i=1 [ -∥u i 0 ∥ L ∞ (R) -1, ∥u i 0 ∥ L ∞ (R) + 1 ] . ii) Convergence Assume that u i ε ,
u i ∈ L ∞ ( (0, T ) × R ) ∩ L ∞ ( (0, T ); BV (R) ) ∩ C ( [0, T ); L 1 loc (R)
) , (1.51) strongly in C ([0, T ); L 1 loc (R)).

Moreover, u i satises, for all T > 0 and for i = 1, • • • , d, the following inequalities

u i L ∞ ((0,T )×R) u i 0 L ∞ (R) , (1.52) u i L ∞ ((0,T );BV (R)) u i 0 BV (R) , (1.53) u i (t, •) -u i (s, •) L 1 (R) ( λ i L ∞ ((0,T )×R×K 0 ) u i 0 BV (R)
)

|t -s|, for all s, t ∈ [0, T ), (1.54) and the following equality

u i (t, •) = u i (t, •) = u i (t, •), except at most on a countable set in R, for all t ∈ [0, T ), (1.55)
where u i and u i are, respectively, the upper relaxed semi-limit and the lower relaxed semilimit dened in (1.44) and (1.45).

iii) Global existence of weak discontinuous viscosity solution Let u ε be the solution of (1.41), constructed in (i). Suppose that assumption (1.38) is satised. Then u =

( u i ) i=1,••• ,d and u = ( u i ) i=1,••• ,d
, are respectively discontinuous viscosity sub-solution and super-solution of system (1.36) in the sense of Ishii denition. Generally, (without assumption (1.38)) u and u, are both almost everywhere in space discontinuous viscosity sub-solution and super-solution of system (1.36) and moreover verify equality (1.55).

The key point to establish this theorem is the uniform BV estimate on u i ε (1.49). We rst consider the parabolic regularization of (1.36) and we show that the smooth solution admits the L ∞ bound (1.48) and the fundamental BV estimate (1.49). These estimates will allow us to pass to the limit when the regularization vanishes. Then we will show that the relaxed semi-limits u i and u i are, respectively, sub-and super-solutions of (1.36). These estimates also imply that the set of the discontinuous points, with respect to x, of the solution u is at most countable. Taking into account the nite speed propagation property of (1.36) and the time continuous estimate (1.50), it is then possible to show this property uniformly in time, which proves in particular (1.55).

Recall that in the framework of non-decreasing solutions, the Hamilton-Jacobi system (1.36) becomes a classical transport system. For such types, we can mention Bianchini and Bressan, who had proved in [START_REF] Bianchini | Vanishing viscosity solutions of nonlinear hyperbolic systems[END_REF] a striking result of global existence and uniqueness of a solution for general non-conservative (d × d) strictly hyperbolic systems, including diagonal systems. The key step in their proof was an a priori estimate on the total variation of the approximate solutions proved relying on small total variation of initial data. Their existence result is a generalization of Glimm's result [START_REF] Glimm | Solutions in the large for nonlinear hyperbolic systems of equations[END_REF], proved in the case of conservation laws. Let us mention that an existence result has also been obtained by LeFloch and Liu [START_REF] Lefloch | Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form[END_REF][START_REF] Lefloch | Existence theory for nonlinear hyperbolic systems in nonconservative form[END_REF] in the non-conservative case. After that, El Hajj and Monneau proved in [START_REF] Hajj | Some niqueness results for diagonal hyperbolic systems equation with large and monotone data[END_REF] a global existence and uniqueness result for strictly hyperbolic diagonal systems, with the assumption ∂λ i ∂u i

0 (i = 1, • • • , d)
, by widening the regularity assumption and considering 20 1.5. Global existence of discontinuous solution for a system modeling isentropic gas dynamics continuous solutions. That was a generalization to the work done by Lax [START_REF] Lax | Hyperbolic systems of conservation laws and the mathematical theory of shock waves[END_REF], in the case of (2×2) strictly hyperbolic systems, where the existence of Lipschitz solutions was proved.

Theorem 1.4.3. (Global existence of non-decreasing viscosity solution)

Assume that (1.37) and (1.38) are satised. Suppose that u i 0 ∈ L ∞ (R) and the function (1.52), (1.53) and (1.54).

u i 0 is non-decreasing for i = 1, • • • , d. Then, system (1.36) has a discontinuous viscosity solution u = ( u i ) i=1,••• ,d , such that for i = 1, • • • , d, u i satises (1.51),
1.5 Global existence of discontinuous solution for a system modeling isentropic gas dynamics Techniques borrowed from hyperbolic conservation laws are used to accurately capture the complicated surface motion that satises the global entropy condition for propagating fronts. Here, we analyze the coupling of this level set formulation to a system of conservation laws for compressible gas dynamics. Based on the existence result done in [START_REF] Hajj | Some niqueness results for diagonal hyperbolic systems equation with large and monotone data[END_REF] and considering some monotonicity assumptions on the initial data, we study the system presented illustrating mass and momentum conservation laws for propagation of gas.

           ∂ t ρ + ∂ x (ρu) = 0 in (0, T ) × R, ∂ t (ρu) + ∂ x (ρu 2 + p(ρ)) = 0, with p(ρ) = (γ-1) 2 4γ ρ γ in (0, T ) × R, u(0, x) = u 0 (x) and ρ(0, x) = ρ 0 (x) 0, for x ∈ R, (1.56)
where ρ is the density, u is the speed and p(ρ) is the pressure given by a simple power law for an exponent γ > 1. First, we assume the following conditions, with θ = γ-1 2 u 0 , ρ 0 ∈ L ∞ (R) and u 0 ± ρ θ 0 are non-decreasing functions with θ = γ-1 2 . (1.57)

We will prove the following result.

Theorem 1.5.1. Assume (1.57) is veried, with ρ 0 0 and T > 0. Then system (1.56)

has a solution (u, ρ) ∈ L ∞ ( (0, T ) × R
) in distributional sense, with ρ 0 and

u, ρ θ ∈ L ∞ ( (0, T ) × R ) ∩ L ∞ ( (0, T ); BV (R) ) ∩ C ( [0, T ); L 1 loc (R)
) .

( Existence and uniqueness result for C 1 solution was proved by T. T. Li in [68, pp. 35-41] in the case where ρ 0 > 0. Notice that in the previous theorem, we only assume that ρ 0 0, which allows us to consider solutions with vacuum. In this case, we can mention the work of Lions et al in [START_REF] Lions | Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates[END_REF] where the existence of a solution was obtained for ρ 0 0 with any u 0 , ρ 0 ∈ L ∞ (R) and γ > 1. This result was an extension to that of DiPerna in [START_REF] Diperna | Convergence of approximate solutions to conservation laws[END_REF][START_REF] Diperna | Compensated compactness and general systems of conservation laws[END_REF]. Another result for vacuum state was presented by Mercier [START_REF] Mercier | Étude de diérents aspects des EDP hyperbolique: persistance d'onde de choc dans la dynamique des uides compressibles, modélisation du trac routier, stabilité des lois deconservation scalaires[END_REF]. In addition to that, El Hajj and Monneau [START_REF] Hajj | Some niqueness results for diagonal hyperbolic systems equation with large and monotone data[END_REF][START_REF] Hajj | Global continuous solutions for diagonal hyperbolic systems with large and monotone data[END_REF] presented an application to their work in the continuous case on gas dynamics. To more explain the physical model, we consider a bounded channel (see Figure 2.1) that contains a certain number of parallel edge dislocations and bounded by walls that are impenetrable by dislocations (that means the plastic deformation in the walls is zero). This channel has a nite width in the x direction and an innite extension in the y direction.

The dislocation lines -→ ξ are supposed to be perpendicular to x and y and the Burgers vector -→ b is parallel to the x direction.

The whole fabrication is set in an innite crystal where dislocations can move under the action of some constant exterior stress eld τ ̸ = 0, and/or under the action of the stress eld created by dislocations themselves. Inside the material, the short and long range dislocation-dislocation interactions result in the internal stresses.

In our work, we derive our mathematical model on dislocation dynamics with shortrange interaction. All our computations are inspired by GCZ [START_REF] Groma | Spatial correlations and higher-order gradient terms in a continuum description of dislocations dynamics[END_REF] where we suppose a particular conguration of the dislocation lines that will be explained later.

Let θ + and θ -be the densities of the positive and negative dislocations respectively.

Denote by

ϕ = θ + + θ -,
the total dislocation density and by

ψ = θ + -θ -,
the sign dislocation density. In the case of no short range dislocation-dislocation interactions, the model satised by ϕ and ψ is the following

     ∂ϕ ∂t (x, t) + ∂ ∂x [ψ(x, t){τ sc (x, t) + τ ext }] = 0, ∂ψ ∂t (x, t) + ∂ ∂x [ϕ(x, t){τ sc (x, t) + τ ext }] = 0, (2.1)
where τ ext is the external stress, and τ sc is the self-consistent stress given by

τ sc (x, t) = ∫ ψ(y, t)τ ind (x -y, t)dy, ( 2.2)
with τ ind (x i -x j ) is the shear stress created at x i by a positive dislocation located at x j ; τ ind (x) is proportional to 1

x .

In the presence of short range dislocation-dislocation interactions, the model satised by ϕ and ψ can be written as

     ∂ϕ ∂t (x, t) + ∂ ∂x [ψ(x, t){τ sc (x, t) + τ ext -τ f + τ b }] = 0, ∂ψ ∂t (x, t) + ∂ ∂x [ϕ(x, t){τ sc (x, t) + τ ext -τ f + τ b }] = 0, (2.3)
where τ f is the local ow stress given by

τ f = AC 2 b √ ϕ, ( 2.4)
with A = µ/[2π(1 -ν)], µ is the shear modulus, ν is the Poisson ratio, b is the modulus of the Burgers vector, and C = ∫ ∫ x(x 2 -y 2 ) (x 2 + y 2 ) 2 p a (x, y)dx dy. The stress contribution τ f is related to the correlation function p a which in physical terms characterizes the polarization of dipoles of dislocations of opposite signs. Moreover, τ b is the back stress and is given by

τ b = - AD ⃗ b ϕ ∇ψ, ( 2.5)
where

D = ∫ ∫ x 2 (x 2 -y 2 ) (x 2 + y 2 ) 2 p(x,
τ e = τ ext -τ f .
In a special physical setting depending on the internal material structure and the frame of reference, we may assume ⃗ b = (1, 0) and A = D = 1. This will also simplify the presentation of the mathematical model. Plug (2.5) into (2.3), we obtain, after setting all constants to be 1,

       ∂ϕ ∂t (x, t) + ∂ ∂x [ ψ(x, t) { τ sc (x, t) + τ e - 1 ϕ ∂ψ ∂x }] = 0, ∂ψ ∂t (x, t) + ∂ ∂x [ ϕ(x, t) { τ sc (x, t) + τ e - 1 ϕ ∂ψ ∂x }] = 0, (2.6)
in the case of a constrained channel deforming in simple shear, i.e. a channel of width L in the x direction and innite extension in the y direction, bounded by walls that are impenetrable by dislocations (the plastic deformation at the walls is zero). The slip direction corresponds to the x direction, and the layer is sheared by a constant shear stress.

A simplied model.

Our interest lies in a simplied model where we suppose a particular conguration of the dislocation lines. It is a simplied version of the system studied by Van der Giessen et al [START_REF] Yemov | Multiple slip in a strain gradient plasticity model motivated by a statistical mechanics description od dislocations[END_REF] using both discrete dislocation and continuum plasticity approaches. The simplications are originated from the assumption of the activation of a single slip system, hence reactions between dislocations may be neglected, and the boundary conditions reduce to no ux conditions for the dislocation uxes at the boundary walls.

The simplicity is a result of the homogeneity in the y direction. That means the problem is assumed to be invariant by translation in the y direction. In order to better explain this point, we suppose that, if (S) is a cross sectional surface perpendicular to the dislocation lines, then the arrangement of dislocation points in (S) is invariant by translation in the y direction. In this case, we deduce that studying the dynamics of dislocation points on the line (L) (see Figure 2.2) gives the complete information of the dynamics of dislocation lines in the channel. As a summary, we can write that dynamics in (L) ⇒ dynamics in (S) ⇒ dynamics in the bounded channel. i.e. any dislocation interaction in the system is of short-range nature and hence it is described by the ow stress τ f and the gradient dependent stress τ b . Assuming moreover that τ ext = 0, system (2.6) becomes

       ∂ϕ ∂t + ∂ ∂x [ ψ { -τ f - 1 ϕ ∂ψ ∂x }] = 0, ∂ψ ∂t + ∂ ∂x [ ϕ { -τ f - 1 ϕ ∂ψ ∂x }] = 0. (2.7) Since τ f scales as √ ϕ, (2.7) can be rewritten        ∂ϕ ∂t + ∂ ∂x [ ψ { - √ ϕ - 1 ϕ ∂ψ ∂x }] = 0, ∂ψ ∂t + ∂ ∂x [ ϕ { - √ ϕ - 1 ϕ ∂ψ ∂x }] = 0. (2.8) Writing (2.8) in terms of θ ± , we obtain        θ + t = [( √ θ + + θ -+ θ + x -θ - x θ + + θ - ) θ + ] x , θ - t = [ - ( √ θ + + θ -+ θ + x -θ - x θ + + θ - ) θ - ] x .
(2.9)

We now pose

ρ ± x = θ ± , ρ = ρ + -ρ -, κ = ρ + + ρ -,
where ρ + , ρ -are the unknown scalars, that we denote for simplicity by ρ ± . Here, the dierence ρ + -ρ -represents the plastic deformation.

Subtract the two equations of (2.9) and integrate with respect to the space variable, we obtain

θ + t -θ - t = [( √ θ + + θ -+ θ + x -θ - x θ + + θ - ) ( θ + + θ -) ] x , hence ∫ x 0 ( θ + t -θ - t ) dy = ( √ θ + + θ -+ θ + x -θ - x θ + + θ - ) ( θ + + θ -) , thus we get ∂ ∂t ∫ x 0 ( θ + -θ -) dy = ( √ θ + + θ -+ θ + x -θ - x θ + + θ - ) ( θ + + θ -) , therefore ∂ ∂t ∫ x 0 ( ρ + x -ρ - x ) dy = ( √ ρ + x + ρ - x + ρ + xx -ρ - xx ρ + x + ρ - x ) ( ρ + x + ρ - x ) , then ∂ ∂t ∫ x 0 ρ x dy = ( √ κ x + ρ xx κ x ) κ x , consequently ρ t = κ x √ κ x + ρ xx .
(2.10)

Now, let's add the two equations of (2.9) and integrate with respect to the space variable, we obtain

θ + t + θ - t = [( √ θ + + θ -+ θ + x -θ - x θ + + θ - ) ( θ + -θ -) ] x , hence ∫ x 0 ( θ + t + θ - t ) dy = ( √ θ + + θ -+ θ + x -θ - x θ + + θ - ) ( θ + -θ -) , thus we get ∂ ∂t ∫ x 0 ( θ + + θ -) dy = ( √ θ + + θ -+ θ + x -θ - x θ + + θ - ) ( θ + -θ -) , therefore ∂ ∂t ∫ x 0 ( ρ + x + ρ - x ) dy = ( √ ρ + x + ρ - x + ρ + xx -ρ - xx ρ + x + ρ - x ) ( ρ + x -ρ - x ) , then ∂ ∂t ∫ x 0 κ x dy = ( √ κ x + ρ xx κ x ) ρ x , subsequently κ t = ρ x √ κ x + ρ x ρ xx κ x ,
and we conclude that

κ t κ x = ρ t ρ x .
(2.11)

After joining (2.10) and (2.11), we obtain the following system

{ κ t κ x = ρ t ρ x , ρ t = ρ xx + κ x √ κ x .
(2.12)

Remark that, in order to do the above calculations, θ + and θ -are assumed to be suciently regular. Consider the domain

I T = I × (0, T ), I = (0, 1), T > 0.
Regularizing equation (2.9) by adding εθ ± xx and writing down the equations in ρ and κ, we get (for

T > 0)    κ t = εκ xx + ρ x ρ xx κ x + ρ x √ κ x on I T , ρ t = (1 + ε)ρ xx + κ x √ κ x on I T .
(2.13)

The initial data are ρ(x, 0) = ρ 0 (x) and κ(x, 0) = κ 0 (x) x ∈ I, (2.14) and the boundary data are of Dirichlet type

ρ(0, t) = ρ(1, t) = κ(0, t) = 0 and κ(1, t) = 1 t > 0.
(

The Dirichlet boundary conditions are derived from the fact that the physical model is balanced. Thus, it starts with the same number of positive and negative dislocations. To be more precise, let n + and n -be the total number of positive and negative dislocations respectively at t = 0. We assume that there is neither annihilation nor creation of dislocations inside the material. Hence there is a conservation of n + and n -with respect to time. This could be mathematically formulated as follows

ρ(1, t) -ρ(0, t) = ∫ 1 0 ρ x (x, t) dx = ∫ 1 0 (θ + (x, t) -θ -(x, t)) dx = n + -n -= 0.
Also, the walls of the channel are impenetrable by dislocations, then the dislocation uxes at the boundary must be zero, which enables us to deduce that

ρ t = κ t = 0 on ∂I × [0, T ],
hence ρ and κ are constants along the boundary walls. This gives inspiration that the convenient boundary conditions of our system are the Dirichlet boundary conditions.

Groma, Czikor and Zaiser [START_REF] Groma | Spatial correlations and higher-order gradient terms in a continuum description of dislocations dynamics[END_REF] have stepped forward in modelling the consequence of short range dislocation-dislocation interactions. Considering the motion of individual dislocations, they have established a continuum description of the dislocation densities.

However, for a system of straight parallel dislocations, Groma and Balogh [START_REF] Groma | Link between the individual and continuum approaches of the description of the collective behavior of dislocations[END_REF][START_REF] Groma | Investigation of dislocation pattern formation in a two-dimensional self-consistent eld approximation[END_REF] have demonstrated a continuum description derived from the equations of motion of individual dislocations. The disadvantage of this result was neglecting short range dislocationdislocation interactions and their eect on the boundary, for getting a closed set of equations, hence, banning the mathematical formulation of the real mechanism occurring near the boundary of a material. From another side and in the case of three dimensions, a continuum description of a system of curved dislocations was formulated by El-Azab [START_REF] Azab | Statistical mechanics treatment of the evolution of dislocation distributions in single crystals[END_REF],

Monneau [START_REF] Monneau | A transport formulation for moving fronts and applications to dislocation dynamics[END_REF], M. Koslowski et al [START_REF] Koslowski | Dislocation structures and the deformation of materials[END_REF] and M. Zaiser and T. Hochrainer [START_REF] Hochrainer | A continuum theory of dislocation motion[END_REF][START_REF] Zaiser | Fundamental of a continuum theory of dislocations[END_REF][START_REF] Zaiser | Some steps towards a continuum representation of 3d dislocation systems[END_REF]]. Yet, there are still many opened problems in studying the theory of dislocation dynamics in the three dimensional case.

As a conclusion, Groma, Czikor and Zaiser model deals with relatively close dislocations due to their existence in a small bounded length. Thus, long range dislocationdislocation interactions are zero. However, in order to study the dynamics of dislocation densities under the eect of long range interactions, we use the model of Groma and Balogh [START_REF] Groma | Link between the individual and continuum approaches of the description of the collective behavior of dislocations[END_REF][START_REF] Groma | Investigation of dislocation pattern formation in a two-dimensional self-consistent eld approximation[END_REF], which will be presented in the next section.

Groma-Balogh model

Inspired by Groma and Balogh [START_REF] Groma | Link between the individual and continuum approaches of the description of the collective behavior of dislocations[END_REF][START_REF] Groma | Investigation of dislocation pattern formation in a two-dimensional self-consistent eld approximation[END_REF], we present the modelization process of our system in the two dimensional case which will be reduced to a one dimensional coupled system upon taking into consideration the periodicity of the domain and the dependence of dislocation densities on a single variable x = x 1 + x 2 , where (x 1 , x 2 ) are the coordinates of a point in R 2 (see gure 2.3). For simplication, suppose that there exists N dislocations of positive type as well as of negative one, and denote by ⃗ r ± i the position of a dislocation of type ± in the plane ( ⃗ b, ⃗ b ⊥ ) where ⃗ b ⊥ is a vector orthogonal to Burgers vector ⃗ b. For every vector ⃗ r, dene the discrete density function of type ± by

θ ±,d (⃗ r) = N ∑ i=1 δ(⃗ r -⃗ r ± i ).
Remark that for every i = 1, ..., N , we have

d dt δ(⃗ r -⃗ r ± i ) = -∇. [ d dt (⃗ r ± i )δ(⃗ r -⃗ r ± i )
] ,

with d dt (⃗ r ± i ) = ±( ⃗ bv)
where v is the dislocation speed. Indeed, let ϕ be any test function

< d dt δ(⃗ r -⃗ r ± i ), ϕ(⃗ r) > D ′ ,D = d dt ϕ(⃗ r ± i ) = d dt (⃗ r ± i ).∇ ⃗ r ϕ(⃗ r ± i ) = - d dt (⃗ r ± i ) < ∇ ⃗ r δ(⃗ r -⃗ r ± i ), ϕ(⃗ r) > D ′ ,D =-< d dt (⃗ r ± i )∇ ⃗ r δ(⃗ r -⃗ r ± i ), ϕ(⃗ r) > D ′ ,D .
(

The speed is proportional to Peach-Koehler force exerted on the dislocation. In the case of null exterior constraints, this force is simply the one created by the elastic eld generated by the dislocation itself (see M. Peach and J. S. Koehler [START_REF] Peach | The forces exerted on dislocations and the stress elds produced by them[END_REF]). From what precedes, we get

d dt θ ±,d = ∓∇.
[ ⃗ bvθ ±,d ] .

(2.17)

Suppose that the continuous density of dislocations θ ± , which is considered as a regularization of the discrete density θ ±,d , veries the same evolution equation

d dt θ ± = ∓∇. [ ⃗ bvθ ±, ] .
Our next step is to calculate Peach-Koehler force v. Consider a crystal with periodic deformation where its total displacement

U = (U 1 , U 2 ) : R × R + → R 2 can be decomposed into a 1-periodic displacement u = (u 1 , u 2
) and a linear displacement A(t) t X, where X = (x 1 , x 2 ) is a vector and A(t) is a given 2 × 2 matrix representing the shear stress

A(t) = ( A 11 (t) A 12 (t) A 21 (t) A 22 (t)
) .

The displacement U is then given by

U (X, t) = u(X, t) + A(t) t X,
and the total strain is dened by

ε(U ) = 1 2 (∇U + t ∇U ) = 1 2 ( ∇u + t ∇u + A(t) + t A(t)
) .

The coecients of ∇u are (∇u

) ij = ∂u i ∂x j , i, j ∈ {1, 2}. The total stain ε(U ) is decomposed into two parts ε(U ) = ε e (U ) + ε p ,
where ε e (U ) is the elastic deformation and ε p is the plastic deformation given by

ε p = ε 0 (ρ + -ρ -) .
(2.18)

such that ρ + and ρ -are scalars representing the displacement of the dislocations in the directions + ⃗ b and -⃗ b respectively. The density of dislocation of type ± is given by θ ± = ⃗ b.∇ρ ± 0 and the dierence ρ + -ρ -characterizes the plastic deformation. Here, ε 0 denotes the shear matrix dened in the case of a single sliding direction by

ε 0 = 1 2 ( ⃗ b ⊗ ⃗ b ⊥ + ⃗ b ⊥ ⊗ ⃗ b ) , ( 2.19)
where

( ⃗ b ⊗ ⃗ b ⊥ ) ij = b i b ⊥ j .
The stress is then given by σ = Λ : ε e (U ).

(

In other words, the elasticity constraints σ ij satisfy

σ ij = ∑ k,l∈{1,2} Λ ijkl ε e kl (U ) i, j ∈ {1, 2}, with Λ = (Λ ijkl ) ijkl , i, j, k, l = 1, 2.
Here, Λ ijkl are the elastic constant coecients of the material, satisfying the following symmetric property

Λ ijkl = Λ jikl = Λ ijlk = Λ klij ,
and, for every symmetric matrix ε = (ε ij ) ij and m > 0, the following coercivity hypothesis ∑ ijkl=1,2

Λ ijkl ε ij ε kl m ∑ ij=1,2 ε 2 ij .
(2.21)

The stress σ satises Euler-Lagrange equation div σ = 0.

Taking into consideration the elastic energy dened by

E el = 1 2 ∫ R 2 (Λ : ε e (u)) : ε e (u),
we get, by referring to [START_REF] Alvarez | Dislocation dynamics: short-time existence and uniqueness of the solution[END_REF], that Peach-Koehler force is

v = ∇ ρ ± E el = -(σ : ε 0 ). (2.22)
Hence, the functions ρ ± and u are solutions of the following coupled system (see Groma, Balogh [START_REF] Groma | Link between the individual and continuum approaches of the description of the collective behavior of dislocations[END_REF][START_REF] Groma | Investigation of dislocation pattern formation in a two-dimensional self-consistent eld approximation[END_REF] and Groma [START_REF] Groma | Link between the microscopic and mesoscopic length-scale description of the collective behavior of dislocations[END_REF])

               div σ = 0 in R 2 × (0, T ), σ = Λ : (ε(U ) -ε p ) in R 2 × (0, T ), ε(U ) = 1 2 (∇u + t ∇u + A(t) + t A(t)) in R 2 × (0, T ), ε p = ε 0 (ρ + -ρ -) in R 2 × (0, T ), (ρ ± ) t = ±(σ : ε 0 ) ⃗ b.∇ρ ± in R 2 × (0, T ), (2.23)
that is equivalent to

                                   ∑ j=1,2 ∂σ ij ∂x j = 0 in R 2 × (0, T ), σ ij = ∑ k,l∈{1,2} Λ ijkl (ε kl (U ) -ε p kl ) in R 2 × (0, T ), ε ij (U ) = 1 2 ( (∇u) ij + (∇u) ij + A ij (t) + A ji (t) ) in R 2 × (0, T ), ε p ij = ε 0 ij (ρ + -ρ -) in R 2 × (0, T ), (ρ ± ) t = ±   ∑ i,j∈{1,2} σ ij ε 0 ij  ⃗ b.∇ρ ± in R 2 × (0, T ), (2.24)
where the unknowns of the system are ρ ± and u.

           (ρ + ) t = -C 1 ( (ρ + -ρ -) + C 2 ∫ 1 0 (ρ + -ρ -) + L(t)
)

Dρ + in R × (0, T ), (ρ -) t = C 1 ( (ρ + -ρ -) + C 2 ∫ 1 0 (ρ + -ρ -) + L(t) ) Dρ - in R × (0, T ), (2.26)
where

L(t) = -(λ+2µ) (λ+µ) (A 12 (t) + A 21 (t)), C 1 = (λ+µ)µ λ+2µ and C 2 = µ (λ+µ) .
Proof Plugging (2.25) into the rst equation of (2.23) gives

div (2µε(U ) + λtr(ε(U ))I d ) = div (2µε p + λtr(ε p )I d ) .
By means of (2.18), we get

µ∆u + (λ + µ)∇(div u) = µ      ∂(ρ + -ρ -) ∂x 2 ∂(ρ + -ρ -) ∂x 1      .
Using the fact that x =

x 1 + x 2 yields 2µ      ∂ 2 u 1 ∂x 2 ∂ 2 u 2 ∂x 2      + (λ + µ)      ∂ 2 (u 1 + u 2 ) ∂x 2 ∂ 2 (u 1 + u 2 ) ∂x 2      = µ      ∂(ρ + -ρ -) ∂x ∂(ρ + -ρ -) ∂x      .
Now, by adding the above two equations, we obtain

∂ 2 (u 1 + u 2 ) ∂x 2 = µ λ + 2µ ( ∂(ρ + -ρ -) ∂x
) .

Regarding the periodicity of u and integrating the above equation

∂(u 1 + u 2 ) ∂x = µ λ + 2µ ( (ρ + -ρ -) - ∫ 1 0 (ρ + -ρ -)
) .

(

Using

(σ : ε 0 ) = σ 12 = 2µ(ε e (U )) 12 = µ ( ∂(u 1 + u 2 ) ∂x + A 12 (t) + A 21 (t) -(ρ + -ρ -)
) ,

and (2.27) results in

(σ : ε 0 ) = - (λ + µ)µ λ + 2µ ( (ρ + -ρ -) + µ 2(λ + µ) ∫ 1 0 (ρ + -ρ -) + L(t)
)

,

where

L(t) = - (λ + 2µ) (λ + µ) (A 12 (t) + A 21 (t)). Thus, for ⃗ b = (1, 0), system (2.23
) can be transformed to (2.26). Without loss of generality, the two positive constants C 1 and C 2 will be considered 1. 2

We realize that Groma and Balogh model describes the dynamics of dislocation densities in a unique direction of propagation. For an extended study of dislocation dynamics in multi-directions of propagation, refer to S. Yemov [START_REF] Yemov | Discrete dislocation and nonlocal crystal plasticity modelling[END_REF] and S. Yemov and E. Van der Giessen [START_REF] Yemov | Multiple slip in a strain gradient plasticity model motivated by a statistical mechanics description od dislocations[END_REF]. Also, we have established a model that deals with dislocation dynamics in multi-directions of propagation as well as gas dynamics (see Chapter 5). ena when stress elds are applied. The typical thickness of a dislocation line is of order of 10 -9 m and its typical length is of order of 10 -6 m. Mathematically, this theory was originally developed by Volterra [START_REF] Volterra | Sur l'équilibre des corps élastiques multiplement connexes[END_REF] in 1907. In 1934, Orowan [START_REF] Orowan | Kristallplastizitat[END_REF], Polanyi [START_REF] Polanyi | Uber eine art gitterstorung die einem kristall plastisch machen konnte[END_REF] and Taylor [START_REF] Taylor | The mechanism of plastic deformation of crystals[END_REF] introduced the principal explanation of macroscopic plastic deformation in crystals as a result of the motion of dislocations. The rst direct observation of dislocations using electron microscope goes back to Hirsch, Horne and Whelan [START_REF] Hirsch | Direct observations of the arrangement and motion of dislocations in aluminium[END_REF] and Bollmann [START_REF] Bollmann | Interference eects in the electron microscopy of thin crystal foils[END_REF] in 1956. For a recent study of the physical theory of dislocations, we refer the reader to Nabarro [START_REF] Nabarro | Theory of crystal dislocations[END_REF] and Hirth and Lothe [START_REF] Hirth | Theory of dislocations[END_REF]. However, for a mathematical study of various models on dislocations, we refer to [START_REF] Cannone | Global existence of non-linear and non-local transport equations describing the dynamics of dislocation densities[END_REF][START_REF] Azab | Statistical mechanics treatment of the evolution of dislocation distributions in single crystals[END_REF][START_REF] Hajj | Well-posedness theory for a non-conservative Burgers type system arising in dislocation dynamics[END_REF][START_REF] Hajj | A convergent scheme for a non-local coupled system modeling dislocations densities dynamics[END_REF][START_REF] Monneau | A transport formulation for moving fronts and applications to dislocation dynamics[END_REF].

In our work, we are interested in the mathematical study of a model on dislocation densities developed by Groma, Czikor and Zaiser (GCZ for short) in [START_REF] Groma | Spatial correlations and higher-order gradient terms in a continuum description of dislocations dynamics[END_REF] that describes the short-range interactions. This model is written as a coupled system of nonlinear parabolic equations on a bounded domain in one-space dimension. Several variants of the GCZ models have been treated in the work of [START_REF] Hajj | Dislocation dynamics: from microscopic models to macroscopic crystal plasticity[END_REF][START_REF] Hajj | Derivation and study of dynamical models of dislocation densities[END_REF][START_REF] Ibrahim | Existence and uniqueness for a non-linear parabolic/Hamilton-Jacobi coupled system describing the dynamics of dislocation densities[END_REF] where particular assumptions on the exterior stress eld have been involved. In fact, in their work, they only considered either constant or bounded space-time dependent stresses. This case was far from the real stress in [START_REF] Groma | Spatial correlations and higher-order gradient terms in a continuum description of dislocations dynamics[END_REF] which scales as a nonlinear term (that we precise later) depending on the dislocation density. To our knowledge, our result is the rst theoretical one of its type, treating the real physical stress eld where we overcome the nonlinear diculty.

Our main result is to prove the short time existence and uniqueness using a xed point argument and a comparison principle on the gradient of the solution.

This paper is organized as follows. Section 3.2 is devoted to the derivation and the general setting of our problem. The principal notations and results are detailed in Section -→ ξ are supposed to be perpendicular to x and y. The Burgers vector -→ b is parallel to x-direction. The direction of Burgers vector determine the classication (positive or negative) of edge dislocations. On our work, we derive our mathematical model on dislocation dynamics with short-range interaction. All our computations are inspired by GCZ [START_REF] Groma | Spatial correlations and higher-order gradient terms in a continuum description of dislocations dynamics[END_REF] where we suppose a particular conguration of the dislocation lines.

As we stated in the modelization chapter, we set θ + and θ -to be the densities of the 3.2. The modeling positive and negative dislocations respectively. Denote by ϕ = θ + + θ -, the total dislocation density and by

ψ = θ + -θ -,
the sign dislocation density.

In the case of short range dislocation-dislocation interactions, the model satised by ϕ and ψ can be written as

     ∂ϕ ∂t (x, t) + ∂ ∂x [ψ(x, t){τ sc (x, t) + τ ext -τ f + τ b }] = 0, ∂ψ ∂t (x, t) + ∂ ∂x [ϕ(x, t){τ sc (x, t) + τ ext -τ f + τ b }] = 0, (3.1)
where the self-consistent stress τ sc , the external stress τ ext , the local ow stress τ f and the back stress τ b are dened in the modelization chapter.

In a special physical setting depending on the internal material structure and the frame of reference, we may assume ⃗ b = (1, 0) and we set all constants to be 1. This will also simplify the presentation of the mathematical model to obtain

       ∂ϕ ∂t (x, t) + ∂ ∂x [ ψ(x, t) { τ sc (x, t) + τ ext -τ f - 1 ϕ ∂ψ ∂x }] = 0, ∂ψ ∂t (x, t) + ∂ ∂x [ ϕ(x, t) { τ sc (x, t) + τ ext -τ f - 1 ϕ ∂ψ ∂x }] = 0. (3.2)
We consider a constrained channel of width L in the x direction and innite extension in the y direction, bounded by walls that are impenetrable by dislocations (the plastic deformation at the walls is zero). The slip direction corresponds to the x direction, and the layer is sheared by a constant shear stress.

A simplied model

The system (3.2) is a simplied version of a system studied by Van der Giessen and co-workers using both discrete dislocation and continuum plasticity approaches. The simplications stem out from the fact that only a single slip system is assumed to be active, such that reactions between dislocations need not to be considered, and that the boundary conditions reduce to "no ux" conditions for the dislocation uxes at the boundary walls.

Recall that the system envisaged is particularly simple also because it is homogeneous in the y direction. In other words, the problem is assumed to be invariant by translation in the y direction.

In this case the long-range self-consistent stress eld τ sc is zero. That means, any dislocation interaction in the system is of short-range nature and hence it is described by the ow stress τ f and the gradient dependent stress τ b . Assuming moreover that τ ext = 0,

system (3.2) becomes        ∂ϕ ∂t + ∂ ∂x [ ψ { -τ f - 1 ϕ ∂ψ ∂x }] = 0, ∂ψ ∂t + ∂ ∂x [ ϕ { -τ f - 1 ϕ ∂ψ ∂x }] = 0. (3.3) Since τ f scales as √ ϕ, (3.3) can be rewritten        ∂ϕ ∂t + ∂ ∂x [ ψ { - √ ϕ - 1 ϕ ∂ψ ∂x }] = 0, ∂ψ ∂t + ∂ ∂x [ ϕ { - √ ϕ - 1 ϕ ∂ψ ∂x }] = 0. (3.4) Writing (3.4) in terms of θ ± , we obtain        θ + t = [( √ θ + + θ -+ θ + x -θ - x θ + + θ - ) θ + ] x , θ - t = [ - ( √ θ + + θ -+ θ + x -θ - x θ + + θ - ) θ - ] x . ( 3.5)
We now pose

ρ ± x = θ ± , ρ = ρ + -ρ -, κ = ρ + + ρ -,
where ρ + , ρ -are the unknown scalars, that we denote for simplicity by ρ ± . Here, the dierence ρ + -ρ -represents the plastic deformation.

Similar to the procedure done in the modeling of GCZ model in Chapter 2, subtracting the two equations of (3.5) and integrate with respect to the space variable, gives

ρ t = κ x √ κ x + ρ xx .
On the other hand, adding the two equations of (3.5) and integrating with respect to the space variable results in

κ t κ x = ρ t ρ x . System (3.5) now becomes { κ t κ x = ρ t ρ x , ρ t = ρ xx + κ x √ κ x . (3.6)
Remark that, in order to do the above calculations, θ + and θ -are assumed to be sufciently regular, say C 2+α, 2+α 2 , where C l,l/2 is the Hölder space that will be dened in

T > 0)    κ t = εκ xx + ρ x ρ xx κ x + ρ x √ κ x on I T , ρ t = (1 + ε)ρ xx + κ x √ κ x on I T . (3.7)
The initial data are ρ(x, 0) = ρ 0 (x) and κ(x, 0) = κ 0 (x) x ∈ I, (3.8) and the boundary data are of Dirichlet type

ρ(0, t) = ρ(1, t) = κ(0, t) = 0 and κ(1, t) = 1 t > 0.
(3.9)

The Dirichlet boundary condition is derived from the fact that the physical model is balanced. Thus, it starts with the same number of positive and negative dislocations. To be more precise, let n + and n -be the total number of positive and negative dislocations respectively at t = 0. We assume that there is neither annihilation nor creation of dislocations inside the material. Hence there is a conservation of n + and n -with respect to time. This could be mathematically formulated as follows

ρ(1, t) -ρ(0, t) = ∫ 1 0 ρ x (x, t) dx = ∫ 1 0 (θ + (x, t) -θ -(x, t)) dx = n + -n -= 0.
Also, the walls of the channel are impenetrable by dislocations, then the dislocation uxes at the boundary must be zero, which enables us to deduce that

ρ t = κ t = 0 on ∂I × [0, T ],
hence ρ and κ are constants along the boundary walls. This gives inspiration that the convenient boundary conditions of our system are the Dirichlet boundary conditions.

Notations and the main results

Notation

Let T > 0 and I T is the cylinder I × (0, T ) with I = (0, 1); I is the closure of I; I T is the closure of I T ; ∂I is the boundary of I. The indicator function 1 1 A of the set A ⊆ R is used when convenient. The letter c will denote constant taking dierent values, to be determinate in each case. We denote by D k z (u) = ∂ k u ∂z k , where u is a function depending on the parameter z and k ∈ N. We will use the following spaces

• The Lebesgue spaces L p (I T ), 1 p +∞ with norms ∥ • ∥ L p (I T ) ; • The parabolic Sobolev space Y , dened by : Y = W 2,1 p (I T ) = {u ∈ L p (I T ); D r t D s x u ∈ L p (I T ) for 2r + s 2}, equipped with the norm ∥u∥ Y = ∑ 2r+s 2 ∥D r t D s x u∥ L p (I T ) ;
• The Hölder space C l,l/2 (I T ), l > 0 a non-integral positive number, is the Banach space of functions v(x, t) that are continuous in I T , together with all derivatives of the form D r t D s x v for 2r + s < l, and have a nite norm |v| (l)

I T = ⟨v⟩ (l) I T + ∑ [l] j=0 ⟨v⟩ (j) I T ,
where ⟨v⟩ (0)

I T = |v| (0) I T = ∥v∥ ∞,I T , ⟨v⟩ (j)
I T = ∑ 2r+s=j |D r t D s x v| (0) I T , ⟨v⟩ (l)
I T = ⟨v⟩ (l)
x,I T + ⟨v⟩

(l/2)
t,I T , and ⟨v⟩ (l)

x,I T = ∑ 2r+s=[l] ⟨D r t D s x v⟩ (l-[l])
x,I T , ⟨v⟩

(l/2) t,I T = ∑ 0<l-2r-s<2 ⟨D r t D s x v⟩ ( l-2r-s 2 ) t,I T , with ⟨v⟩ (α) x,I T = inf{c; |v(x, t) -v(x ′ , t)| c|x -x ′ | α , (x, t), (x ′ , t) ∈ I T }, 0 < α < 1, ⟨v⟩ (α) t,I T = inf{c; |v(x, t) -v(x, t ′ )| c|t -t ′ | α , (x, t), (x, t ′ ) ∈ I T }, 0 < α < 1.
In addition to these notations, let us give a brief review on the compatibility conditions used afterwards. Consider the following typical problem in parabolic theory

     u t = εu xx + f on I T u(x, 0) = ϕ on I u = Φ on ∂I × (0, T ), (3.10)
where T > 0 and ε > 0.

Compatibility condition of order 0. Let ϕ ∈ C( Ī) and Φ ∈ C(∂I × [0, T ]). We say that the compatibility condition of order 0 is satised if

ϕ ∂I = Φ t=0 . (3.11) Compatibility condition of order 1. Let ϕ ∈ C 2 ( Ī), Φ ∈ C 1 (∂I × [0, T ]) and f ∈ C(I T ).
We say that the compatibility condition of order 1 is satised if (3.11) is satised and in addition we have

(εϕ xx + f ) ∂I = ∂Φ ∂t t=0 .
(3.12)

The main result

We rst present a result of short time existence and uniqueness of solutions to problem Let ρ and κ ∈ C ∞ (I), such that ρ (0) = ρ (1) = 0 and κ (0) = 0, κ (1) = 1 with

κ x > |ρ x | on I. (3.13) Assume { (1 + ε)ρ xx + κ x √ κ x = 0 on ∂I, (1 + ε)κ xx + ρ x √ κ x = 0 on ∂I. (3.14)
Then there exists a short time T > 0 and a unique solution (ρ, κ) of problem (3.7)-(3.9), satisfying

(ρ, κ) ∈ ( C 3+α, 3+α 2 (I T ) ∩ C ∞ (I × (0, T )) ) 2 ,
with 0 < α < 1 and

κ x > |ρ x | on I × (0, T ).
(3.15)

A comparison principle

In this section, and for symmetry reasons, we take I = (-1, 1) and

β δ (x) := √ x 2 + δ 2 , x, δ ∈ R.
Some basic properties on the positive function β δ will be frequently utilized. Mainly

{ β δ (x)β ′ δ (x) = x, β 3 δ (x)β ′′ δ (x) = δ 2 .
Proposition 3.4.1. Let (ρ, κ) be a regular solution of (3.7) on the compact I T with κ x > 0, and the initial data (ρ 0 , κ 0 ) satises

κ 0 x β γ 0 (ρ 0 x ), γ 0 ∈ (0, 1). (3.16)
Then there exists a C 1 positive function γ : [0, T ] → ]0, +∞[ such that

κ x β γ (ρ x ) on I T . (3.17)
In all what follows, we use the following notation

M := κ x -β γ (ρ x ), (3.18)
where the function γ is to be determined in a way that κ x β γ (ρ x ) on I T (i.e. M 0).

Proof

Assuming the regularity of ρ, κ and γ, we compute

     M t = κ xt -β ′ γ (ρ x )ρ xt -Γ, M x = κ xx -β ′ γ (ρ x )ρ xx , M xx = κ xxx -β ′′ γ (ρ x )ρ 2 xx -β ′ γ (ρ x )ρ xxx , ( 3.20)
where

Γ = γγ ′ √ ρ 2 x + γ 2 . (3.21)
Dierentiating (3.7) with respect to x, we easily obtain

       κ xt = εκ xxx + ρ 2 xx κ x + ρ x ρ xxx κ x - ρ x ρ xx κ xx κ 2 x + ρ xx √ κ x + ρ x κ xx 2 √ κ x , ρ xt = (1 + ε)ρ xxx + 3 2 κ xx √ κ x .
M t = εM xx + AM x + B M + L + L 2 -Γ, (3.23)
with

                               A = - ρ x ρ xx κ 2 x + ρ x 2 √ κ x - 3 2 β ′ γ (ρ x ) √ κ x , B = ρ 2 xx κ 2 x + ρ xx √ κ x - β ′ γ (ρ x )ρ xxx κ x , L = β γ (ρ x )ρ 2 xx κ 2 x - β ′ γ (ρ x )ρ x ρ 2 xx κ 2 x , L = εβ ′′ γ (ρ x )ρ 2 xx + L 3 , L ! = β ′ γ (ρ x )ρ x ρ xx 2 √ κ x + β γ (ρ x )ρ xx √ κ x - 3 2 (β ′ γ (ρ x )) 2 ρ xx √ κ x . (3.24)
By estimating the term L in (3.24), We write

L = ρ 2 xx κ 2 x (β γ (ρ x ) -ρ x β ′ γ (ρ x )) = ρ 2 xx κ 2 x γ 2 √ ρ 2 x + γ 2 0. (3.25)
The estimate of L 2 requires to estimate L 3 rst. In fact we have

L ! = ρ xx 2 √ κ x ( β ′ γ (ρ x )ρ x + 2β γ (ρ x ) -3(β ′ γ (ρ x )) 2 κ x
) .

A simple computation shows that:

β ′ γ (ρ x )ρ x + 2β γ (ρ x ) = 3ρ 2 x + 2γ 2 β(ρ x ) ,
therefore, by elementary properties of the function β γ , we get

L ! = ρ xx 2 √ κ x ( 3ρ 2 x + 2γ 2 β(ρ x ) -3(β ′ γ (ρ x )) 2 κ x ) = ρ xx 2 √ κ x ( 3ρ x ( ρ x -β ′ γ (ρ x )κ x ) + 2γ 2 β γ (ρ x ) ) = ρ xx 2 √ κ x ( 3ρ 2 x ( β γ (ρ x ) -κ x ) + 2γ 2 β γ (ρ x ) β 2 γ (ρ x )
)

.

By using the denition of M , we nally arrive to

L ! = - ( 3ρ 2 x ρ xx 2β ′ γ (ρ x ) √ κ x ) M + γ 2 ρ xx β γ (ρ x ) √ κ x , hence L = εβ ′′ γ (ρ x )ρ 2 xx - ( 3ρ 2 x ρ xx 2β ′ γ (ρ x ) √ κ x ) M + γ 2 ρ xx β γ (ρ x ) √ κ x . ( 3
M t εM xx + AM x + B M + εβ ′′ γ (ρ x )ρ 2 xx + γ 2 ρ xx β γ (ρ x ) √ κ x -Γ, (3.27)
where

B = B - 3ρ 2 x ρ xx 2β ′ γ (ρ x ) √ κ x .
To study of the term εβ ′′ γ (ρ x )ρ 2 xx in (3.27), we can easily write

εβ ′′ γ (ρ x )ρ 2 xx = εκ x β ′′ γ (ρ x )ρ 2 xx κ x = ε(M + β γ (ρ x ))β ′′ γ (ρ x )ρ 2 xx κ x = ( εβ ′′ γ (ρ x )ρ 2 xx κ x ) M + εγ 2 ρ 2 xx (ρ 2 x + γ 2 )κ x . Therefore (3.27) gives M t εM xx + AM x + BM + εγ 2 ρ 2 xx (ρ 2 x + γ 2 )κ x + γ 2 ρ xx β γ (ρ x ) √ κ x -Γ, (3.28)
where

B = B + εβ ′′ γ (ρ x )ρ 2 xx κ x . (3.29)
Finally, we use 0 < γ < 1 to estimate the term

γ 2 ρ xx β γ (ρ x ) √ κ x in (3.28). Then β γ(t) (ρ x (x, t)) = √ ρ 2 x (x, t) + γ 2 (t) α(t) ( ρ 2 x (x, t) + γ 2 (t) ) 3/4 ,
where

α(t) = 1 ( ∥ρ x (•, t)∥ 2 L ∞ (I) + 1 ) 1/4 .
We now compute

γ 2 |ρ xx | β γ (ρ x ) √ κ x γ 2 |ρ xx | α(t)(ρ 2 x + γ 2 ) 3/4 √ κ x = γ|ρ xx | (ρ 2 x + γ 2 ) 1/2 √ κ x • γ α(t)(ρ 2 x + γ 2 ) 1/4 ,
therefore, by Young's inequality, we get

γ 2 |ρ xx | β γ (ρ x ) √ κ x εγ 2 ρ 2 xx (ρ 2 x + γ 2 )κ x + γ 2 4εα 2 (t)(ρ 2 x + γ 2 ) 1/2 . (3.30)
We want to choose γ such that the dierential inequality in M is homogeneous. In fact, using (3.28) and (3.30), we can infer that

M t εM xx + AM x + BM - γ 2 4εα 2 (t)(ρ 2 x + γ 2 ) 1/2 -Γ. (3.31)
It is worth noticing that the term

εγ 2 ρ 2 xx (ρ 2 x + γ 2 )κ x in (3.
γ(t) = γ 0 e - ∫ t 0 ds 4εα 2 (s) (3.32)
we nally deduce (see (3.31) and the denition (3.21) of Γ) that M t εM xx + AM x + BM, (3.33) and, on the other side, we can easily justify our claim 0 < γ < 1 as γ 0 ∈ (0, 1). 2

Proof

Since ρ and κ are constants on the boundary ∂I × [0, T ], we obtain   

εκ xx + ρ x ρ xx κ x + ρ x √ κ x = 0 on ∂I × [0, T ], (1 + ε)ρ xx + κ x √ κ x = 0 on ∂I × [0, T ], therefore M x = √ κ x 1 + ε β ′ γ (ρ x )M on ∂I × [0, T ]. (3.34) Indeed, M x = κ xx -β ′ γ (ρ x )ρ xx = - ρ x √ κ x 1 + ε + β ′ γ (ρ x ) κ x √ κ x 1 + ε = √ κ x 1 + ε β ′ γ (ρ x ) ( -ρ x β ′ γ (ρ x ) + κ x ) = √ κ x 1 + ε β ′ γ (ρ x ) (-β γ (ρ x ) + κ x ) = √ κ x 1 + ε β ′ γ (ρ x )M.
Here we would like to show that the minimum of M (•, t) is not attained on the boundary and then to use (3.28) to show its positivity. However, the above boundary equation (3.34) carries no information that violates the presence of the minimum on ∂I. For this reason, we carefully multiply M by a suitable positive function having large values on the boundary. In particular, we consider M dened by

M (x, t) = cosh(θx)M (x, t), θ ∈ R.
Elementary computations show

M x = ΘM on ∂I × [0, T ], (3.35)
with

Θ = ( θ tanh(θx) + √ κ x 1 + ε β ′ γ (ρ x )
) .

The boundedness of

√ κ x 1 + ε β ′ γ (ρ x
) on I T permits the existence of θ large enough so that Θ(1, t) > 0 and Θ(-1, t) < 0 for all t ∈ [0, T ]. Hence, by (3.35), the function M (•, t) can not have a positive minimum on ∂I. 2

Proof of Proposition 3.4.1

First, we write the partial dierential inequality satised by M

M t εM xx + ( - ρ x ρ xx κ 2 x + ρ x 2 √ κ x - 3 2 β ′ γ (ρ x ) √ κ x -2θε tanh(θx) ) M x + [ ρ 2 xx κ 2 x + ρ xx √ κ x - β ′ γ (ρ x )ρ xxx κ x - 3ρ 2 x ρ xx 2β ′ γ (ρ x ) √ κ x + εβ ′′ γ (ρ x )ρ 2 xx κ x -θ tanh(θx) ( - ρ x ρ xx κ 2 x + ρ x 2 √ κ x - 3 2 β ′ γ (ρ x ) √ κ x ) + εθ 2 ( 2 tanh 2 (θx) -1 ) ] M . (3.36)
Due to the regularity of M , we may nd a curve t → (x 0 (t), t) such that

min x∈I M (x, t) = M (x 0 (t), t) = m(t) t ∈ [0, T ].
Without loss of generality, we may always assume κ 0

x > √ (ρ 0 x ) 2 + γ 2 0 .
In fact, it suces to adjust γ 0 in (3.16). Therefore m(0) > 0 and x 0 (0) ∈ I.

Again, the regularity of M ensures x 0 (t) ∈ I for all 0 t t 0 T.

We claim that t 0 = T . Indeed, if not, we get x 0 (t 0 ) ∈ ∂I. Let us show that this can not be true. Indeed, since x 0 (t) ∈ I for 0 t < t 0 we directly obtain M x (x 0 (t), t) = 0 and M xx (x 0 (t), t) 0.

Moreover, using (3.36), we obtain, for some constant c ∈ R, the following ordinary dierential inequality involving m m ′ cm for 0 t < t 0 , and therefore, using Grönwall's inequality, we get m(t) m(0)e ct for 0 t < t 0 .

(3.37)

Since m(0) > 0, the above inequality gives m(t 0 ) > 0. Consequently, (see Lemma 3.4.3)

x 0 (t 0 ) ∈ I and the claim holds true. Now as we get x 0 (t) ∈ I for all t ∈ [0, T ], the inequality (3.37) also holds true for all t ∈ [0, T ] with m(0) > 0. Hence, we can infer that M 0 on I T and therefore M 0 on I T . 2

Short time existence and uniqueness

In this section we prove the short time existence and uniqueness for (3.7), (3.8) and (3.9).

The main idea is to nd a solution of a truncated system where we carefully truncate the gradients ρ x and κ x . To be more precise, we consider, for real values 0 a b, the two real valued functions I a and ∈J a,b dened by

I a (x) = x1 1 {|x| a} + a1 1 {x a} -a1 1 {x -a} , and J a,b (x) = √ x1 1 {a x b} + √ b1 1 {x b} + √ a1 1 {x a} .
It is easily seen that I a is a truncation of the identity function, while J a,b is a truncation of √

x. Remark that both functions are bounded Lipschitz continuous on R, and this property will be repeatedly used hereafter in this section. The value p > 3 is taken to emphasis some regularity properties on the solution. Indeed, it is well known (see [START_REF] Ladyzenskaja | Linear and quasilinear equations of parabolic type, Translated from the Russian[END_REF] for the details) that if p > 3 then Y is continuously embedded in a parabolic Hölder space

Y → C 1+α, 1+α 2 (I T ), α = 1 - 3 p , ( 3.38)
with the following fundamental estimate, valid for u = 0 on the parabolic boundary

∂ p (I T ) = (∂I × [0, T ]) ∪ (I × {0}), that reads ∥u x ∥ L ∞ (I T ) cT p-3 2p ∥u∥ Y , ( 3.39)
where c = c(p) > 0 is a constant depending only on p.

Recall that in [62, Section 1], for l > 0, the parabolic Hölder space C l,l/2 (I T ) is the Banach space of functions v(x, t) that are continuous in I T , together with all derivatives of the form D r t D s x v for 2r + s < l, and have a nite norm |v| (l)

I T = ⟨v⟩ (l) I T + ∑ [l] j=0 ⟨v⟩ (j) I T ,
Another very useful inequality in our study is a Sobolev estimate for parabolic equations (see [START_REF] Ibrahim | Dynamics of dislocation densities in a bounded channel. Part 1. Smooth solutions to a singular coupled parabolic system[END_REF]). To state this estimate, we consider solutions u ∈ Y , u = 0 on ∂ p (I T ),

of u t = εu xx + f, f ∈ L p (I T ) called the source term, (3.40)
then we have

∥u∥ L p (I T ) T + ∥u x ∥ L p (I T ) √ T + ∥u xx ∥ L p (I T ) + ∥u t ∥ L p (I T ) c∥f ∥ L p (I T ) , (3.41)
where c = c(ε, p) > 0 is a constant depending only on p and ε.

Now, we may state the main proposition of this section Proposition 3.5.1. (Fixed point argument) Let p > 3 and let

ρ 0 , κ 0 ∈ C ∞ (I),
be two given functions such that ρ 0 (0) = ρ 0 (1) = κ 0 (0) = 0 and κ 0 (1) = 1. Suppose furthermore that

     κ 0 x γ 0 on I, ∥D s x ρ 0 ∥ L ∞ (I) M 0 , s = 1, 2, ∥D s x κ 0 ∥ L ∞ (I) M 0 , s = 1, 2, (3.42)
with 0 < γ 0 M 0 . Then there exists a unique solution (ρ, κ) ∈ Y 2 of (3.7), (3.8) and (3.9) where

T = T (M 0 , γ 0 , ε, p), 0 < T < 1.
Moreover, this solution satises

   γ 0 2 κ x 2M 0 on I T , |ρ x | 2M 0 on I T . (3.43)

Proof

Throughout the proof, and in various estimates we suppose that 0 < T < 1. This is in no way a problem since we have to choose T small enough to construct our solution.

The proof uses a xed point argument on a subspace of Y . Looking at (3.43), we can articially modify (3.7) using suitable truncations. To simplify our computations, we set

I := I 2M 0 and J := J γ 0 2 ,2M 0 ,
and we consider

   κ t = εκ xx + ρ xx I(ρ x ) (γ 0 /2) + (κ x -γ 0 /2) + + ρ x J (κ x ) on I T , ρ t = (1 + ε)ρ xx + κ x J (κ x ) on I T , ( 3
Y ρ = {u ∈ Y ; ∥u x ∥ L p (I T ) λ, u = ρ 0 on ∂ p (I T )},
and

Y κ = {u ∈ Y ; ∥u x ∥ L p (I T ) λ, u = κ 0 on ∂ p (I T )}.
Dene the application

Ξ : Y ρ × Y κ -→ Y ρ × Y κ , (ρ, κ) -→ Ξ(ρ, κ) = (ρ, κ), where (ρ, κ) is the solution of    κ t = εκ xx + ρ xx I(ρ x ) (γ 0 /2) + (κ x -γ 0 /2) + + ρ x J (κ x ) on I T , ρ t = (1 + ε)ρ xx + κx J (κ x ) on I T . (3.45)
The existence and uniqueness of (ρ, κ) In a second step, we use (3.41) basically on the functions ρ = ρ -ρ 0 and κ = κ -κ 0 , to gain the L p bounds on ρ x and κ x if we choose T small enough. The above steps ensure that the application Ξ is well dened at least for suciently small time. We now show that Ξ is a contraction. In fact, let Ξ(ρ, κ) = (ρ, κ) and Ξ(r, k) = (r, k). The couple (ρ -r, κ -k) is the solution of the following system

∈ Y ρ × Y κ solution of (3.
{ (κ -k) t = ε(κ -k) xx + F 1 on I T , (ρ -r) t = (1 + ε)(ρ -r) xx + F 2 on I T , ( 3.46)
where

F 1 = ρ xx I(ρ x ) (γ 0 /2) + (κ x -γ 0 /2) + - r xx I(r x ) (γ 0 /2) + ( kx -γ 0 /2) + + ρ x J (κ x ) -r x J ( kx ),
and

F 2 = κx J (κ x ) -kx J ( kx ), with (ρ -r, κ -k) = (0, 0) on ∂ p (I T ).
In the remaining part of the proof, the constant c > 0 will always denote a quantity that depends on all constants in Proposition 3.5.1 but independent of T . Using (3.46) and (3.41), we deduce that

{ ∥κ -k∥ Y c∥F ∥ L p (I T ) , ∥ρ -r∥ Y c∥F ∥ L p (I T ) .
(3.47)

We now estimate the term ∥ρ -r∥ Y . In fact, since J is bounded Lipschitz continuous, we can infer that

|F 2 | |κ x ||J (κ x ) -J ( kx )| + |J ( kx )||κ x -kx | Lip(J )|κ x ||κ x -kx | + √ 2M 0 |κ x -kx |, therefore ∥F 2 ∥ L p (I T ) c(∥κ x ∥ L p (I T ) + 1)∥(κ -κ′ ) x ∥ L ∞ (I T ) .
By (3.39) and the fact that ∥κ x ∥ L p (I T ) λ, we deduce that (

∥F 2 ∥ L p (I T ) c(λ + 1)T p-3 2p ∥κ -κ′ ∥ Y . ( 3
We need to estimate also the term ∥κ -k∥ Y . We write

F = A + A + A ! + A " , with                          A = I(ρ x ) (γ 0 /2) + (κ x -γ 0 /2) + (ρ xx -r xx ) , A = r xx (I(ρ x ) -I(r x )) (γ 0 /2) + (κ x -γ 0 /2) + , A ! = r xx I(r x ) ( 1 (γ 0 /2) + (κ x -γ 0 /2) + - 1 (γ 0 /2) + ( kx -γ 0 /2) + ) , A " = ρ x (J (κ x ) -J ( kx )) + J ( kx )(ρ x -r x ). (3.50) We estimate the L p norms of A i , i = 1, 2, 3, 4. First remark that the coecient of (ρ xx -r xx ) in A 1 is bounded in L p (I T ) by (3.49), hence we have ∥A 1 ∥ L p (I T ) cT p-3 2p ∥κ -k∥ Y . (3.51)
Also remark that r solves the second equation of (3.45) with κ replaced by k, hence (after possibly using the Sobolev estimate (3.41) for r = r -ρ 0 ) we deduce that

∥r xx ∥ L p (I T ) c(M 0 + λ √ 2M 0 ),
therefore, using (3.39) and the fact that I is Lipschitz, we get

∥A ∥ L p (I T ) cT p-3
2p ∥ρ -r∥ Y .

(3.52)

For A ! , we use the same arguments as for A , and we are lead to

∥A ! ∥ L p (I T ) cT p-3
2p ∥κ -k∥ Y .

(3.53)

Here we have used that the function x → 1 (γ 0 /2)+(x-γ 0 /2) + is Lipschitz continuous over R. For A " , we notice that the rst term in the right hand side is treated as in A , while the second term is treated as in A . Hence

∥A " ∥ L p (I T ) cT p-3 2p ∥κ -k∥ Y .
κ x √ κ x ∈ C α,α/2 (I T ).
This simple observation suggests that we may have a better regularity for the solution ρ obtained in Proposition 3.5.1, which, by its turn, can also lead to a better regularity on κ due to the coupling in (3.7).

Proposition 3.5.3. (Bootstrap argument) Under the same hypothesis of Proposition 3.5.1 and if, in addition, the functions ρ 0 , κ 0 satisfy the condition

{ (1 + ε)ρ 0 xx + κ 0 x √ κ 0 x = 0 on ∂I, (1 + ε)κ 0 xx + ρ 0 x √ κ 0 x = 0
on ∂I, (3.57) then the solution (ρ, κ) obtained in Proposition 3.5.1 satises Proof of Proposition 3.5.3

(ρ, κ) ∈ (C 3+α, 3+α 2 (I T ) ∩ C ∞ ( Ī × (0, T ))) 2 . ( 3
     0 = κ t (x, 0) = εκ xx (x, 0) + ρ x (x, 0)ρ xx (x, 0) κ x (x, 0) + ρ x (x, 0) √ κ x (x, 0) for x ∈ ∂I, 0 = ρ t (x, 0) = (1 + ε)ρ xx (x, 0) + κ x (x, 0) √ κ x (x, 0), for x ∈ ∂I.
The proof follows the idea of Remark 3.5.2. In fact, since

κ x ∈ C α,α/2 (I T ) with γ 0 2 κ x 2M 0 , then the continuous function κ x √ κ x satises κ x (x, t) √ κ x (x, t) -κ x (y, t) √ κ x (y, t) |κ x (x, t)| √ κ x (x, t) - √ κ x (y, t) + √ κ x (y, t) |κ x (x, t) -κ x (y, t)| √ 2M 0 √ γ 0 |κ x (x, t) -κ x (y, t)| + √ 2M 0 |κ x (x, t) -κ x (y, t)| c|x -y| α .
Similarly, we obtain

κ x (x, t) √ κ x (x, t) -κ x (x, s) √ κ x (x, s) c|t -s| α/2 .
These arguments show that the source term κ x √ κ x of the second equation of (3.7) is of class C α,α/2 (I T ). Remark 3.5.4 with 0 < β = α < 2 and the compatibility conditions (3.57), ensures that ρ ∈ C α+2, α+2 2 (I T ), which, by its turn, and thanks to similar computations as above, shows that the the source term ρxρxx κx + ρ x √ κ x of the rst equation of (3.7) is of class C α,α/2 (I T ). Finally, we also deduce that κ ∈ C α+2, α+2 2 (I T ) and then κ x ∈ C 1+α, 1+α 2 (I T ). We now repeat exactly the same ideas but with this new regularity on κ x , taking into consideration that still 0 < β = α + 1 < 2, and hence Remark 3.5.4 is still applicable. Therefore, we are nally lead to (ρ, κ) ∈ (C 3+α, 3+α 2 (I T )) 2 . It is worth noticing that this is the maximum parabolic Hölder regularity that could be obtained up to the boundary since the compatibility condition (3.57) is not sucient when β > 2. Indeed, higher order compatibility assumptions are required if we aim to achieve more regularity up to the boundary.

To get the interior C ∞ regularity, we need to carefully overcome the problem of compatibility at t = 0, x ∈ ∂I. Indeed, let 0 < δ < T , and dene any test function we can easily see that (r, κ) satisfy a parabolic system where higher order compatibility conditions on the initial data are satised. Hence, the parabolic Hölder regularity can be innitely applied for ρ and κ. Accordingly

ϕ δ ∈ C ∞ [0, T ] by ϕ δ (t) =              0 if 0 t δ 3 , ϕ δ (t) ∈ (0, 1) if δ 3 t 2δ 3 ,
1 if 2δ 3 t T. ( 3
(ρ, κ) ∈ C ∞ (I T ).
From (3.59) and (3.60) we deduce that The main diculties that we faced in our work were the BV space which is a space of discontinuous functions, the presence of the absolute value in the equation which makes it hard to pass to the limit and the strong coupling between the two equations.

(ρ, κ) = (ρ, κ) on [ 2δ 3 , T ] , ∀ 0 < δ < T. Thus (ρ, κ) ∈ (C ∞ ( Ī × (0, T )))
Previous results done on Eikonal systems are available only for a single equation or in the Lipschitz continuous space.

Introduction and main results

Presentation and physical motivations

A dislocation is a crystal defect which corresponds to a discontinuity in the crystalline structure organization. This concept has been introduced by Polanyi, Taylor and Orowan in 1934 as the main explanation, at the microscopic scale, of plastic deformation. A dislocation creates around it a perturbation that can be seen as an elastic eld. Under an exterior strain, a dislocation moves according to its Burgers vector which characterizes the intensity and the direction of the defect displacement (see Hirth and Lothe [START_REF] Hirth | Theory of dislocations[END_REF] for an introduction to dislocations).

Here, we are interested in the dynamics of dislocation densities. More precisely, we consider a particular type called the edge dislocations where the line holding the dislocations is perpendicular to the Burgers vector. In a simple geometric setting where dislocation lines are assumed to be parallel, it can be seen that the dislocation distribution in any 2-d plane, perpendicular to these lines, are as shown in In this model (see Figure 4.1), the dynamics of the dislocation densities has been introduced by Groma and Balogh [START_REF] Groma | Investigation of dislocation pattern formation in a two-dimensional self-consistent eld approximation[END_REF] as a coupled system, namely a transport problem where the velocity is given by the elasticity equations. We assume ⃗ b = (1, 0). If the 2-d domain is 1-periodic in x 1 and x 2 , and if the dislocation densities depend only on the variable x = x 1 + x 2 (where (x 1 , x 2 ) is the coordinate of a generic point in R 2 ), then the 2-d model of [START_REF] Groma | Investigation of dislocation pattern formation in a two-dimensional self-consistent eld approximation[END_REF] reduces to the following system of 1-d coupled non-local Hamilton-Jacobi (HJ for short) equations

           ∂ρ + ∂t (x, t) = - ( (ρ + -ρ -)(x, t) + α ∫ 1 0 (ρ + -ρ -)(y, t)dy + a(t) ) ∂ρ + ∂x (x, t) in R × (0, T ), ∂ρ - ∂t (x, t) = ( (ρ + -ρ -)(x, t) + α ∫ 1 0 (ρ + -ρ -)(y, t)dy + a(t) ) ∂ρ - ∂x (x, t) in R × (0, T ), (4.1)
where ρ + , ρ -are the unknown scalars, that we denote for simplicity by ρ ± . We refer the reader to El Hajj and Forcadel [38, Lemme 3.1] for more modeling details. Here, the dierence (ρ + -ρ -) represents the plastic deformation, and the space derivatives ∂ρ ± ∂x represent the dislocation densities of ± dislocations respectively. The constant α depends on the elastic coecients and the material size, while the function a(t) represents the exterior shear stress eld, which is assumed to satisfy the following regularity a ∈ L ∞ (0, T ).

(4.2)
From a physical viewpoint, the dislocation density has a nonnegative sign ∂ρ ± ∂x 0. In our work, however, we do not consider this assumption but rather we allow the density to change sign. The initial conditions associated to system (4.1) are dened as follows

ρ ± (x, 0) = ρ ± 0 (x) = P ± 0 (x) + L 0 x on R, (4.3)
where P ± 0 are 1-periodic functions. In particular, ρ + 0 -ρ - 0 is 1-periodic. From equation (4.3), we can see that the total initial density of the positive dislocations is equal to that of the negative dislocations, which in its turn equals L 0 . We thus have a model for a periodic distribution of the ± dislocations, with a spatial period of length 1. In fact, the use of the periodic boundary conditions is a way of regarding what is going on in the interior of the material away from its boundary.

We now present a brief review on the study of system (4.1). In [START_REF] Hajj | Well-posedness theory for a non-conservative Burgers type system arising in dislocation dynamics[END_REF], and based on an energy estimate, the global existence and uniqueness of a solution in the class of nondecreasing W 1,2 loc (R×[0, +∞)) functions has been established. In [START_REF] Hajj | A convergent scheme for a non-local coupled system modeling dislocations densities dynamics[END_REF], the authors used the notion of viscosity solutions (initially introduced by Crandall and Lions [START_REF] Crandall | Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre[END_REF][START_REF] Crandall | Two approximations of solutions of Hamilton-Jacobi equations[END_REF] to deal with Hamilton-Jacobi equations) in order to solve (4.1). More precisely, they relied on the extended notion of viscosity solutions in the case of a system of equations as given by Ishii and Koike [START_REF] Ishii | Viscosity solution for monotone systems of second-order elliptic PDEs[END_REF][START_REF] Ishii | Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games[END_REF] in order to show the global existence and uniqueness of a Lipschitz solution. They also proposed a convergent numerical scheme, and proved a Crandall-Lions error type estimate between the continuous solution and the numerical one.

It is worth mentioning that, for an equivalent 2-d model, a global existence result was proved in Cannone et al. [START_REF] Cannone | Global existence of non-linear and non-local transport equations describing the dynamics of dislocation densities[END_REF], by making use of a special entropy inequality. Afterwards, a local existence and uniqueness result in Hölder spaces was shown in [START_REF] Hajj | Short time existence and uniqueness in Hölder spaces for the 2D dynamics of dislocation densities[END_REF], basing on some commutator estimates.

For a similar model (see Groma, Csikor and Zaiser [START_REF] Groma | Spatial correlations and higher-order gradient terms in a continuum description of dislocations dynamics[END_REF]), to that of Groma-Balogh, describing moreover the boundary layer eects, we refer the reader to Ibrahim et al. [START_REF] Ibrahim | Existence and uniqueness for a non-linear parabolic/Hamilton-Jacobi coupled system describing the dynamics of dislocation densities[END_REF][START_REF] Ibrahim | Dynamics of dislocation densities in a bounded channel. Part 2. Existence of weak solutions to a singular parabolic/Hamilton-Jacobi strongly coupled system[END_REF] where a result of existence and uniqueness has been established under the framework of viscosity solutions for HJ equations and the entropy solution for nonlinear hyperbolic equations.

Let us recall that our work focuses on the dynamics of dislocation densities. However, following a dierent approach, let us quote some recent results on the dynamics of dislocation lines, taken individually, that are represented by non-local HJ equations. In this direction, we address the reader to [START_REF] Alvarez | Dislocation dynamics: short-time existence and uniqueness of the solution[END_REF] for local existence results, and to [START_REF] Barles | Global existence results and uniqueness for dislocation equations[END_REF][START_REF] Barles | Uniqueness results for nonlocal Hamilton-Jacobi equations[END_REF][START_REF] Boudjerada | Global existence results for eikonal equation with BV initial data[END_REF][START_REF] Boudjerada | Existence result for a one-dimensional eikonal equation[END_REF] for global existence results.

Our main contribution is to work in BV space to overcome, physically, the random distribution of dislocations in a crystal. This originates our result from previous ones which were done relying on continuous data or in the frame work of strictly hyperbolic systems. Thus, inspired by our previous work [START_REF] Boudjerada | Global existence results for eikonal equation with BV initial data[END_REF][START_REF] Boudjerada | Existence result for a one-dimensional eikonal equation[END_REF] done in the context of dislocations lines, we establish a global existence result of discontinuous viscosity solutions of (4.1) by assuming (4.2), (4.3) and the following regularity on the initial data

P ± 0 ∈ L ∞ (T) ∩ BV (T), (4.4)
where T = R/Z is the [0, 1) periodic interval, and BV (T) is the space of functions of bounded variations given by

BV (T) = {f ∈ L 1 (T) | T V [0, 1](f ) < +∞}, with T V [0, 1](f ) the total variation of f dened as T V [0, 1](f ) = sup {∫ 1 0 f (x)ϕ ′ (x)dx; ϕ ∈ C 1 c (0, 1) and ∥ϕ∥ L ∞ (0,1) 1 } . ( 4.5)
In all what follows, we take the space BV (T) endowed with the semi-norm |f | BV (T) = T V [0, 1](f ). It is worth noticing that BV functions are integrable functions whose distributional derivative is a nite Radon measure.

We now briey present the key steps to prove our existence result. First, we consider an associated local problem which is obtained by freezing the integral term

           ∂ρ + ∂t (x, t) = - ( (ρ + -ρ -)(x, t) + L(t) ) ∂ρ + ∂x (x, t) in R × (0, T ), ∂ρ - ∂t (x, t) = ( (ρ + -ρ -)(x, t) + L(t) ) ∂ρ - ∂x (x, t) in R × (0, T ), (4.6)
with the initial conditions (4.3) and assuming L ∈ L ∞ (0, T ). Then, we regularize, by a classical convolution argument, the function L(•) and the initial conditions in (4.3). This approximation leads to the study, for every 0 < ε < 1, of the following system

                   ∂ρ + ε ∂t (x, t) = - ( (ρ + ε -ρ - ε )(x, t) + L ε (t) ) ∂ρ + ε ∂x (x, t) in R × (0, T ), ∂ρ - ε ∂t (x, t) = ( (ρ + ε -ρ - ε )(x, t) + L ε (t) ) ∂ρ - ε ∂x (x, t) in R × (0, T ), ρ ± ε (x, 0) = P ± 0,ε (x) + L 0 x in R, (4.8)
where L ϵ and P ± 0,ε are the regularization of the functions L and P ± 0 respectively (see (4.50) for the denitions of L ϵ and P ± 0,ε ). Next, after adding the viscosity term η ∂ 2 ρ ± ε ∂x 2 to the approximated problem, we show that the obtained equation admits a global in time regular solution ρ ± ε , which veries some η-uniform Lipschitz estimates. Then, we prove (η, ε)-uniform BV estimate, and some other (η, ε)-uniform a priori estimates. After that, by passing to the limit as η → 0 (thanks to the stability result for continuous viscosity solutions), we establish the global existence result of Lipschitz continuous viscosity solution of the approximated problem (4.8). Thanks to the local L ∞ uniform estimate on ρ ± ε inherited from the maximum principle, we show that the relaxed semi-limits of Barles and Perthame [START_REF] Barles | Exit time problems in optimal control and vanishing viscosity method[END_REF][START_REF] Barles | Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations[END_REF]

ρ ± (x, t) = lim sup ⋆ ρ ± ε (x, t) = lim sup ϵ-→0 (y,s)-→(x,t) ρ ± ε (y, s), ( 4.9)
and

ρ ± (x, t) = lim inf ⋆ ρ ± ε (x, t) = lim inf ϵ-→0 (y,s)-→(x,t) ρ ± ε (y, s), ( 4.10)
are, respectively, discontinuous viscosity sub-and super-solutions of (4.6). Finally, reaching some L ∞ and BV estimates, we will be able to prove that ρ ± (•, t) = ρ ± (•, t) almost everywhere in R, for all t > 0. Consequently, we ensure the existence of functions ρ ± , dened as a strong limit of ρ ± ε in C([0, T ); L 1 loc (R)), that is almost everywhere discontinuous viscosity solution of (4.6). This turns out to be possible thanks to a uniform BV bound obtained on ρ ± ε , and the nite speed of propagation property enjoyed by the equation.

Let us mention that, the BV estimate proved below is not standard, given that system (4.6) is hyperbolic but not necessarily strictly hyperbolic, since we have no sign property on ρ + -ρ -. Related to our analysis, we also obtain, as a consequence, the global existence of a discontinuous viscosity solution of (4.6) for non-decreasing initial data. Moreover, using a xed point argument, we get a similar result for the non-local system describing the dynamics of dislocation densities.

Main results

In this subsection, we rst present, in Theorem 4.1.1, a global existence result of a weak discontinuous viscosity solution (see Denition 4.5.1) of (4.6). As a consequence, we show, in Theorem 4.1.2, that this solution is a classical discontinuous viscosity solution of (4.6) in the case of non-decreasing solutions. Moreover, based on these two theorems, we present, in Theorem 4.1.4, some similar results for a model related to the dynamics of dislocation densities.

We now announce our rst main result. Suppose that assumptions (4.3), (4.4) and (4.7) are satised. Then the following points hold i) Existence and uniqueness of approximated problem There exists a unique Lipschitz continuous viscosity solution ρ ± ε of (4.8) such that

ρ ± ε (x, t) = P ± ε (x, t) + L 0 x,
where P ± ε are 1-periodic functions (with respect to the space variable). Moreover, for all T > 0, we have the following uniform a priori estimates

max ± ( ∥P ± ε ∥ L ∞ (T×(0,T )) ) M 0 , (4.11) ∂ρ ± ε ∂x L ∞ ((0,T );L 1 (T)) |ρ ± 0 | BV (T) , (4.12) ∂ρ ± ε ∂t L ∞ ((0,T );L 1 (T)) [ 2M 0 + ∥L∥ L ∞ (0,T ) ] |ρ ± 0 | BV (T) , (4.13)
with

M 0 = max ± ( ∥P ± 0 ∥ L ∞ (T) ) + |L 0 |∥L∥ L ∞ (0,T ) T. (4.14)
ii) Sub-and super-solutions of the limit problem Let ρ ± ε be the solution of (4.8) constructed in (i), then the relaxed semi-limits ρ ± and ρ ± , dened by (4.9), (4.10), are respectively discontinuous viscosity sub-and super-solutions of (4.6).

iii) Convergence and existence of weak solution Assume that ρ ± ε satises (4.11), (4.12) and (4.13). Then, up to the extraction of a subsequence, the functions ρ ± ε converge, as ε tends to zero, to a function

ρ ± ∈ L ∞ loc (R × (0, T )) ∩ L ∞ ((0, T ); BV (T)) ∩ C([0, T ); L 1 loc (R)), (4.15)
strongly in C([0, T ); L 1 loc (R)). Moreover

ρ ± (x, t) = P ± (x, t) + L 0 x, ( 4.16)
where P ± are 1-periodic functions (with respect to the space variable) and ρ ± satisfy, for all T > 0, the following estimates

max ± ( ∥P ± ∥ L ∞ (T×(0,T )) ) M 0 , (4.17) ∥ρ ± ∥ L ∞ ((0,T );BV (T)) |ρ ± 0 | BV (T) , (4.18) ∥ρ ± (•, t) -ρ ± (•, s)∥ L 1 (T) ([ 2M 0 + ∥L∥ L ∞ (0,T ) ] |ρ ± 0 | BV (T) ) |t -s|, for all s, t ∈ [0, T ), (4.19)
and the following equality

ρ ± (•, t) = ρ ± (•, t) = ρ ± (•, t), for all t ∈ [0, T ), (4.20)
except on a countable set in D ⊂ R.

The key point to establish this theorem is the uniform BV estimate (4.12) on ρ ± ϵ . We rst consider the parabolic regularization of (4.6) and we show that the Lipschitz continuous viscosity solution admits the L ∞ bound (4.11), and the fundamental BV estimate (4.12). These estimates allow us, by relying on the stability property of viscosity solutions (see Barles [10,Theorem 4.1]), to pass to the limit when the regularization vanishes, and then to show that the relaxed semi-limits ρ ± and ρ ± are, respectively, suband super-solutions of (4.6). Moreover, these estimates also imply that the solution ρ ± (•, t) is continuous except at most on a countable set D t ⊂ R. Taking into account the nite speed of propagation property of (4.6) and the time continuous estimate (4.13), it is then possible to construct the set D out of D t ; hence proving (4.20).

The solution constructed in the previous theorem can be seen as a discontinuous viscosity solution but in some weak sense, since it only veries an almost everywhere equality in space between ρ ± and ρ ± , which is reected by (4.20). In what follows, such a solution will be called a weak discontinuous viscosity solution of (4.6).

Recall that in the framework of non-decreasing solutions, the non-linear Eikonal system (4.6) becomes a non-linear transport system. Therefore, related to our analysis in Theorem 4.1.1, it is then possible to get the following theorem as a by-product. 1. Let us mention that, considering the framework of non-decreasing (or non-increasing) solutions and the particular case where L(t) ≡ 0, system (4.6) can be seen as a hyperbolic system. This kind of system was already studied by Bianchini and Bressan in [START_REF] Bianchini | Vanishing viscosity solutions of nonlinear hyperbolic systems[END_REF], where an existence and uniqueness result was proved only within the framework of strictly hyperbolic systems and assuming that the initial data has small total variation. In general, our system is hyperbolic but not necessarily strictly hyperbolic since we have no sign property on ρ + -ρ -. Moreover, if we require such a property we would need, to apply the classical maximum principle, a more regular solution (namely Lipschitz). This takes us outside the framework of discontinuous solutions and brings us back to what has been treated in [START_REF] Hajj | A convergent scheme for a non-local coupled system modeling dislocations densities dynamics[END_REF]Theorem 1.1] in the case of Lipschitz continuous viscosity solution.

2. We point out that our results can be easily extended to generalized rst order HJ systems with suitable assumptions on the hamiltonian in order to ensure the BV estimate and the maximum principle. Here, the problem lies in the fact that the initial data is discontinuous, and hence we can not establish a comparison principle result. On the other hand, it is possible to have such a result if we consider uniformly continuous data (refer to the result of Ishii and Koiki [START_REF] Ishii | Viscosity solution for monotone systems of second-order elliptic PDEs[END_REF]).

A brief review of some related literature about hyperbolic systems

From a mathematical point of view, in the case of non-decreasing solutions, system (4.1) is related to other similar models such as transport equations based on vector elds with low regularity. These models were, for instance, studied by Diperna and Lions in [START_REF] Diperna | Ordinary dierential equations, transport theory and Sobolev spaces[END_REF]. They proved the existence and uniqueness of a solution (in the renormalized sense), for vector elds in L 1 ((0, +∞); W 1,1 loc (R N )) whose divergence in L 1 ((0, +∞); L ∞ (R N )). This study was generalized by Ambrosio [START_REF] Ambrosio | Transport equation and Cauchy problem for BVvector elds[END_REF], who considered vector elds in L 1 ((0, +∞); BV loc (R N ))

with bounded divergence. We notice that, in our work, we only deal velocity vector elds in L ∞ ((0, +∞), BV (T)).

We now recall some well-known results for a class of 2 × 2 strictly hyperbolic systems in one space dimension. We remark that our system (4.1) is hyperbolic, but not necessarily strictly hyperbolic since we have no sign property on ρ + -ρ -. Lax [START_REF] Lax | Hyperbolic systems of conservation laws and the mathematical theory of shock waves[END_REF] proved the existence and uniqueness of non-decreasing and smooth solutions for diagonalized 2 × 2 strictly hyperbolic systems. This result was improved in [START_REF] Hajj | Some niqueness results for diagonal hyperbolic systems equation with large and monotone data[END_REF][START_REF] Hajj | Global continuous solutions for diagonal hyperbolic systems with large and monotone data[END_REF] by considering more general hyperbolic systems. Let us mention that, in the case of general strictly hyperbolic systems, Bianchini and Bressan [START_REF] Bianchini | Vanishing viscosity solutions of nonlinear hyperbolic systems[END_REF] proved a striking global existence and uniqueness result, assuming that the initial data has small total variation. This approach is mainly based on a careful analysis of the vanishing viscosity approximation. An existence result has been rst proved by Glimm [START_REF] Glimm | Solutions in the large for nonlinear hyperbolic systems of equations[END_REF] in the special case of conservative equations. Moreover, an existence result has been also obtained by LeFloch, Liu [START_REF] Lefloch | Existence theory for nonlinear hyperbolic systems in nonconservative form[END_REF] and LeFloch [START_REF] Lefloch | Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form[END_REF][START_REF] Lefloch | Graph solutions of nonlinear hyperbolic systems[END_REF],

in the non-conservative case.

When the system is hyperbolic and symmetric (this corresponds to the case α = a = 0 in our system (4.1)), a result of local existence and uniqueness in C([0, T );

H s (R N )) ∩ C 1 ([0, T ); H s-1 (R N )), s > N 2 +

Organization of the paper

This paper is organized as follows. In Section 4.2, we regularize the approximated problem (4.8) by adding the viscosity term η ∂ 2 ρ ± ε ∂x 2 . Then, we show that this equation admits a global in time regular solution ρ ± ε , which veries some η-uniform Lipschitz estimates. In Section 4.3, we prove the (η, ε)-uniform BV estimate (4.12), and some other (η, ε)-uniform a priori estimates. In Section 4.4, we prove, by passing to the limit as η → 0 (thanks to the stability result for continuous viscosity solutions), the global existence result of Lipschitz continuous viscosity solution of the approximated problem (4.8) (announced in Theorem 4.1.1 (i)). In Section 4.5, we pass to the limit as ε → 0 and we prove Theorem 4.1.1 (ii). Here, we use the stability result for discontinuous viscosity solutions and the nite speed propagation property of the equation. Section 4.6 is devoted to the proof of the main results Theorem 4.1.1 (iii), and Theorem 4.1.2. This will be achieved by using our (η, ε)-uniform a priori estimates, in particular the BV estimate, and again the nite speed propagation property of the equation. Finally, in Section 4.7, we present the proof of Theorem 4.1.4 as an application to a model of dislocation dynamics in crystals.

Global solution for parabolic regularized equation

In this section, we recall some preliminary global existence and uniqueness results for a parabolic system obtained by regularizing problem (4.6), with smoothed data v ± 0 and smoothed function L. More precisely, we consider, for 0 < η 1, the following system

       ∂v η ± ∂t (x, t) -η ∂ 2 v η ± ∂x 2 (x, t) = ∓ ( (v + η -v - η )(x, t) + L(t) ) ∂v η ± ∂x (x, t) in R × (0, T ), v ± η (x, 0) = v ± 0 (x) = Q ± 0 (x) + L 0 x in R, (4.21)
where Q ± 0 are 1-periodic functions, such that

Q ± 0 ∈ C ∞ (T) and L ∈ C ∞ (0, T ). (4.22)
We also regularize the non-linear term ∂vη ± ∂x replacing it by function β δ

( ∂vη ± ∂x ) , 0 < δ 1, where β δ is dened as follows β δ (x) = √ x 2 + δ 2 . (4.23)
This brings us, nally, to consider the following parabolic system Assume that (4.22) holds. Then, for all T > 0, we have i) (Existence and uniqueness of (4.24)) System (4.24) admits a unique solution v ±,δ η , such that

       ∂v η ±,δ ∂t (x, t) -η ∂ 2 v η,δ ± ∂x 2 (x, t) = ∓ ( (v +,δ η -v -,δ η )(x, t) + L(t) ) β δ ( ∂v η ±,δ ∂x (x, t) ) in R × (0, T ), v ±,δ η (x, 0) = v ± 0 (x) in R.
v ±,δ η (x, t) = Q ±,δ η (x, t) + L 0 x,
where Q ±,δ η are 1-periodic functions (with respect to the space variable), belonging to C ∞ (T × (0, T )) and satisfying the following estimates

max ± ( ∥Q ±,δ η ∥ L ∞ (T×(0,T )) ) N δ 0 , (4.25) max ± ( ∂v η ±,δ ∂x L ∞ (T×(0,T )) ) max ± ( ∂v 0 ± ∂x L ∞ (T)
)

, ( 4.26)
where

N δ 0 = max ± ( ∥Q ± 0 ∥ L ∞ (T) ) + β δ (L 0 )∥ L∥ L ∞ (0,T ) T. (4.27) ii) (Convergence)
Up to the extraction of a subsequence, the functions v ±,δ η converge, as δ goes to 0, to v ± η locally and strongly in

C([0, T ); W 1,∞ (R)). Moreover v ± η (x, t) = Q ± η (x, t) + L 0 x,
such that Q ± η are 1-periodic functions; v ± η is the unique solution of (4.21) (in the distributional sense) belonging to the space C([0, T ); W 1,∞ loc (R)) and satises, for all 0 < η 1, the following η-uniform estimates

max ± ( ∥Q ± η ∥ L ∞ (T×(0,T )) ) N 0 , (4.28) max ± ( ∂v η ± ∂x L ∞ (T×(0,T )) ) max ± ( ∂v 0 ± ∂x L ∞ (T) ) , ( 4.29)
where

N 0 = max ± ( ∥Q ± 0 ∥ L ∞ (T) ) + |L 0 |∥ L∥ L ∞ (0,T ) T. (4.30)
To prove Theorem 4.2.1, we need to recall the following lemma.

Lemma 4.2.2. (Estimates for the heat semi-group) Let S η (t) = e ηt∆ be the heat semi-group. Then, for all functions f ∈ L ∞ (R), we have the following estimates

∥S η (t)f ∥ L ∞ (R) ∥f ∥ L ∞ (R)
for all t > 0, and

∥∂ x S η (t)f ∥ L ∞ (R) γ 0 t -1
The proof of this lemma is a direct application of the classical version of the L r -L p estimates for the heat semi-group (see A. Pazy [77, Lemma 1.1.8, Theorem 6.4.5]).

Proof of Theorem 4.2.1

The proof is outlined in the following four steps.

Step 1 (Rewriting the equation) Assume that (4.22) holds and take v δ η =

( v +,δ η v -,δ η ) and v 0 = ( v + 0 v - 0 )
. Thus, equation (4.24) can be written in the following integral form

   v δ η (x, t) = S η (t)v 0 (x) + B(v δ η , v δ η )(t), v ±,δ η (x, 0) = v ± 0 (x), (4.31)
where S η (t) = e ηt∆ , and

B(u, v)(t) = ∫ t 0 S η (t -s) ( β δ (∂ x v 1 ) 0 0 β δ (∂ x v 2 ) ) (( -1 1 1 -1 ) ( u 1 u 2 ) + ( -L(t) L(t)
)) ds,

taking u = ( u 1 u 2 ) and v = ( v 1 v 2 )
. Our next step is to apply xed point argument. For that reason, some estimates are needed to be veried on B and S η (t)v 0 .

Step 2 (Estimates on B and S η (t)v 0 )

For a detailed demonstration, let us consider the Banach spaces L ∞ ((0, T ); W 1,∞ (T))) 2 and L ∞ ((0, T ); W 1,∞ (T)). Recall that for u = (u

1 u 2 ) and v = (v 1 v 2 ), we have              ∥u∥ (L ∞ ((0,T );W 1,∞ (T))) 2 = max(∥u 1 ∥ L ∞ ((0,T );W 1,∞ (T)) , ∥u 2 ∥ L ∞ ((0,T );W 1,∞ (T)) ), ∥u 1 ∥ L ∞ ((0,T );W 1,∞ (T)) = sup 0<t T ∥u 1 ∥ W 1,∞ (T) , ∥u 1 ∥ W 1,∞ (T) = ∥u 1 ∥ L ∞ (T) + ∥∂ x u 1 ∥ L ∞ (T) . (4.32) Notice that B(u, v)(t) is a vector of two components B 1 (u, v)(t) = ∫ t 0 S η (t -s)(-u 1 + u 2 -L(t))β δ (∂ x v 1 )ds,
and

B 2 (u, v)(t) = ∫ t 0 S η (t -s)(u 1 -u 2 + L(t))β δ (∂ x v 2 )ds. Thus ∥B(u, v)(t)∥ (L ∞ ((0,T );W 1,∞ (T))) 2 = max(∥B 1 (u, v)(t)∥ L ∞ ((0,T );W 1,∞ (T)) , ∥B 2 (u, v)(t)∥ L ∞ ((0,T );W 1,∞ (T)) ).
We remark that

∥B 1 (u, v)(t)∥ W 1,∞ (T) ∫ t 0 ∥S η (t -s)(-u 1 + u 2 -L(t))β δ (∂ x v 1 )∥ L ∞ (T) ds + ∫ t 0 ∥∂ x S η (t -s)(-u 1 + u 2 -L(t))β δ (∂ x v 1 )∥ L ∞ (T) ds. (4.33)
By applying Lemma 4.2.2 and using Young's convolution inequality, we get, for all 0 < t T , the following estimate

∥B(u, v)(t)∥ (L ∞ ((0,T ),W 1,∞ (T))) 2 (4.34) (T + 2T 1 2 γ 0 )(2∥u∥ (L ∞ ((0,T );W 1,∞ (T))) 2 + ∥L∥ L ∞ (0,T ) )(δ + ∥v∥ (L ∞ ((0,T );W 1,∞ (T))) 2 ).
Finally, using the classical properties of heat semi-group, we obtain Step 4 (Regularity, estimates and convergence)

∥S η (t)v 0 ∥ (L ∞ ((0,T );W 1,∞ (T))) 2 ∥v 0 ∥ (L ∞ ((0,T );W 1,∞ (T))) 2 .
Estimates (4.25)-(4.26) directly follow by applying the maximum principle for parabolic equations (see Lieberman [START_REF] Lieberman | Second order parabolic dierential equations[END_REF]Th 2.10]). In turn, these estimates are more than sucient to get the strong compactness, consequently, the strong convergence in C((0, T ); W 1,∞ (T)). Using the L p -regularity of parabolic equations and the fact that β δ is regular, we can show, by classical bootstrap argument, that Q ±,δ η belongs to C ∞ (R×(0, Proof. First, we prove the above estimate on v η ±,δ ; the solution of (4.24) constructed in Theorem 4.2.1 (i), and then, thanks to the uniqueness of solution, we deduce the announced result by 4.3. A priori uniform estimates on the solution of (4.21)

T * ))∩L ∞ ((0, T * ); W 1,∞ loc (R)). In particular, it belongs to C ∞ (T × (0, T * )) ∩ L ∞ ((0, T * ); W 1,∞ (T)) by periodicity of Q ±,δ η .
passing to the limit as δ tends to 0. Indeed, upon dierentiating the rst equation in (4.24) with respect to x, and then multiplying by

β ′ δ ( ∂vη ±,δ ∂x )
, where β δ is dened in (4.23), we obtain ∂ ∂t

( β δ ( ∂v η ±,δ ∂x )) = η ∂ 2 ∂x 2 ( ∂v η,δ ± ∂x ) β ′ δ ( ∂v η ±,δ ∂x ) ∓ ∂ ∂x ( ( (v +,δ η -v -,δ η ) + L(t) ) β δ ( ∂v η ±,δ ∂x )) β ′ δ ( ∂v η ±,δ ∂x ) .
Integrating over the spatial variable in (0, 1), we get ∂ ∂t

[∫ 1 0 β δ ( ∂v η ±,δ ∂x ) dx ] = η ∫ 1 0 ∂ ∂x ( ∂ 2 v η,δ ± ∂x 2 ) β ′ δ ( ∂v η ±,δ ∂x ) dx I 1 ∓ ∫ 1 0 ∂ ∂x ( ( (v +,δ η -v -,δ η ) + L(t) ) β δ ( ∂v η ±,δ ∂x )) β ′ δ ( ∂v η ±,δ ∂x ) dx I 2 . (4.37)
We note that these computations are all justied because v ±,δ

η ∈ C ∞ (R×(0, T )).
In what follows, we will show that the terms I 1 and I 2 are bounded uniformly in η and δ.

Step 1 (Estimate of I 1 ) Upon integrating by parts, and since ∂vη ±,δ ∂x are 1-periodic functions with respect to the space variable, we have

I 1 = -η ∫ 1 0 ( ∂ 2 v η ±,δ ∂x 2 ) 2 β ′′ δ ( ∂v η ±,δ ∂x ) dx.
Thanks to the convexity of β δ , we know that

( ∂ 2 v η ±,δ ∂x 2 ) 2 β ′′ δ ( ∂v η ±,δ ∂x ) 0,
and consequently

I 1 0. (4.38)
Step 2 (Estimate of I 2 ) To estimate I 2 , we integrate by parts and we use the fact that v +,δ η -v -,δ η and ∂vη ±,δ ∂x are 1-periodic functions, in order to get

I 2 = ± ∫ 1 0 ( (v +,δ η -v -,δ η ) + L(t) ) β δ ( ∂v η ±,δ ∂x ) β ′′ δ ( ∂v η ±,δ ∂x ) ∂ 2 v η ±,δ ∂x 2 dx.
Furthermore, by basic properties of β δ dened in (4.23), we can see that

∂ 2 v η ±,δ ∂x 2 β δ ( ∂v η ±,δ ∂x ) β ′′ δ ( ∂v η ±,δ ∂x ) = δ ∂ ∂x ( arctan [ 1 δ ∂v η ±,δ ∂x ]) ,
which shows that

I 2 = ±δ ∫ 1 0 ( (v +,δ η -v -,δ η ) + L(t) ) ∂ ∂x ( arctan [ 1 δ ∂v η ±,δ ∂x ]) dx.
Again, integrating by parts the above equality, we obtain

I 2 = ∓δ ∫ 1 0 ( ∂v η +,δ ∂x - ∂v η -,δ ∂x ) arctan [ 1 δ ∂v η ±,δ ∂x ] dx.
Thereafter, using (4.26), we deduce that

I 2 δπ max ± ( ∂v 0 ± ∂x L ∞ (T)
) .

(4.39)

Step 3 (Passage to the limit) Integrating (4.37) in time on (0, t), for 0 < t < T , we get

∫ 1 0 β δ ( ∂v η ±,δ ∂x ) dx ∫ 1 0 β δ ( ∂v 0 ± ∂x ) dx + T δπ max ± ( ∂v 0 ± ∂x L ∞ (T)
)

,
where we have used (4.38) and (4.39). By passing to the limit δ → 0 in the previous estimate, and using the fact that ∂vη ±,δ ∂x converges locally uniformly towards ∂vη ± ∂x , we obtain

∫ 1 0 ∂v η ± ∂x dx ∫ 1 0 ∂v 0 ± ∂x dx,
which leads to the desired result.

2

The following estimate provides the compactness in time of the solution v η (given in Theorem 4.2.1 (ii)) uniformly with respect to η.

Lemma 4.3.2. (Estimate on the time derivative of the solution)

Assume that (4.22) holds, and let W -1,∞ (T) be the dual space of W 1,1 (T). Then, for all 0 < η 1 and T > 0, the solution v η of (4.21), given by Theorem 4.2.1 (ii), satises the following estimate

∂v η ± ∂t L ∞ ((0,T );W -1,∞ (T))
M , (4.40) where M depends only on T , L 0 ,

∂v 0 ± ∂x L ∞ (T) , ∥Q ± 0 ∥ L ∞ (T) and ∥ L∥ L ∞ (0,T ) .
Proof. Roughly speaking, the idea is to control the term ∂v η,δ ± ∂t using the right hand side of (4.24), and then to pass to the limit δ → 0 in order to control ∂vη ± ∂t . The proof is given by duality. Multiplying (4.24) by ϕ ∈ L 1 ((0, T ); W 1,1 (T)) and integrating on T × (0, T ), we get ∫ T×(0,T )

ϕ ∂v η,δ ± ∂t dxdt = I 1 η ∫ T×(0,T ) ϕ ∂ 2 v η,δ ± ∂x 2 dxdt ∓ I 2 ∫ T×(0,T ) ϕ ( (v +,δ η -v -,δ η ) + L(t) ) β δ ( ∂v η ±,δ ∂x ) dxdt .
Performing integrating by parts in I 1 , we obtain, for all 0 < η 1

|I 1 | ∫ T×(0,T ) ∂ϕ ∂x ∂v η ±,δ ∂x ∂ϕ ∂x L 1 (T×(0,T )) ∂v η ±,δ ∂x L ∞ (T×(0,T )) ∥ϕ∥ L 1 ((0,T );W 1,1 (T)) B 0 , (4.41)
where we have used inequality (4.26) in the second line with

B 0 = max ± ( ∂v 0 ± ∂x L ∞ (T)
) .

Similarly, using (4.25), we have

|I 2 | ≤ ∥ϕ∥ L 1 (R×(0,T )) ∥(Q +,δ η -Q -,δ η ) + L(•)∥ L ∞ (T×(0,T )) β δ ( ∂v η ±,δ ∂x ) L ∞ (R×(0,T )) ≤ ∥ϕ∥ L 1 ((0,T );W 1,1 (T)) (2N δ 0 + ∥ L∥ L ∞ (0,T ) )β δ (B 0 ).
ϕ ∂v η ±,δ ∂t C δ ∥ϕ∥ L 1 ((0,T );W 1,1 (T)) ,
where C δ := B 0 + (2N δ 0 + ∥ L∥ L ∞ (0,T ) )β δ (B 0 ) and N δ 0 is dened in (4.27). Then, by duality, we deduce that

∂v η ±,δ ∂t L ∞ ((0,T );W -1,∞ (T)) C δ .
Finally, passing to the limit in the previous inequality, as δ goes to 0, and using the weakly-⋆ convergence in L ∞ ((0, T ); W -1,∞ (T)), we get

∂v η ± ∂t L ∞ ((0,T );W -1,∞ (T)) lim inf δ→0 ∂v η ±,δ ∂t L ∞ ((0,T );W -1,∞ (T))

M ,

where

M = lim δ→0 C δ = B 0 + (2N 0 + ∥ L∥ L ∞ (0,T ) )B 0 ,
and N 0 dened in (4.30). This terminates the proof. 2

Global continuous solution for regularized Eikonal system

This section is devoted to the proof of Theorem 4.1.1 (i). For the sake of a clear presentation, we divide it into two subsections. In the rst subsection, we prove the global existence of Lipschitz continuous viscosity solution of the Eikonal equation (4.43), by passing to the limit η → 0 in equation (4.21). Using this result, we prove Theorem 4.1.1 (i) in the next subsection.

4.4.1 Passage to the limit when the parameter η tends to 0

In this subsection, we state a global in time existence and uniqueness result for the following system

       ∂v ± ∂t (x, t) = ∓ ( (v + -v -)(x, t) + L(t) ) ∂v ± ∂x (x, t) in R × (0, T ), v ± (x, 0) = v ± 0 (x) = Q ± 0 (x) + L 0 x in R. (4.43)
We prove that this equation admits a unique Lipschitz continuous viscosity solution v ± , when subjected to Lipschitz continuous initial data. Moreover, we show that v ± is the limit of v ± η when η → 0, where v ± η is the solution of (4.21) given by Theorem 4.2.1 (ii). We also show that v ± satises the Lipschitz estimates (4.28), (4.29) and, if (4.22) holds, the BV estimate (4.36).

Before showing the main theorem, let us recall the denition of the continuous viscosity solution for system (4.21) with 0 ≤ η 1, that was proposed by Ishii and Koike in [START_REF] Ishii | Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games[END_REF].

• v ± (x, 0) v ± 0 (x),
• for all k ∈ {+, -} and for any test-

function ϕ ∈ C 2 (R × (0, T )) such that v k -ϕ reaches a local maximum at a point (x 0 , t 0 ) ∈ R × (0, T ), we have ∂ϕ ∂t (x 0 , t 0 ) + k ( (v + η -v - η )(x 0 , t 0 ) + L(t 0 ) ) ∂ϕ ∂x (x 0 , t 0 ) -η ∂ 2 ϕ ∂x 2 (x 0 , t 0 ) 0.
In a similar way, a function v ± ∈ C(R × [0, T )) is a viscosity super-solution of (4.21) if and only if

• v ± (x, 0) v ± 0 (x),
• for all k ∈ {+, -} and for any test-

function ϕ ∈ C 2 (R × (0, T )) such that v k -ϕ reaches a local minimum at point (x 0 , t 0 ) ∈ R × (0, T ), we have ∂ϕ ∂t (x 0 , t 0 ) + k ( (v + η -v - η )(x 0 , t 0 ) + L(t 0 ) ) ∂ϕ ∂x (x 0 , t 0 ) -η ∂ 2 ϕ ∂x 2 (x 0 , t 0 ) 0.
Finally, a continuous function v ± is a viscosity solution of (4.21) if and only if it is a sub-and a super-solution of (4.21).

We have the following existence and uniqueness result for the local equation (4.43). Assume that (4.22) holds. Then, for all T > 0, we have i) Problem (4.43) admits a unique Lipschitz continuous viscosity solution v ± , such that

v ± (x, t) = Q ± (x, t) + L 0 x, (4.44)
where Q ± are 1-periodic functions (with respect to the space), belonging to W 1,∞ (T × (0, T )) and satisfying the following estimates:

max ± ( ∥Q ± ∥ L ∞ (T×(0,T )) ) N 0 , (4.45) max ± ( ∂v ± ∂x L ∞ (T×(0,T )) ) max ± ( ∂v 0 ± ∂x L ∞ (T)
) , (4.46) where N 0 is dened in (4.30).

ii) The solution v ± satises

∫ T ∂v ± ∂t (x, t) dx (2N 0 + ∥ L∥ L ∞ (0,T ) ) ∫ T ∂v 0 ± ∂x (x) dx, for all t ∈ [0, T ), (4.47)
and the following BV estimate

∫ T ∂v ± ∂x (x, t) dx ∫ T ∂v 0 ± ∂x (x) dx, for all t ∈ [0, T ). (4.48)
To prove this theorem, we need the following stability lemma. Let v ± η be the solution of (4.21), constructed in Theorem 4.2.1 (ii) as the limit of v ±,δ η when δ → 0. Then we have i) The function v ± η is a continuous viscosity solution of (4.21). ii) If v ± η converge locally uniformly, as η goes to 0, to v ± , then v ± is a Lipschitz continuous viscosity solution of (4.43).

To prove this lemma, it is sucient to apply the stability result for continuous viscosity solutions (see Barles [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF]Th 2.3]), and the fact that all classical C 2 solutions of (4.24) are also viscosity solutions.

We recall the following compactness lemma.

(θ n ) n is a uniformly bounded sequence in L ∞ ((0, T ); X) and (∂ t θ n ) n is uniformly bounded in L r ((0, T ); Y ) for r > 1, then (θ n ) n is relatively compact in C((0, T ); B).
X = W 1,∞ (K), B = L ∞ (K) and Y = W -1,∞ (K) = (W 1,1 (K)) ′ , we can show that v ± η is relatively compact in L ∞ (K × (0, T )).
Hence, as η → 0, we can extract a subsequence, still denoted by v ± η , that converge locally uniformly to some limit v ± . Then, by Lemma 4.4.3 (ii), we can see that v ± is a Lipschitz continuous viscosity solution of (4.43), and consequently the existence follows. However, the uniqueness of this solution is an independent result valid for Lipschitz continuous viscosity solutions of (4.43) with Lipschitz velocity (see for instance Crandall et al. [START_REF] Crandall | On existence and uniqueness of solutions of Hamilton-Jacobi equations[END_REF]).

Finally, using the lower semi-continuity of the L ∞ norm on L ∞ (T × (0, T )), we pass to the limit in estimates (4.28) and (4.29) in order to get (4.45) and (4.46). Proof of ii): From estimate (4.36), we deduce that (v ± η (•, t)) η are uniformly bounded in BV (T) and then, thanks to the lower semi-continuity of the total variation, we get

T V [0, 1](v ± (•, t)) lim inf η-→0 T V [0, 1](v ± η (•, t)) ∫ T ∂v 0 ± ∂x (x)
∥v ± (•, t) -v ± (•, s)∥ L 1 (T) (2N 0 + ∥ L∥ L ∞ (0,T ) ) ∂v 0 ± ∂x L 1 (T)
|t -s| for all t, s 0.

Proof of Theorem 4.1.1 (i)

In this subsection, we prove the global existence and uniqueness of a Lipschitz continuous viscosity solution of equation (4.6), after the regularization of the function L, and the initial data ρ ± 0 given in (4.3). This amounts to prove Theorem 4.1.1 (i). Here, the approximation brings us to consider, for every ϵ > 0, the following equation

       ∂ρ ± ε ∂t (x, t) = ∓ ( (ρ + ε -ρ - ε )(x, t) + L ε (t) ) ∂ρ ± ε ∂x (x, t) in R × (0, T ), ρ ± ε (x, 0) = ρ ± 0,ε (x) = P ± 0,ε (x) + L 0 x in R, (4.49)
where

P ± 0,ε (x) = P ± 0 ⋆ η ϵ (x) and L ϵ (t) = L ⋆ η ϵ (t) ∀ x ∈ R, t ∈ R. (4.50)
Here, the function L is the extension by 0 of the function L over R, and η ϵ is the standard mollier dened as follows

η ϵ (•) = 1 ϵ η ( • ϵ ) , such that η ∈ C ∞ c (R), supp{η} ⊆ (-1, 1), η 0, and ∫ R η = 1.
By assumptions (4.4), (4.7), and by classical properties of the mollier, we can see that, for all ϵ > 0, the functions P ± 0,ε and L ϵ satisfy assumption (4.22). Thus, we can apply Theorem 4.4.2 with L = L ϵ and v ± 0 = ρ ± 0,ε in order to prove that equation (4.49) admits a unique Lipschitz continuous viscosity solution

ρ ± ε (x, t) = P ± ε (x, t) + L 0 x, (4.51)
where P ± ε is 1-periodic function (with respect to the space). Moreover, for all T > 0, we have the following uniform a priori estimates max

± ( ∥P ± ε ∥ L ∞ (T×(0,T )) ) M ε 0 , (4.52) ∂ρ ± ε ∂x (•, t) L 1 (T) ∂ρ ± 0,ε ∂x L 1 (T)
, for all t ∈ [0, T ),

(4.53) ∂ρ ± ε ∂t (•, t) L 1 (T) [ 2M ε 0 + ∥L ε ∥ L ∞ (0,T ) ] ∂ρ ± 0,ε ∂x L 1 (T)
, for all t ∈ [0, T ),

(4.54)
with

M ε 0 = max ± ( ∥P ± 0,ε ∥ L ∞ (T) ) + |L 0 |∥L ε ∥ L ∞ (0,T ) T. (4.55)
We also have the following estimate (depending on ϵ) for the third estimate)

max ± ( ∂ρ ± ε ∂x L ∞ (T×(0,T )) ) max ±   ∂ρ ± 0,ε ∂x L ∞ (T)   . ( 4
∥P ± 0,ε ∥ L ∞ (T) ∥P ± 0 ∥ L ∞ (T) ∥L ϵ ∥ L ∞ (0,T ) ∥L∥ L ∞ (0,T ) , ∂ρ ± 0,ε ∂x L 1 (T) T V [0, 1](ρ ± 0 ) = |ρ ± 0 | BV (T) ,
which, together with (4.52), (4.53) and (4.54) imply (4.11), (4.12) and (4.13).

(ii).

In order to present it, we start by passing to the limit ε → 0 in equation (4.49), using estimate (4.11) and the stability result of discontinuous viscosity solution. This is based on the technical method of relaxed semi-limits introduced by Barles and Perthame ( [START_REF] Barles | Exit time problems in optimal control and vanishing viscosity method[END_REF][START_REF] Barles | Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations[END_REF]). This section will be divided in two subsections.

First, in Subsection 4.5.1, we recall some useful results for viscosity solutions, and then in Subsection 4.5.2, we give the proof of Theorem 4.1.1 (ii).

Some useful results for viscosity solutions

In the following, we denote by f ⋆ and f ⋆ the respective upper and lower semi-continuous envelopes of a locally bounded function f dened on [0, T ) × Ω, where Ω is an open subset of R n , and given by f ⋆ (X, t) = lim sup

(Y,s)→(X,t) f (Y, s) and f ⋆ (X, t) = lim inf (Y,s)→(X,t)
f (Y, s). Assume that L is bounded on (0, T ) and ρ ± 0 is locally bounded on R. Assume moreover that

L + sub = L - super = L ⋆ and L - sub = L + super = L ⋆ .
(1) (Discontinuous viscosity sub-solution)

An upper semi-continuous function ρ ± on R × [0, T ) is a discontinuous viscosity sub-solution of (4.6) if it satises

(i) ρ ± (x, 0) (ρ ± 0 ) ⋆ (x),
(ii) for all k ∈ {+, -} and for any test-

function ϕ ∈ C 1 (R × (0, T )) such that ρ k -ϕ reaches a local maximum at a point (x 0 , t 0 ) ∈ R × (0, T ), we have ∂ϕ ∂t (x 0 , t 0 ) + k ( (ρ + -ρ -)(x 0 , t 0 ) + L k sub (t 0 )
) ∂ϕ ∂x (x 0 , t 0 ) 0.

(4.58)

(2) (Discontinuous viscosity super-solution)

A lower semi-continuous function

ρ ± on R × [0, T ) is a discontinuous viscosity super-solution of (4.6) if it satises (i) ρ ± (x, 0) (ρ ± 0 ) ⋆ (x),
(ii) for all k ∈ {+, -} and for any test-

function ϕ ∈ C 1 (R × (0, T )) such that ρ k -ϕ reaches a local minimum at a point (x 0 , t 0 ) ∈ R × (0, T ), we have ∂ϕ ∂t (x 0 , t 0 ) + k ( (ρ + -ρ -)(x 0 , t 0 ) + L k super (t 0 ) ) ∂ϕ ∂x (x 0 , t 0 ) 0. (4.59)
(3) (Discontinuous viscosity solution)

Finally, we say that a locally bounded function ρ ± dened on R×[0, T ) is a discontinuous viscosity solution of (4.6) if its upper semi-continuous (respectively lower semi-continuous) envelope is a viscosity sub-solution (respectively super-solution).

We also need to recall the following stability result of discontinuous viscosity solutions. Assume that, for ϵ > 0, ρ ± ε is an upper semi-continuous viscosity sub-solution (resp. a lower semi-continuous viscosity super-solution) of

∂ρ ± ε ∂t (x, t) = ∓ ( (ρ + ε -ρ - ε )(x, t) + L ε (t) ) ∂ρ ± ε ∂x (x, t) in R × (0, T ),
where (L ϵ ) ϵ is a sequence of uniformly locally bounded functions on (0, T ). Assume moreover that the functions ρ ± ε are uniformly locally bounded on R × (0, T ). Then, the functions ρ ± dened by (4.9) (resp. ρ ± dened by (4.10)) is a viscosity sub-solution (resp. super-solution) of the system

∂ρ ± ∂t (x, t) = ∓ ( (ρ + -ρ -)(x, t) + L ± sub (t) ) ∂ρ ± ∂x (x, t) in R × (0, T ),
(resp. of the system

∂ρ ± ∂t (x, t) = ∓ ( (ρ + -ρ -)(x, t) + L ± super (t) ) ∂ρ ± ∂x (x, t) in R × (0, T )),
where

L + sub = L - super = L and L - sub = L + super = L,
and

L(t) = lim inf ⋆ L ϵ (t), L(t) = lim sup ⋆ L ϵ (t).
Remark 4.5.3. The previous stability result allows us to pass to the limit (in the viscosity sense) in the non-linear term, by only imposing L ∞ uniform bounds on the solution.

We end up this subsection by showing the following nite speed of propagation property, valid for continuous viscosity solutions of system (4.43).

Lemma 4.5.4. (Finite speed propagation property) Under the assumption (4.22), if v ± is the unique continuous viscosity solution of (4.43), given by Theorem 4.4.2 (i), then there exists a positive constant C > 0 such that v ± satises, for all h 0, the following estimate

inf |y-x| Ct v ± (y, h) v ± (x, t + h) sup |y-x| Ct v ± (y, h), for all (x, t) ∈ R × [0, T -h), ( 4.60)
where

C = 2N 0 + ∥ L∥ L ∞ (0,T )
, and N 0 is dened in (4.30).

Proof. Let us start by proving the right hand side of (4.60), in viscosity sense, namely

v ± (x, t + h) sup |y-x| Ct v ± (y, h). ( 4.61)
We set v ±,h (x, t) = v ± (x, t + h). Then, we can see that

∂v ±,h ∂t (x, t) = ∂v ± ∂t (x, t + h) = ∓ ( (v +,h -v -,h )(x, t) + Lh (t) ) ∂v ±,h ∂x (x, t) ,
where Lh (t) = L(t + h). Thus, v ±,h is a viscosity solution of the following system

∂w ± ∂t (x, t) = ∓ ( (w + -w -)(x, t) + Lh (t) ) ∂w ± ∂x (x, t) , with w ± (x, 0) = v ± (x, h). Since ∥(v +,h -v -,h ) + Lh ∥ L ∞ (R×(0,T )) 2N 0 + ∥ L∥ L ∞ (0,T ) = C, we can deduce that v ±,h is viscosity sub-solution of the following system ∂w ± ∂t = C ∂w ± ∂x , with w ± (x, 0) = v ± (x, h). ( 4.62)
Furthermore, if we denote

α ± (x, t) = sup |y-x| Ct v ± (y, h),
then, by Lax-Oleinik formula (See [10, Lemma 2.1]), we know that α ± is the unique continuous viscosity solution of (4.62). By the comparison principle (see [17, Th 1.1]), we deduce that

v ± (x, t + h) α ± (x, t) on R × (0, T ),
which implies (4.61). The same arguments lead to the inequality

inf |y-x| Ct v ± (y, h) v ± (x, t + h),
by considering the equation

∂w ± ∂t = -C ∂w ± ∂x .

(ii)

We only show the proof for the sub-solution, as the case for super-solution follows similarly. Let ρ ± ϵ be the solution of (4.49), constructed in Theorem 4.1.1 (i). We have to prove that the relaxed semi-limit ρ ± , dened in (4.9), is a discontinuous viscosity sub-solution of (4.6), in the sense of Denition 4.5.1. We proceed in two steps.

Step 1 (Meaning of the initial data) It is sucient to prove the following inequality

ρ ± (x, 0) (ρ ± 0 ) ⋆ (x) = (P ± 0 ) ⋆ (x) + L 0 x for all x ∈ R. (4.63)
From the denition of ρ ± , we know that there exists a sequence (ϵ n , x ϵn , t ϵn ) → (0, x, 0), such that when n → +∞

ρ ± (x, 0) = lim n-→+∞ ρ ± ϵn (x ϵn , t ϵn ). Let C ϵn = 2M ϵn 0 + ∥L ϵn ∥ L ∞ (0,T ) and C = 2M 0 + ∥L∥ L ∞ (0,T ) , (4.64)
where M ϵ 0 and M 0 are dened in (4.55) and (4.14), respectively. Using Lemma 4.5.4, with h = 0 and t = t ϵn , we get

ρ ± ϵn (x ϵn , t ϵn ) sup |y-xϵ n | Cϵ n tϵ n ρ ± ϵn (y, 0) sup |y-xϵ n | Cϵ n tϵ n ( P ± 0,ϵn (y) + L 0 y ) sup |y-xϵ n | Ctϵ n ( (P ± 0 ⋆ η ϵn )(y) + L 0 y ) sup |y-xϵ n | Ctϵ n (∫ R P ± 0 (z)η ϵn (y -z)dz + L 0 y ) sup |y-xϵ n | Ctϵ n ( sup |z-y| ϵn P ± 0 (z) + L 0 y ) ,
where we have used, in the second and the third lines, the denition of the functions ρ ± 0,ϵn , L ϵn in (4.50), and some of the basic properties of the molliers. Furthermore, the convergence, as n → +∞, of (ϵ n , x ϵn , t ϵn ) n to (0, x, 0), implies that for all ζ > 0,

P ± 0 (z) + sup |y-x| ζ(1+C) L 0 y. (4.65)
Now, we pass to the limit as n → +∞ in the previous inequality to obtain

ρ ± (x, 0) sup |z-x| ζ(2+C) P ± 0 (z) + sup |y-x| ζ(1+C) L 0 y.
Then, we pass to the limit as ζ → 0 to complete the proof of (4.63).

Step 2 (Meaning of the system) We will show that ρ ± is a discontinuous viscosity sub-solution of the following system

∂ρ ± ∂t ± ( (ρ + -ρ -) + L ± sub ) ∂ρ ± ∂x = 0 in R × (0, T ), (4.66)
where L + sub = L ⋆ and L - sub = L ⋆ (see Denition 4.5.1). Indeed, using the denition of L ϵ in (4.50)

and the classical properties of the molliers, we know that

∥L ϵ ∥ L ∞ (0,T ) ∥L∥ L ∞ (0,T ) .
This proves, according to estimate (4.11), and the stability result of discontinuous viscosity solution (announced in Theorem 4.5.2), that ρ ± is a discontinuous viscosity sub-solution of the equation

∂ρ ± ∂t ± ( (ρ + -ρ -) + L ± sub ) ∂ρ ± ∂x = 0 in R × (0, T ),
where L + sub = L and L sub = L. Moreover, we also know that there exists a sequence (ϵ n , t ϵn ) → (0, t), as n goes to +∞, such that

L(t) = lim n-→+∞ L ϵn (t ϵn ).
Using (4.50), together with the fact that η ϵn 0 and

∫ R η ϵn = 1, we get L ϵn (t ϵn ) = ∫ T 0 L(τ )η ϵn (t ϵn -τ ) dτ sup |τ -tϵ n | ϵn L(τ ).
Thanks to the convergence, as n → +∞, of (ϵ n , t ϵn ) n to (0, t), we can deduce, as in (4.65), that for every ζ > 0 there exists n ζ > 0 such that, for all n n ζ , we have

L ϵn (t ϵn ) sup |τ -t| 2ζ L(τ ).
Finally, we pass to the limit as n → +∞, ζ → 0, successively, in order to we get L(t) L ⋆ (t). In the same way, we can prove that L ⋆ (t) L(t). This proves that ρ ± is a discontinuous viscosity sub-solution of (4.66).

Preliminary results

In the following lemma, we recall some basic properties of bounded BV -functions.

Lemma 4.6.1. (Properties of BV -functions [START_REF] Ambrosio | Minimizing movements[END_REF]) Let f be a bounded BV -function on I = (a, b) ⊂ R. Then, the following hold i) f is continuous except at most on a countable set.

ii) The right and left limits

f (x + ) = lim y→x, y>x f (y), f (x -) = lim y→x, y<x f (y),
both exist at every point x ∈ I. Moreover, there exists a unique right-continuous function f r (resp. left-continuous function f l ) coinciding with f except at most on a countable set.

iii) There exists a pair of non-decreasing and bounded functions

f 1 , f 2 such that f = f 1 -f 2 .
The next lemma shows a local estimate that is valid for sequences of non-decreasing functions strongly convergent in L 1 (-2, 2).

Lemma 4.6.2. (Sequences of non-decreasing functions) i) Sequences of non-decreasing functions strongly convergent in L 1 (-2, 2) Let (ϕ ϵ ) ϵ be a sequence of non-decreasing functions, dened on [-2, 2], such that ϕ ϵ → ϕ strongly in L 1 (-2, 2), as ε → 0, where ϕ is a non-decreasing dened function on [-2, 2]. Then, for all 0 < δ 1, there exists ϵ δ > 0, such that, for every 0 < ϵ ϵ δ , the following estimate holds

-δ + ϕ(x -δ) ϕ ϵ (x) δ + ϕ(x + δ), ∀x ∈ [- 1, 1]. (4.67)
ii) Sequences of non-decreasing functions strongly convergent in C([0, T ); L 1 (-2, 2)) Let (ϕ ϵ ) ϵ be a sequence of functions, dened on R × [0, T ), such that, for all t ∈ [0, T ), the function ϕ ϵ (•, t) is non-decreasing on [-2, 2]. Assume, moreover, that ϕ ϵ → ϕ strongly in C([0, T ); L 1 (-2, 2)), as ε → 0, with, for all t ∈ [0, T ), the function ϕ(•, t) is dened and nondecreasing on [-2, 2]. Then, for all 0 < δ 1, there exists ϵ δ T > 0 such that, for every 0 < ϵ ϵ δ T , the following estimate holds

-δ + ϕ(x -δ, t) ϕ ϵ (x, t) δ + ϕ(x + δ, t), ∀x ∈ [-1, 1], ∀t ∈ [0, T ). (4.68)
Proof Proof of i) We will just prove the right hand side of estimate (4.67). Since ϕ ϵ -→ ϕ strongly in L 1 (-2, 2), then for all 0 < δ 1, there exits ϵ δ > 0, such that, for all 0 < ϵ ϵ δ , we have

∥ϕ ϵ -ϕ∥ L 1 (-2,2) δ 2 . (4.69)
Furthermore, since ϕ ϵ and ϕ are non-decreasing functions, then, for all x ∈ [-1, 1], δ ∈ (0, 1] and y ∈ (x, x + δ), we have ϕ ϵ (x) -ϕ(x + δ) ϕ ϵ (y) -ϕ(y).

Integrating with respect to y over the interval (x, x + δ), we get

ϕ ϵ (x) -ϕ(x + δ) 1 δ ∥ϕ ϵ -ϕ∥ L 1 (-2,2) .
From (4.69), the above estimate implies that, for all 0 < δ 1 and 0 < ϵ ϵ δ ,

ϕ ϵ (x) δ + ϕ(x + δ), for all x ∈ [-1, 1].
The left hand side of (4.67) is proved in the same way.

Proof of ii) To prove (4.68), we repeat the same procedure as in the proof of i), using the strong

convergence in C([0, T ); L 1 (-2, 2)) of ϕ ϵ . 2
Finally, in the following lemma, we show a local estimate that is valid for sequences of bounded BV -functions, strongly convergent in L 1 (-2, 2).

Lemma 4.6.3. (local estimate for a sequence of BV -functions) Let (ϕ ϵ ) ϵ be a sequence of functions dened on [-2, 2], uniformly bounded in L ∞ (-2, 2)∩BV (-2, 2) and strongly convergent to

ϕ ∈ L ∞ (-2, 2) ∩ BV (-2, 2) in L 1 (-2, 2)
, with ϕ a right-continuous function. Then, there is a subsequence (ϕ ϵ ′ ) ϵ ′ , for any 0 < δ 1, there exists ϵ δ > 0, such that, for all 0 < ϵ ′ ≤ ϵ δ , the following estimate holds

-2δ + ϕ 1 (x -δ) -ϕ 2 (x + δ) ϕ ϵ ′ (x) 2δ + ϕ 1 (x + δ) -ϕ 2 (x -δ), ∀x ∈ [-1, 1], (4.70)
where ϕ 1 and ϕ 2 are two locally bounded, right-continuous and non-decreasing functions on [-2, 2]

satisfying ϕ = ϕ 1 -ϕ 2 .
Proof. The proof of this lemma follows from Lemma 4.6.2 (i), and the properties (ii) and (iii)

of Lemma 4.6.1. Again, we will only show the right hand side of (4.70). Indeed, let (ϕ ϵ ) ϵ be a sequence of uniformly bounded functions in BV (-2, 2) ∩ L ∞ (-2, 2). Then, there exists two bounded, non-decreasing functions ϕ 1 ϵ and ϕ 2 ϵ satisfying

ϕ ϵ = ϕ 1 ϵ -ϕ 2 ϵ , (4.71)
and dened as

ϕ 1 ϵ (x) = T V [-2, x](ϕ ϵ ) + ϕ ϵ (x), ϕ 2 ϵ (x) = T V [-2, x](ϕ ϵ ), (4.72)
where T V [-2, x](ϕ ϵ ) is the total variation of ϕ ϵ on [-2, x]. Furthermore, since (ϕ 1 ϵ ) ϵ is uniformly bounded in L ∞ (-2, 2) ∩ BV (-2, 2), then, by Helly's compactness Theorem (see [START_REF] Ambrosio | Minimizing movements[END_REF]Th 3.23]) and by using the weak-⋆ convergence in L ∞ (-2, 2) ∩ BV (-2, 2), we can extract a subsequence (ϕ

1 ϵ ′ ) ϵ ′ such that ϕ 1 ϵ ′ -→ ϕ 1 strongly in L 1 (-2, 2), with ϕ 1 ∈ L ∞ (-2, 2) ∩ BV (-2, 2
). According to the property (ii) of Lemma 4.6.1, we know that ϕ 1 coincides with a right-continuous function almost everywhere in R. Since the two functions are equal in L 1 (-2, 2), we can choose ϕ 1 as a right-continuous function. Therefore, the function ϕ 1 will be a dened and non-decreasing function on R. Moreover, using the equality (4.71) and the strong convergence of (ϕ ϵ ) ϵ in L 1 (-2, 2), we also get

ϕ 2 ϵ ′ -→ ϕ 2 strongly in L 1 (-2, 2),
where ϕ 2 is a right-continuous and non-decreasing function on (-2, 2) satisfying ϕ 2 = ϕ 1 -ϕ. Thus, by Lemma 4.6.2 (i), we deduce that, for all 0 < δ 1, there exists ϵ δ 1 and ϵ δ 2 such that the following hold for all

0 < ϵ ′ ≤ ϵ δ 1 , ϕ 1 ϵ ′ (x) δ + ϕ 1 (x + δ), ∀ x ∈ [-1, 1],
and for all

0 < ϵ ′ ≤ ϵ δ 2 , -δ + ϕ 2 (x -δ) ϕ 2 ϵ ′ (x), ∀ x ∈ [-1, 1].
Collecting the two previous estimates with equality (4.71), we obtain that, for all 0 < δ 1 and 0

< ϵ ′ ≤ ϵ δ = inf(ϵ δ 1 , ϵ δ 2 ), ϕ ϵ ′ (x) 2δ + ϕ 1 (x + δ) -ϕ 2 (x -δ), ∀x ∈ [-1, 1].
The left hand side of (4.70) is obtained in a similar way.

2

Remark 4.6.4. In the previous lemma, we have decided to choose, for a uniformly bounded sequence in BV (-2, 2), a right-continuous limit among the strong L 1 (-2, 2) limits. We may have also replace it by a left-continuous one, which isn't an issue since, it is only necessary to construct a limit in L 1 (-2, 2), that is dened and non-decreasing everywhere in (-2, 2). Let ρ ± ϵ be the solution of (4.49), constructed in Theorem 4.1.1 (i). We know that

ρ ± ϵ (x, t) = P ± ϵ (x, t) + L 0 x, (4.73)
where P ± ε are 1-periodic functions (with respect to x). Estimates (4.11), (4.12), (4.13), together with the fact that

∂P ± ϵ ∂t = ∂ρ ± ϵ ∂t and ∂P ± ϵ ∂x + L 0 = ∂ρ ± ϵ ∂x , infer that, (P ± ϵ ) ϵ is uniformly bounded in L ∞ ((0, T ); BV (T)) ∩ L ∞ (T × (0, T )) and ( ∂P ± ϵ ∂t ) ϵ is uniformly bounded in L ∞ ((0, T ); L 1 (T)).
Using Simon's lemma (see Lemma 4.4.4) in the particular setting X = BV (T), B = Y = L 1 (T), and the following compact embedding BV (T) → L 1 (T), we can extract a subsequence, denoted by (P ± ϵn ) ϵn , that converges strongly in C([0, T ); L 1 (T)) to some limit P ± , as n → +∞. Now, thanks to estimates (4.11) and (4.12), we can extract yet another subsequence, still denoted by (P ± ϵn ) ϵn , satisfying P ± ϵn -→ P ± , strongly in C([0, T ); L 1 (T)),

P ± ϵn -→ P ± , weakly-⋆ in L ∞ (T × (0, T )), P ± ϵn -→ P ± , weakly-⋆ in L ∞ ((0, T ); BV (T)). (4.74)
Passing to the limit in (4.73), and taking the inferior limit lim inf in estimates (4.11), (4.12), using the lower semi-continuity of ∥ • ∥ L ∞ and | • | BV , we can prove that ρ ± satises (4.15), (4.16), (4.17) and (4.18). Moreover, using estimate (4.13), Remark 4.4.5 and the strong convergence in C([0, T ); L 1 (T)) of P ± ϵn , we can check that ρ ± also satises the L 1 -Lipschitz estimate in time (4.19). Since, for all t ∈ [0, T ), the function P ± (•, t) ∈ L ∞ (T) ∩ BV (T), then, by property (ii) of Lemma 4.6.1, we know that this function coincides with a right-continuous function almost everywhere in T and consequently in L 1 (T). This allows us to consider, in the upcoming arguments, a right-continuous limit with respect to the space variable. Now, we will prove that ρ ± is a discontinuous viscosity solution of (4.6). Since we have already proved, in Theorem 4.1.1 (ii), that ρ ± and ρ ± are respectively discontinuous viscosity sub-and super-solutions, then it is sucient to show that We only show the proof of the rst equality; The second is treated similarly. The remaining part of the proof is divided into two steps.

ρ ± (x, t) = (ρ ± ) ⋆ (x, t) and ρ ± (x, t) = (ρ ± ) ⋆ (x, t) for all (x, t) ∈ R × [0, T ).
Step 1 (Fundamental inequality) We prove the following inequality

ρ ± (x, t) (ρ ± ) ⋆ (x, t) for all (x, t) ∈ R × [0, T ). (4.76) Let x ∈ [-1 2 , 1 2
] and t ∈ [0, T ). In fact, by the denition of ρ ± , we know that there exists a sequence (ϵ m , x ϵm , t ϵm ) → (0, x, t), when m → +∞, such that

ρ ± (x, t) = lim m-→+∞ ρ ± ϵm (x ϵm , t ϵm ).
On the other hand, for all α > 0, we can ensure the existence of m α > 0 such that, for all m m α , we have |x ϵm -x| α and |t ϵm -t| α.

However, using Lemma 4.5.4, with

h α =    t -α if t > 0, 0 if t = 0,
we deduce that, for all α > 0 such that h α 0, and for all m m α ,

ρ ± ϵm (x ϵm , t ϵm ) sup |y-xϵ m | C(tϵ m -hα) ρ ± ϵm (y, h α ) sup |y-xϵ m | 2αC ρ ± ϵm (y, h α ) sup |y-x| α(2C+1) ρ ± ϵm (y, h α ),
where the constant C is dened in (4.64). Moreover, from the maximum principle of the Eikonal system (4.49), and since the initial data is non-decreasing, we know that ρ ± εm are non-decreasing (with respect to x) and therefore, for all m m α ,

ρ ± ϵm (x ϵm , t ϵm ) ρ ± ϵm (x + α(2C + 1), h α ). Now, as ρ ± ϵm (x, t) = P ± ϵm (x, t) + L 0 x,
and since ρ ± ϵm are non-decreasing functions (with respect to x) satisfying (4.11), (4.12), (4.13), we can extract a subsequence, still denoted by (ρ ± ϵm ) ϵm , that converges in the sense of (4.74) to ρ ± . Note that the convergence in (4.74) is valid on the 1-periodic interval and in particular on (-2, 2). Since ρ ± is the sum of a 1-periodic and a linear function, we can easily check that the function ρ ± (•, t) is dened and non-decreasing on [-2, 2]. By the previous inequality and Lemma 4.6.2 (ii), we obtain that, for all 0 < α ≤ α 0 = α 0 (C, t), where

α 0 (C, t) =      min ( 1 2(2C+1) , t 2 ) if t > 0, 1 2(2C+1) , if t = 0, there exists m α T > 0, such that, for every m m α T , ρ ± ϵm (x ϵm , t ϵm ) ρ ± (x + 2α(C + 1), h α ) + α.
Passing to the limit m → +∞ and then α → 0, we obtain (4.76) for all x ∈ [-1 2 , 1 2 ]. Once again, since ρ ± is the sum of a 1-periodic and a linear function, we nally get (4.76) for all x ∈ R.

Step 2 (Conclusion) It remains to show that

(ρ ± ) ⋆ (x, t) ρ ± (x, t) for all (x, t) ∈ R × [0, T ). (4.77) Let x ∈ [-1 2 , 1 2
] and t ∈ [0, T ). In fact, from the denition of (ρ ± ) ⋆ , we know that there exists a sequence (x ϵm , t ϵm ) → (x, t) when m → +∞, such that

(ρ ± ) ⋆ (x, t) = lim m-→+∞ ρ ± (x ϵm , t ϵm ).
Similarly (as in Step 1), for all α > 0, there exists m α > 0 such that, for all m m α , |x ϵm -x| α and |t ϵm -t| α.

However, using Lemma 4.6.2 (ii), we know that, for all 0 < α 1 2 , there exists k α T > 0 and a subsequence 0 < α k α such that, for every k k α T ,

ρ ± (x ϵm , t ϵm ) ρ ± α k (x ϵm + α, t ϵm ) + α sup α k ≤α,|s-t| α |y-x| 2α ρ ± α k (y, s) + α.
Using the periodicity property, and passing to the limit m → +∞ then α → 0, we obtain (4.77).

(iii)

Let ρ ± ϵ be the solution of (4.49), constructed in Theorem 4.1.1 (i). As explained at the beginning of the proof of Theorem 4.1.2, we can extract a subsequence ρ ± ϵn (x, t) = P ± ϵn (x, t) + L 0 x, where P ± ϵn is a 1-periodic function that converges, in the sense of (4.74), to a limit P ± with ρ ± = P ± + L 0 x satisfying (4.15), (4.17), (4.18) and (4.19). Moreover, for all t ∈ [0, T ), ρ ± (•, t) is a right-continuous function on R. It remains to show equality (4.20). For a better presentation, we will perform this in three steps.

Step 1 (Regularity in time estimate) Let T > 0, and set γ c = 2(C + 1), where C is dened in (4.64). First, we will show that there are two bounded and non-decreasing functions ρ

± 1 , ρ ± 2 satisfying ρ ± = ρ ± 1 -ρ ±
2 , and the following inequalities

-2h + ρ ± 1 (x -hγ c , t) -ρ ± 2 (x + hγ c , t) ρ ± (x, t + h) ρ ± (x, t + h) 2h + ρ ± 1 (x + hγ c , t) -ρ ± 2 (x -hγ c , t) (4.78)
valid for all x ∈ [-

) 2h + ρ ± 1 (x + hγ c , t) -ρ ± 2 (x -hγ c , t).
Accordingly, we pass to the limit as n → +∞ to get (4.80). Similarly, using the nite speed of propagation property, specically the left inequality in (4.60), and the fact that ρ ± ϵm (•, t) is uniformly bounded in L ∞ (-2, 2) ∩ BV (-2, 2), we can prove the left inequality in (4.78), namely,

-2h + ρ ± 1 (x -hγ c , t) -ρ ± 2 (x + hγ c , t) ρ ± (x, t + h). (4.83)
Step 2 (Right and left continuity) Let T > 0 and t ∈ [0, T ). Since ρ ± 1 (•, t), ρ ± 2 (•, t) are bounded and non-decreasing functions on [-2, 2], then, from property (ii) of Lemma 4.6.1, we know that, the right and left limits of these functions exist at every point x ∈ (-2, 2). This implies that, for all α > 0 and x ∈ [- 1 2 , 1 2 ], there exists h α t > 0, such that, for all 0 < z h α t , we have

ρ ± 1 (x + z, t) α 4 + ρ ± 1,r (x, t) ρ ± 2 (x + z, t) α 4 + ρ ± 2,r (x, t) and -ρ ± 1 (x -z, t) α 4 -ρ ± 1,l (x, t) -ρ ± 2 (x -z, t) α 4 -ρ ± 2,l (x, t) (4.84)
where ρ ± 1,r (•, t), ρ ± 2,r (•, t), are right-continuous functions on (-2, 2), and ρ ± 1,l (•, t), ρ ± 2,r (•, t) are left-continuous functions on (-2, 2). Note that, as a consequence of Heine-Cantor Theorem, the choice of the constant h α t does not depend on x. Now, let T > 0, t ∈ [0, T ) and α > 0. We observe that, if we denote hα t,T = min

( h α t γ c , 1 2(2C + 1) , α 4 , T -t 2 ) ,
then, for all 0 < h hα t,T , the assumption (4.79) holds. Therefore, we obtain Step 3 (Link between ρ ± and ρ ± ) Let T > 0, x ∈ [-1 2 , 1 2 ], t ∈ [0, T ) and α > 0. In Step 2, we have shown that there exists a positive constant hα t,T , such that, for all 0 < h hα

ρ ± (x, t + h) 2h + ρ ± 1 (x + hγ c , t) -ρ ± 2 (x -hγ c , t) α + ρ ± 1,r (x, t) -ρ ± 2,l (x,
t,T ρ ± (x, t + h) α + ρ ± 1,r (x, t) -ρ ± 2,l (x, t), -ρ ± (x, t + h) α -ρ ± 1,l (x, t) + ρ ± 2,r (x, t). (4.87) Since ∪ t∈[0,T ) [t, t + hα t,T ] is a cover of [0, T 2 ]
, then there is a nite number N α of ordered intervals, satisfying

∪ 0 i N α [τ α i , τ α i + hα τ α i ,T ] ⊃ [ 0, T 2
] with τ α 0 = 0 and τ α i+1 = τ α i + hα

τ α i ,T for i = 1, . . . , N α -1.
This expression, joint with (4.87), show that for all x ∈ [-1 2 , 1 2 ], τ ∈ [0, T 2 ], and for any positive constant α ∈ Q, there exists an index 0 j N α , such that

ρ ± (x, τ ) α + ρ ± 1,r (x, τ α j ) -ρ ± 2,l (x, τ α j ), -ρ ± (x, τ ) α -ρ ± 1,l (x, τ α j ) + ρ ± 2,r (x, τ α j ). (4.88)
Moreover, from property (ii) of Lemma 4.6.1, we know that, for any positive constant α ∈ Q, and 0 i N α , the functions ρ ± 1,r (•,

τ α i ), ρ ± 1,l (•, τ α i ) (resp. ρ ± 2,r (•, τ α i ), ρ ± 2,l (•, τ α i )) coincide with ρ ± 1 (•, τ α i ) (resp. ρ ± 2 (•, τ α i )) except on a countable set on [-1 2 , 1 2
], denoted by D α i . Now, we dene the following countable set

D = ∪ α∈Q ∪ 0 i N α D α i .
Thanks to (4.88), we can see that, for all x ∈ [-1 2 , 1 2 ]\D, τ ∈ [0, T 2 ], and for all positive constant α ∈ Q, there exist indexes 0 j N α , such that Adding the previous inequalities, we deduce that, for all rational number α > 0, x ∈ [-1 2 , 1 2 ]\D and τ ∈ [0, T 2 ], 0 ρ ± (x, τ ) -ρ ± (x, τ ) 2α.

ρ ± (x, τ ) α + ρ ± 1 (x, τ α j ) -ρ ± 2 (x, τ α j ) α + ρ ± (x, τ α j ), -ρ ± (x, τ ) α -ρ ± 1 (x, τ α j ) + ρ ± 2 (x, τ α j ) α -ρ ± (x, τ α j ).
Passing to the limit α → 0; replacing T by 2T , and using the 1-periodicity assumption, we get ρ ± (•, τ ) = ρ ± (•, τ ) except at most on a countable set in R, for all τ ∈ [0, T ]. To do so, it is sucient to use the right continuity of the functions ρ ± (•, τ ), ρ ± 1 (•, τ ), and Lemma 4.6.2 (i). Indeed, for α > 0, the right continuity of the functions ρ ± (•, τ ), ρ ± 1 (•, τ ) implies that, for all x ∈ [- 1 2 , 1 2 ], there exists α 1 τ > 0, such that, for all 0 < δ α 1 τ , we have

ρ ± (x, τ ) α + ρ ± (x + δ, τ ) = α + ρ ± 1 (x + δ, τ ) -ρ ± 2 (x + δ, τ ) 2α + ρ ± 1 (x, τ ) -ρ ± 2 (x + δ, τ ), (4.91)
where ρ ± 1 , ρ ± 2 are the right-continuous non-decreasing functions given in (4.82). However, using Lemma 4.6.2 (i), we know that, for all 0 < δ 1 2 , there exists k α τ > 0, and a subsequence 0 < ϵ k δ, such that for every k k α τ ,

ρ ± 1 (x, τ ) δ 2 + ρ ± 1,ϵ k (x + δ 2 , τ ), -ρ ± 2 (x + δ, τ ) δ 2 -ρ ± 2,ϵ k (x + δ 2 , τ ) , ( 4
ρ ± (x, τ ) 2α + δ + ρ ± ϵ k (x + δ 2 , τ ) 2α + δ + sup ϵ k ≤δ,|s-τ | δ |y-x| 2δ ρ ± ϵ k (y, s).
To complete the proof, we pass to the limit as δ → 0 and then α → 0, to get ρ ± (x, τ ) ρ ± (x, τ ) rstly for all x ∈ [-1 2 , 1 2 ] and then, according to the periodicity, for all x ∈ R. Similarly, we can show that ρ ± (x, τ ) ρ ± (x, τ ), which, joint with (4.90), proves the desired result.

2

Note that, ρ ± and ρ ± satisfy (4.16), after replacing P ± by P ± and P ± , respectively. Note also that, equality (4.20) implies these semi-relaxed limits satisfy estimates (4.17), (4.18) and (4.19).

Application to dislocations densities dynamics

In this section, we study the non-local system (4.1) describing the dynamics of dislocation densities. This is a simplied 1-d model derived from the original one of Groma and Balogh [START_REF] Groma | Investigation of dislocation pattern formation in a two-dimensional self-consistent eld approximation[END_REF],

initially introduced in 2-d setting (see El Hajj and Forcadel [38,Lemme 3.1] for modeling details).

Indeed, we consider the following non-local Eikonal system We consider, for ϵ > 0, the following regularized problem of (4.93)

       ∂ρ ± ∂t (x, t) = ∓ ( (ρ + -ρ -)(x, t) + α ∫ 1 0 (ρ + -ρ -)(y, t)dy + a(t) ) ∂ρ ± ∂x (x, t) in R × (0, T ), ρ ± (x, 0) = ρ ± 0 (x) = P ± 0 (x) + L 0 x in R,
       ∂ρ ± ϵ ∂t (x, t) = ∓ ( (ρ + ϵ -ρ - ϵ )(x, t) + α ∫ 1 0 (ρ + ϵ -ρ - ϵ )(y, t)dy + a ϵ (t) ) ∂ρ ± ϵ ∂x (x, t) in R × (0, T ), ρ ± ϵ (x, 0) = ρ ± 0,ϵ (x) = P ± 0,ϵ (x) + L 0 x in R, (4.94)
where P ± 0,ϵ and a ϵ (•) are the regularization of P ± 0 and a(•), respectively, as dened in (4.50). We dene, for all ϵ > 0 and T > 0, the set

X ϵ T =                            v ± ϵ = Q ± ϵ + L 0 x, such that Q ± ϵ ∈ L ∞ (T × (0, T )), and
∥Q ± ε ∥ L ∞ (T×(0,T )) I ∂v ± ε ∂x L ∞ (T×(0,T )) J ε ∂v ± ε ∂x L ∞ ((0,T );L 1 (T)) |ρ ± 0 | BV (R) ∂v ± ε ∂t L ∞ ((0,T );L 1 (T)) K                            with I = 2 ( max ± ( ∥P ± 0 ∥ L ∞ (T) ) + |L 0 |∥a∥ L ∞ (0,T ) T ) , J ε = max ±   ∂ρ ± 0,ε ∂x L ∞ (T)   and K = max ± ([ 2I + ∥a∥ L ∞ (0,T ) ] |ρ ± 0 | BV (R)
) .

Now, for v ± ϵ ∈ X ϵ T , we dene ρ ± ϵ = G(v ± ϵ ) as the unique Lipschitz continuous viscosity solution of the following local problem

       ∂ρ ± ϵ ∂t (x, t) = ∓ ( (ρ + ϵ -ρ - ϵ )(x, t) + α ∫ 1 0 (v + ϵ -v - ϵ )(y, t)dy + a ϵ (t) ) ∂ρ ± ϵ ∂x (x, t) in R × (0, T ), ρ ± ϵ (x, 0) = ρ ± 0,ϵ (x) = P ± 0,ϵ (x) + L 0 x in R.
(4.95)

We will show that G : X ϵ T -→ X ϵ T is a strict contraction for T small enough. First, we will prove that G is well dened. Let v ± ϵ ∈ X ϵ T . Using (4.2) and proceeding as in the proof of Theorem 4.1.1

(i), we can show that the system (4.95) admits a unique Lipschitz continuous viscosity solution ρ ± ϵ satisfying (4.51), (4.52), (4.53), (4.54) and (4.56), with

L ϵ (t) = α ∫ 1 0 (v + ϵ -v - ϵ )(y, t)dy + a ϵ (t).
This shows, by Young's inequality and by classical properties of the molliers, that ρ ± ϵ ∈ X ϵ T for

T 1 4α|L 0 | . It remains to show that G is a contraction. Let ρ ±,i ϵ = G(v ±,i ϵ ), i = 1, 2, and set L = max ± (∥v ±,1 ϵ -v ±,2 ϵ ∥ L ∞ (T×(0,T )) ).
We remark that ρ ±,2 ϵ is a viscosity sub-solution of

∂ρ ± ϵ ∂t (x, t) = ∓ ( (ρ + ϵ -ρ - ϵ )(x, t) + α ∫ 1 0 (v +,1 ϵ -v -,1 ϵ )(y, t)dy + a ϵ (t) ) ∂ρ ± ϵ ∂x (x, t) + 2αLJ ε .
Moreover, ρ ±,1 ϵ + 2αLJ ε t is a viscosity solution of the same equation. Using the comparison principle of this local system (see El Hajj and Forcadel [38,Th. 3.2]) and interchanging ρ ±,1

ϵ , ρ ±,2 ϵ , we get max ± (∥ρ ±,1 ϵ -ρ ±,2 ϵ ∥ L ∞ (T×(0,T )) ) 2αJ ε T max ± (∥v ±,1 ϵ -v ±,2 ϵ ∥ L ∞ (T×(0,T )) ).
This shows that, for T small enough, G is a contraction on X ϵ T which is a closed set. Hence, by the xed point theorem, there exists a unique Lipschitz continuous viscosity solution of (4.94) in

X ϵ T , for 0 < T min( 1 4α|L 0 | , 1 4αJε
). By iterating this process, we can construct a solution for all T > 0 (refer to Chapter 5 Section 5.2). The aim is to prove that the relaxed semi-limits ρ ± and ρ ± (dened in (4.9), (4.10)) are, respec- tively, discontinuous viscosity sub-solution and super-solutions of equation (4.93) (ρ + -ρ -)(y, t)dy+a ⋆ (t), respectively.

Note that, Theorem 4.1.4 (i) follows directly from the stability result of discontinuous viscosity solutions (see Theorem 4.5.2) and equality (4.20). We will only present the proof for ρ ± ; the proof for ρ ± is done in the same way. In fact, if ρ ± ϵ is the solution of (4.94), given in the previous subsection, then, using the the stability result, we can see that ρ + (resp. ρ -) is a discontinuous viscosity sub-solutions of the following equation

∂ρ + ∂t (x, t) = - ( (ρ + -ρ -)(x, t) + α ∫ 1 0 ((ρ + -ρ -)(y, t)dy + a(t) ) ∂ρ + ∂x (x, t) , ( resp. ∂ρ - ∂t (x, t) = ( (ρ + -ρ -)(x, t) + α ∫ 1 0 ((ρ + -ρ -)(y, t)dy + a(t) ) ∂ρ - ∂x (x, t)
) ,

where a(t) = lim inf ⋆ a ϵ (t) and a(t) = lim sup ⋆ a ϵ (t). This proves that ρ ± is discontinuous viscosity sub-solution of the following system

∂ρ ± ∂t (x, t) = ∓ ( (ρ + -ρ -)(x, t) + α ∫ 1 0 ((ρ + -ρ -)(y, t)dy + a ± sub (t) ) ∂ρ ± ∂x (x, t) ,
where a + sub = a ⋆ and a - sub = a ⋆ . Similarly, we can prove that ρ ± is discontinuous viscosity super-solution of the following system

∂ρ ± ∂t (x, t) = ∓ ( (ρ + -ρ -)(x, t) + α ∫ 1 0 ((ρ + -ρ -)(y, t)dy + a ± super (t) ) ∂ρ ± ∂x (x, t) ,
where a + super = a ⋆ and a - super = a ⋆ . Therefore, we obtain that ρ ± and ρ ± are, respectively, discontinuous viscosity sub and super-solutions of (4.93). To prove Theorem 4.1.4 (ii), we repeat the same process while using equality (4.75) instead of (4.20), which is only valid in the case of non-decreasing solutions. BV solution for a non-linear Hamilton-Jacobi system This article stands as a generalization of Chapter 4 to the d-dimensional case. Our work was done on a system of Hamilton-Jacobi equations whose velocities are non-signed and strongly coupled.

For our system, we were able to prove the global in time existence of a discontinuous solution by considering BV initial data with large total variation. In this article, we have used a particular denition of the discontinuous viscosity solutions which was proposed by Ishii [START_REF] Ishii | Perron's method for monotone systems of second-order elliptic partial dierential equations[END_REF]. Previous results on such models were done in the quasi-monotone case and in Lipschitz space. Some results also arose for strictly hyperbolic systems with small total variation which is a particular case of our system that includes the case of non-decreasing solution. As a generalization of our previous 2-dimensional Hamilton-Jacobi system, this result can be applied on dislocation dynamics in multi-directions of propagation. However, we will expose throughout this chapter an application on gas dynamics.

Introduction

Physical motivation and setting of the problem

In our work, we are looking for solutions of the form u(t, x) =

( u i (t, x) ) i=1,••• ,d of the following one dimensional Hamilton-Jacobi system    ∂ t u i (t, x) = λ i (t, x, u) ∂ x u i (t, x) in (0, T ) × R, u i (0, x) = u i 0 (x) in R, (5.1)
for T > 0 and i = 1, ..., d, where d ∈ N * . The function u i is real-valued, ∂ t u i and ∂ x u i stand, respectively, for its time and spatial derivatives.

Here, the velocity λ i is assumed to satisfy, for all i = 1, ..., d, the following assumption

λ i ∈ L ∞ ((0, T ) × R × K)
for T > 0 and for all compact K ⊂ R d .

(5.2)

We introduce also the below non-decreasing assumption of λ i with respect to the variable u i

For all u i v i , (r j ) j=1,••• ,d,j̸ =i ∈ R d-1 and (t, x) ∈ (0, T ) × R, we have λ i (t, x, r(u i )) λ i (t, x, r(v i ))
where r(

u i ) = (r 1 , • • • , u i , • • • , r d ) and r(v i ) = (r 1 , • • • , v i , • • • , r d ).
(

It is worth mentioning that this monotonicity assumption on λ i was not needed in the study of dislocation dynamics (refer to [START_REF] Hajj | A convergent scheme for a non-local coupled system modeling dislocations densities dynamics[END_REF]) and gas dynamics (refer to [START_REF] Hajj | Some niqueness results for diagonal hyperbolic systems equation with large and monotone data[END_REF]).

Our study of system (5.1) is motivated by the consideration of a model describing the dynamics of dislocation densities (see [START_REF] Hajj | Global continuous solutions for diagonal hyperbolic systems with large and monotone data[END_REF]Section 5] for more details about the modeling), which is given by

∂ t u i =   ∑ j=1,...,d B ij u j   |∂ x u i | for i = 1, . . . , d, (5.4)
where (B ij ) i,j=1,...,d is a real matrix. This model can be seen as a special case of system (5.1).

The goal of this work is to establish the global existence of discontinuous viscosity solutions of system (5.1) assuming assumption (5.2) and the following regularity on the initial data

u i 0 ∈ L ∞ (R) ∩ BV (R).
(5.5)

We remark that system (5.1) can be seen as the level-set approach system associated to the motion of the front Γ i t := {x : u i (t, x) = 0} with a normal velocity λ i (t, x, u) depending on the solution u and aected by λ j (t, x, [u]) for i ̸ = j (see for instance Barles et al. [START_REF] Barles | Front propagation and phase eld theory[END_REF]).

Many existence and uniqueness results were brightened up on similar Eikonal systems. Let us mention the most known results. First, motivated by dislocation dynamics, we can point out the result done by El Hajj and Boudjerada in [START_REF] Boudjerada | Global existence results for eikonal equation with BV initial data[END_REF] who were able to prove the global existence of discontinuous viscosity BV solutions for scalar one dimensional non-linear and non-local Eikonal equations, including in particular the case d = 1 in system (5.1), where the velocity does not contain the solution. Also, considering dislocation dynamics as a motivation, this result has been extended to a more general non-linear (2 × 2) system in El Hajj et al. [START_REF] Hajj | Global BV solution for a non-local coupled system modeling the dynamics of dislocation densities[END_REF]. Also, an existence and uniqueness result of a Lipschitz viscosity solution was proved by El Hajj and Forcadel in [START_REF] Hajj | A convergent scheme for a non-local coupled system modeling dislocations densities dynamics[END_REF] for the same system. In the case of general (d × d) system, it is worth mentioning the result of Ishii, Koike [START_REF] Ishii | Viscosity solution for monotone systems of second-order elliptic PDEs[END_REF] and Ishii [START_REF] Ishii | Perron's method for monotone systems of second-order elliptic partial dierential equations[END_REF], who had shown the existence and uniqueness of continuous viscosity solutions for Hamilton-Jacobi systems of the form

   ∂ t u i + H i (t, x, u, Du i ) = 0 with u = (u 1 , . . . , u d ) ∈ R d , x ∈ R N
, and t ∈ (0, +∞),

u i (0, x) = u i 0 (x) for x ∈ R N ,
where the Hamiltonian H i is quasi-monotone in u (see the denition in Ishii, Koike [START_REF] Ishii | Viscosity solution for monotone systems of second-order elliptic PDEs[END_REF]Th.4.7]).

The novelty of this result is mainly working in BV space to overcome, physically, the random distribution of dislocations in a crystal and the general form of our strongly coupled system of Hamilton-Jacobi equations. These properties originates our work from previous ones which were done relying on continuous data or in the frame work of strictly hyperbolic systems. Consequently, this work allows to give meaning to system (5.1) in the framework of discontinuous viscosity solutions. This enables us to enlarge the area of applications to touch dislocation dynamics in multi-directions of propagation, indicated in (5.4) (see [START_REF] Hajj | Global continuous solutions for diagonal hyperbolic systems with large and monotone data[END_REF]Section 5]). More precisely, we present a global existence result for the strongly coupled Hamilton-Jacobi system (5.1) considering large BV initial data. This result is obtained without sign restrictions on the velocity λ i and also unconditional monotonicity of the solution. We only consider the case when the initial data and the velocity satisfy the assumptions (5.2) and (5.5) without any better regularity. However, the state of having non-decreasing initial data is presented as a particular case of our work in Theorem 5.1.3. In its full generality, the fundamental issue of uniqueness for global solution remains open. This question is related to the fact that system (5.1) is not only non-linear but it is also non-monotone which means that the comparison principle, that plays a central role in the level-set approach, does not hold and then we cannot apply directly the viscosity solutions theory. Therefore, the uniqueness of solutions cannot be proved via standard viscosity solutions methods. We refer the reader to [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF][START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF][START_REF] Crandall | User's guide to viscosity solutions of second order partial dierential equations[END_REF] for a complete overview of viscosity solutions. We also refer to Barles [11] for an interesting counter-example on the uniqueness of discontinuous viscosity solution.

Let us now state the key steps followed to prove our existence result. First, by a classical convolution argument, we regularize the velocity and the initial data which were announced in system (5.1). This approximation brings us to consider, for every 0 < ε 1 and for i = 1, ..., d,

the following system    ∂ t u i ε (t, x) = λ i ε (t, x, ρ 1 ε ⋆ u ε (t, •)(x)) ∂ x u i ε (t, x) in (0, T ) × R, u i ε (0, x) = u i 0,ε (x) in R, (5.6)
where λ i ε and u i 0,ε are the regularization of the functions λ i and u i 0 respectively, and they are given by

u i 0,ε (x) = u i 0 ⋆ ρ 1 ε (x) and λ i ε (t, x, w) = λi ⋆ ρ d+2 ε (t, x, w) ∀ (t, x, w) ∈ R × R × R d , (5.7)
with λi is an extension in R d+2 of λ i by 0, and ρ n ε for n = 1, d + 2 are the standard molliers dened as follows

             ∂ t ρ + ∂ x (ρu) = 0 in (0, T ) × R, ∂ t (ρu) + ∂ x ( ρu 2 + p(ρ) ) = 0, with p(ρ) = (γ-1) 2 4γ ρ γ in (0, T ) × R, u(0, x) = u 0 (x) and ρ(0, x) = ρ 0 (x) 0, for x ∈ R, (5.19)
where ρ is the density, u is the speed and p(ρ) is the pressure given by a simple power law for an exponent γ > 1. First, we assume the following conditions u 0 , ρ 0 ∈ L ∞ (R) and u 0 ± ρ θ 0 are non-decreasing functions with θ = γ-1 2 .

(5.20)

We will prove the following result.

Theorem 5.1.5. Assume (5.20) is veried, with ρ 0 0 and T > 0. Then system (5.19) has a solution

(u, ρ) ∈ (L ∞ ( (0, T ) × R ) ) 2 in distributional sense, with ρ 0 and u, ρ θ ∈ L ∞ ( (0, T ) × R ) ∩ L ∞ ( (0, T ); BV (R) ) ∩ C ( [0, T ); L 1 loc (R)
) .

( The proof of this theorem, arises directly from the a global existence result of Lipschitz continuous solution proved in [START_REF] Hajj | Some niqueness results for diagonal hyperbolic systems equation with large and monotone data[END_REF] and announced in Theorem 5.5.1. We refer the reader to Section 5.5 for a detailed proof. Let us mention that, in the case ρ 0 > 0, T. T. Li proved in [68, pp. 35-41] an existence and uniqueness result for C 1 solutions. Notice that in the previous theorem, we only assume that ρ 0 0, which allows us to consider discontinuous solutions with vacuum. In connection to Theorem 5.1.5, let us also mention the work of Lions et al. in [START_REF] Lions | Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates[END_REF] where the existence of an entropy solution was obtained for ρ 0 0 with any u 0 , ρ 0 ∈ L ∞ (R) and γ > 1.

This extended a previous result of DiPerna [START_REF] Diperna | Convergence of approximate solutions to conservation laws[END_REF][START_REF] Diperna | Compensated compactness and general systems of conservation laws[END_REF]. We also refer the reader to El Hajj and Monneau [START_REF] Hajj | Some niqueness results for diagonal hyperbolic systems equation with large and monotone data[END_REF] where a global existence result was proved considering continuous solutions with vacuum.

Organization of the paper

The paper is organized as follows: Section 2 is devoted to show the global existence of Lipschitz continuous viscosity solution of the regularized system with uniform BV estimate (Theorem 5.1.1 (i)). In Section 3, we prove Theorem 5.1.1 (ii) by passing to the limit as ε goes to 0, using the compactness argument. In Section 4, we give the proof of Theorem 5.1.3 and Theorem 5.1.1 (iii), using the nite speed propagation property and the discontinuity property of non-decreasing functions. Finally, in Section 5, we present the proof of Theorem 5.1.5 concerning the global existence result of discontinuous solution for gas dynamics system.

Global Lipschitz continuous viscosity solution of the regularized system

This section is devoted to prove Theorem 5.1.1 (i). First, we recall in following subsection some useful results then we give the proof in next subsection.

Some useful results

In this subsection, we recall a global existence and uniqueness result for the following scalar eikonal equation

   ∂ t v(t, x) = c(t, x)|∂ x v(t, x)| in (0, T ) × R, v(0, x) = v 0 (x) in R, (5.22)
where the initial data and the velocity satisfy the following assumptions

v 0 ∈ W 1,∞ (R) and c ∈ W 1,∞ ((0, T ) × R).
ϕ ∈ C 1 ((0, T ) × R), if (t 0 , x 0 ) ∈ (0, T ) × R is a local maximum point of v -ϕ , we have ∂ t ϕ(t 0 , x 0 ) -c(t 0 , x 0 )|∂ x ϕ(t 0 , x 0 )| 0. A function v ∈ C([0, T ) × R
) is a viscosity super-solution of (5.22) if and only if v(0, x) v 0 (x) and for any ϕ ∈ C 1 ((0, T ) × R), if (t 0 , x 0 ) ∈ (0, T ) × R is a local minimum point of v -ϕ , we have

∂ t ϕ(t 0 , x 0 ) -c(t 0 , x 0 )|∂ x ϕ(t 0 , x 0 ))| 0.
A continuous function v is a viscosity solution of (5.22) if and only if it is a sub-and a supersolution of (5.22).

We have the following existence and uniqueness result for the scalar equation (5.22).

Theorem 5.2.2. (Existence and uniqueness of Lipschitz continuous viscosity solution) Assume that (5.23) holds, then, for all T > 0, we have i) The problem (5.22) admits a unique Lipschitz continuous viscosity solution on (0, T ) × R, satisfying the following estimates

∥v∥ L ∞ ((0,T )×R) ∥v 0 ∥ L ∞ (R) , (5.24) ∥∂ x v∥ L ∞ ((0,T )×R) ∥∂ x v 0 ∥ L ∞ (R) e T ∥∂xc∥ L ∞ ((0,T )×R) .
(5.25)

ii) Assume moreover that

∂ x v 0 ∈ L 1 (R), and
∂ x c ∈ L ∞ ((0, T ); L 1 (R)).
(5.26) then v satises the following BV bound

∫ R |∂ x v(x, t)|dx ∫ R |∂ x v 0 (x)|dx, for all t ∈ [0, T ), (5.27)
and also the following bound

∫ R |∂ t v(x, t)|dx ∥c∥ L ∞ ((0,T )×R) ∥∂ x v 0 ∥ L 1 (R)
, for all t ∈ [0, T ).

(5.28)

The point (i), concerning the Lipschitz estimate, has been established by Ley in [START_REF] Ley | Lower-bound gradient estimates for rst-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts[END_REF] . However, the point (ii), concerning the BV estimate, has been proved by El Hajj and Boudjerada in [START_REF] Boudjerada | Global existence results for eikonal equation with BV initial data[END_REF].

Proof of Theorem 5.1.1 (i)

We fulll the proof in two steps.

Step 1. (Local existence and uniqueness) First, we note u 0,ε = (u i

0,ε ) i=1,••• ,d and K 0 = d ∏ i=1 [ -u i 0 L ∞ (R) -1, u i 0 L ∞ (R) + 1
] ,

and we introduce the following constants

µ 1 ε = ∥∂ x ρ 1 ε ∥ L 1 (R) , µ 2 ε = ∥ρ 1 ε ∥ L ∞ (R) , ν 1 ε = ∥∇ρ d+2 ε ∥ (L 1 (R d+2 )) d+2 (1 + µ 2 ε ∥∂ x u 0,ε ∥ (L 1 (R)) d ), ν 2 ε = ∥∇ρ d+2 ε ∥ (L 1 (R d+2 )) d+2 (1 + Λµ 2 ε ∥∂ x u 0,ε ∥ (L 1 (R)) d ), (5.29)
with

∥∂ x u 0,ε ∥ (L 1 (R)) d = d ∑ i=1 ∥∂ x u i 0,ε ∥ L 1 (R) , Λ = d ∑ i=1 ∥λ i ∥ L ∞ ((0,T )×R×K 0 ) , (5.30)
and

∥∇ρ d+2 ε ∥ (L 1 (R d+2 )) d+2 = ∥∂ t ρ d+2 ε ∥ L 1 (R d+2 ) + ∥∂ x ρ d+2 ε ∥ L 1 (R d+2 ) + d ∑ j=1 ∥∂ w j ρ d+2 ε ∥ L 1 (R d+2 ) , where ρ 1 ε (x) and ρ d+2 ε (t, x, w 1 , • • • , w d ) are dened in (5.8).
We dene, for all 0 < ε 1, T >

0 and i = 1, • • • , d, the set X ε,i T =                      ζ ∈ L ∞ ((0, T ) × R), such that ∥ζ∥ L ∞ ((0,T )×R) u i 0,ε L ∞ (R) ∥∂ x ζ∥ L ∞ ((0,T )×R) L ε (T ) ∥∂ x ζ∥ L ∞ ((0,T );L 1 (R)) ∥∂ x u i 0,ε ∥ L 1 (R) ∥∂ t ζ∥ L ∞ ((0,T );L 1 (R)) K                      , where K = λ i L ∞ ((0,T )×R×K 0 ) ∥∂ x u i 0,ε ∥ L 1 (R) and L ε (T ) = ∂ x u i 0,ε L ∞ (R) e ν 1 ε T ∥λ i ∥ L ∞ ((0,T )×R×K 0 ) • Now, for v ε = (v i ε ) i=1,••• ,d ∈ X ε T , where X ε T = ∏ d i=1 X ε,i T we dene G(v ε ) = u ε = (u i ε ) i=1,••• ,d as
the unique Lipschitz continuous viscosity solution of the following uncoupled system

   ∂ t u i ε (t, x) = λ i ε ( t, x, ρ 1 ε ⋆ v ε (t, •)(x) ) ∂ x u i ε (t, x) in (0, T ) × R, u i ε (0, x) = u i 0,ε (x) in R, for i = 1, • • • , d. (5.31)
We will show that G : X ε T -→ X ε T is well dened and a strict contraction for T small enough. First, we will prove that G is well dened. Indeed, by classical properties of the molliers, using Young's inequality and (5.5), we know that (see Ambrosio et al. [

           ∥u i 0,ε ∥ L ∞ (R) ∥u i 0 ∥ L ∞ (R) ∥∂ x u i 0,ε ∥ L ∞ (R) µ 1 ε ∥u i 0 ∥ L ∞ (R) ∥∂ x u i 0,ε ∥ L 1 (R) T V (u i 0 ) = |u i 0 | BV (R) for i = 1, • • • , d, (5.32)
where µ 1 ε is dened in (5.29). Similarly, using (5.2) and (5.32) we can check that,

if v ε ∈ X ε T , then the velocity c i ε (t, x) = λ i ε ( t, x, ρ 1 ε ⋆ v ε (t, •)(x)
)

satises the following estimates

           ∥c i ε ∥ L ∞ ((0,T )×R) ∥λ i ∥ L ∞ ((0,T )×R×K 0 ) ∥∂ x c i ε ∥ L ∞ ((0,T )×R) ν 1 ε ∥λ i ∥ L ∞ ((0,T )×R×K 0 ) ∥∂ t c i ε ∥ L ∞ ((0,T )×R) ν 2 ε ∥λ i ∥ L ∞ ((0,T )×R×K 0 ) for i = 1, • • • , d, (5.33)
where ν 1 ε and ν 2 ε are dened in (5.29).

Estimates (5.32), (5.33) and the regularization properties of the molliers imply that the velocity

λ i ε ( t, x, ρ 1 ε ⋆ v ε (t, •)(x)
) and the initial data u i 0,ε satisfy the assumptions (5.23) and (5.26). There- fore, applying Theorem 5.2.2 with v 0 = u i 0,ε and c = c i ε we deduce that system (5.31) admits a unique Lipschitz continuous viscosity solution u ε = (u i ε ) i=1,••• ,d . Moreover, according to (5.24), (5.27), (5.28), we have for all t ∈ (0, T )

           ∥u i ε (t, •)∥ L ∞ (R) ∥u i 0,ε ∥ L ∞ (R) ∥∂ x u i ε (t, •)∥ L 1 (R) ∥∂ x u i 0,ε ∥ L 1 (R) ∥∂ t u i ε (t, •)∥ L 1 (R) ∥λ i ∥ L ∞ ((0,T )×R×K 0 ) ∥∂ x u i 0,ε ∥ L 1 (R) for i = 1, • • • , d.
(5.34)

Then, from (5.32), (5.33) we can see that u i ε satises the ε-uniform estimates (5.11), (5.12) and (5.13). Using (5.25) we can also see that u i ε satises, for all t ∈ (0, T ), the following Lipschitz estimate

∥∂ x u i ε (t, •)∥ L ∞ ((0,T )×R) ∥∂ x u i 0,ε ∥ L ∞ (R) e T ∥∂xc i ε ∥ L ∞ ((0,T )×R) ∥∂ x u i 0,ε ∥ L ∞ (R) e T ν 1 ε ∥λ i ∥ L ∞ ((0,T )×R×K 0 ) , (5.35)
(which depends on ε). This also shows that u i ε ∈ X ε,i T for all T > 0 and so G is well dened.

It, thus, remains to show that G is a contraction.

Let u ε,ℓ = G(v ε,ℓ ) with u ε,ℓ = (u i ε,ℓ ) i=1,••• ,d and v ε,ℓ = (v i ε,ℓ ) i=1,••• ,d for ℓ = 1, 2. Set L = ∥v ε,2 -v ε,1 ∥ (L ∞ ((0,T )×R)) d = d ∑ i=1 ∥v i ε,2 -v i ε,1 ∥ L ∞ ((0,T )×R))
.

By the properties of the molliers, we can verify that

|λ i ε ( t, x, ρ 1 ε ⋆ v ε,2 (t, •)(x) ) -λ i ε ( t, x, , ρ 1 ε ⋆ v ε,1 (t, •)(x) ) | ν 3 ε ∥λ i ∥ L ∞ ((0,T )×R×K 0 ) ∥v ε,1 -v ε,2 ∥ (L ∞ ((0,T )×R)) d ν 3 ε ∥λ i ∥ L ∞ ((0,T )×R×K 0 ) L, where ν 3 ε = ∥∇ρ d+2 ε ∥ (L 1 (R d+2 )) d+2 . Since u i ε,2 , for i = 1, • • • , d, is a viscosity solution of the following equation    ∂ t u i ε,2 (t, x) = λ i ε ( t, x, ρ 1 ε ⋆ v ε,2 (t, •)(x) ) ∂ x u i ε,2 (t, x) in (0, T ) × R, u i ε (0, x) = u i 0,ε (x) in R,
we can remark, by adding and subtracting λ i ε

( t, x, ρ 1 ε ⋆ v ε,1 (t, •)(x) ) ∂ x u i ε,2 (t, x) , that u i ε,2 is a viscosity sub-solution of    ∂ t u i ε,2 (t, x) = λ i ε ( t, x, ρ 1 ε ⋆ v ε,1 (t, •)(x) ) ∂ x u i ε,2 (t, x) + ν 3 ε L ε (T )∥λ i ∥ L ∞ ((0,T )×R×K 0 ) L in (0, T ) × R, u i ε (0, x) = u i 0,ε (x) in R. In addition u i ε,1 + ν 3 ε L ε (T )∥λ i ∥ L ∞ ((0,T )×R×K 0 )
Lt is a viscosity solution of the same equation. By comparison principle of the scalar eikonal equation (see Barles [10]), we deduce that

u i ε,2 u i ε,1 + ν 3 ε L ε (T )∥λ i ∥ L ∞ ((0,T )×R×K 0 ) LT. Interchanging u i ε,1 and u i ε,2 we get ∥u i ε,1 -u i ε,2 ∥ L ∞ ((0,T )×R) ν 3 ε L ε (T )∥λ i ∥ L ∞ ((0,T )×R×K 0 ) T ∥v 1 ε -v 2 ε ∥ (L ∞ ((0,T )×R)) d . Then ∥u ε,1 -u ε,2 ∥ (L ∞ ((0,T )×R)) d ν 3 ε ΛL ε (T )T ∥v 1 ε -v 2 ε ∥ (L ∞ ((0,T )×R)) d ,
where Λ is dened in (5.30). This shows that, for T small enough, G is a contraction on X ε T which is a closed set. So, by xed point theorem, there exists a unique Lipschitz continuous viscosity solution of (5.6) in X ε T , for all T > 0, such that T T 1 , with

T 1 = min ( 1 4ν 3 ε Λ∥∂ x u 0,ε ∥ (L ∞ (R)) d , ln(2) ν 1 ε Λ ) , where ν 1 ε = ν 3 ε (1 + µ 2 ε ∥∂ x u 0,ε ∥ (L 1 (R)) d ).
Step 2. (Global existence and uniqueness) We are going to prove that the local time solution obtained in Step 1 can be extended to global one. We argue by contradiction. Assume that there exists a maximum time T max such that, we have the existence of solutions of the system (5.6) in the function space W 1,∞ ([0, T max ) × R).

For every δ > 0, we consider the system (5.6) with the initial conditions u δ,i 0,ε (x) = u i ε (T max -δ, x).

(5.36)

We apply for the second time the same technics of Step 1, by taking the above initial data (5.36), to deduce that there exists a time T ⋆ δ such that system (5.6) admits Lipschitz continuous viscosity solution dened until the time

T 0 = (T max -δ) + T ⋆ δ , ( 5.37)
with

T ⋆ δ = min ( 1 4ν 3 ε Λ∥∂ x u δ 0,ε ∥ (L ∞ (R)) d , ln (2)
ν 3 ε (1 + µ 2 ε ∥∂ x u δ 0,ε ∥ (L 1 (R)) d )Λ ) , where u δ 0,ε = (u δ,i 0,ε ) 1,••• ,d .
Notice that the form of T ⋆ δ is just like that of T 1 obtained in the previous step, but with a new initial data (5.36) depending on δ. According to (5.34) and (5.35) we know that ∂ x u δ 0,ε is δ-uniformly bounded in (L ∞ (R)) d ∩ (L 1 (R)) d and therefore there exists a constant

C(ε, Λ, T max , ∥∂ x u 0,ε ∥ (L 1 (R)) d , ∥∂ x u 0,ε ∥ (L ∞ (R)) d ) > 0,
independent of δ such that T ⋆ δ C > 0. Passing to the limit δ → 0, we can see that lim δ→0 T ⋆ δ C > 0. This implies, using (5.37), that T 0 > T max (for small δ). So we have a contradiction and then we can construct a solution u ∈ X ε T for all T > 0. 2 5.3 Passage to the limit when ε tends to 0

In this section we give the proof of Theorem 5.1.1 (ii) using the following compactness lemma.

Proof of Theorem 5.1.1 (ii)

For the sake of a clear presentation, we proceed in two steps.

Step 1. (Convergence

) Let u ε = ( u i ε ) i=1,••• ,d
be the solution of (5.6), constructed in Theorem 5.1.1 (i). From estimates (5.11), (5.12) and (5.13), we can say that, for all compact K 0 ⊂ R,

(u i ε ) ε is uniformly bounded in L ∞ ((0, T ); BV (K 0 )) ∩ L ∞ ((0, T ) × K 0 ) and (∂ t u i ε ) ε
u i ε (h, y) u i ε (t + h, x) sup |y-x| tΛ u i ε (h, y), for all (t, x) ∈ [0, T -h) × R, (5.42)
where Λ is dened in (5.30).

Proof of Lemma 5.4.2

Let us start by proving the right hand side of (5.42), in viscosity sense, namely

u i ε (t + h, x) sup |y-x| tΛ u i ε (h, y).
(5.43)

Let u i ε,h (t, x) = u i ε (t + h, x
). Then, we can see that

∂ t u i ε,h (t, x) = ∂ t u i ε (t + h, x) = λ i ε ( t + h, x, ρ 1 ε ⋆ u ε (t + h, •)(x) ) |∂ x u i ε (x, t + h)| = λ i ε,h (t, x, [u ε ]) |∂ x u i ε,h (t, x)|. Now, from (5.33) we have λ i ε,h (t, x, [u ε ]) ∥λ i ∥ L ∞ ((0,T )×R×K 0 ) Λ,
where Λ is dened in (5.30). This implies that u i ε,h is viscosity sub-solution of the following equation

∂ t w = Λ|∂ x w|, w(0, x) = u i ε (h, x).
( Let (ϕ ε ) ε be a sequence of dened functions on [0, T ) × R such that, for all t ∈ [0, T ), the function ϕ ε (t, •) is non-decreasing on R. Assume, moreover, that ϕ ε → ϕ strongly in C([0, T ); L 1 loc (R)), as ε → 0, with, for all t ∈ [0, T ), the function ϕ(t, •) is dened and non-decreasing on R. Then, for all a > 0 and 0 < δ a 2 , there exists ε δ a,T > 0, such that, for every 0 < ε ε δ a,T , the following estimate holds

-δ + ϕ(t, x -δ) ϕ ε (t, x) δ + ϕ(t, x + δ), ∀x ∈ [-a, a], ∀t ∈ [0, T ).
to a function u i . Since u i εm is non-decreasing (with respect to x), then for all t ∈ [0, T ) the limit u i (t, •) can be considered non-decreasing and dened on R. By the previous inequality and Lemma 5.4.3, we obtain that, for all 0 < α ≤ α a , where α a (t, ∥c∥ L ∞ (R×(0,T )) ) =

     min ( t 2 , a 2(2Λ+1) ) if t > 0, a 2(2Λ+1) , if t = 0,
there exists m α a,T > 0, such that, for every m m α a,T , we have u i εm (t εm , x εm ) u i (h α , x + 2α(Λ + 1)) + α.

Passing to the limit m → +∞ and then α → 0, we obtain (5.46).

Step 2. It remains to show that (u i ) ⋆ (t, x) u i (t, x).

(5.47)

Consider a > 0, x ∈ [-a 2 , a 2 ] and t ∈ [0, T ). In fact, from the denition of (u i ) ⋆ we know that there exists a sequence (t εm , x εm ) → (t, x), when m → +∞, such that

(u i ) ⋆ (t, x) = lim m-→+∞ u i (t εm , x εm ).
Similarly, as in Step 1, we can state that, for all α > 0, there exists m α > 0, such that, for all m m α , we have |x εm -x| α and |t εm -t| α.

However, using Lemma 5.4.3, we know that, for all 0 < α a 2 , there exists k α a,T > 0 and a subsequence 0 < ε α k α such that, for every k k α a,T , u i (t εm , x εm ) u i εα k (t εm , x εm + α) + α sup εα k ≤α,|s-t| α |y-x| 2α u i α k (s, y) + α.

Passing to the limit m → +∞ and then α → 0, we obtain (5.47). 2

In the next subsection, we will give meaning to the initial data of the discontinuous viscosity sub-solution and super-solution introduced in Theorem 5.1.1 (iii) and Theorem 5.1.3, that is reected by the points (1)-(i) and ( 2)-(i) in Denition 5.4.1.

Meaning of the initial data

As a rst step, we deal with the case of Theorem 5.1.1 (iii) and then as a second step we treat the case of Theorem 5.1.3.

Step 1. (Initial data in Theorem 5.1.1 (iii)) We only prove the result for the sub-solution case, the super-solution case is proved analogously. Let u ε be the solution of (5.6), constructed in Theorem 5.1.1 (i). We have to prove that the relaxed semi-limit (u i ) ⋆ = u i satises (1)-(i) in Denition 5.4.1. It is sucient to prove the following inequality u i (0, x) (u i 0 ) ⋆ (x) for all x ∈ R, i = 1, • • • , d.

(5.48)

From the denition of u i , we know that there exists a sequence (ε n , t εn , x εn ) → (0, 0, x) as n → +∞, such that u i (0, x) = lim

u i 0 (z)
)

,
where we have used in the second and the third lines the denition of the functions u i 0,εn in (5.7) and also the classical properties of the molliers. Furthermore, the convergence, as n → +∞, of (ε n , t εn , x εn ) n to (0, 0, x), implies that for all α > 0, there exists n α > 0, such that, for all n n α , we have ε n α, |x εn -x| α and t εn α.

Thus, for every α > 0 and n n α , we get u i εn (t εn , x εn ) sup |z-x| α(2+Λ) u i 0 (z).

(5.49)

First, we pass to the limit, n → +∞, in the previous inequality to obtain u i (0, x) sup |z-x| α(2+Λ) u i 0 (z).

Then, we pass to the limit, α → 0, to complete the proof of (5.48).

Step 2. (Initial data in Theorem 5. 1.3) In the framework of non-decreasing initial data, we have from (5.45), u i (0, x) = (u i ) ⋆ (0, x) and u i (0, x) = (u i ) ⋆ (0, x). Taking into account what was done before in Step 1, we get that the functions (u i ) ⋆ and (u i ) ⋆ satisfy, respectively (1)-(i) and Assume that λ i is locally bounded on (0, T ) × R × R d for all T > 0. Let λ i ε be the standard regularization of the functions λ i dened in (5.7). Noting λ i (t, x, r) = lim sup ε-→0 (s,y,w)-→(t,x,r) λ i ε (s, y, w), and λ i (t, x, r) = lim inf ε-→0 (s,y,w)-→(t,x,r) λ i ε (s, y, w).

Then, we have λ i (t, x, r) (λ i ) ⋆ (t, x, r) and (λ i ) ⋆ (t, x, r) λ i (t, x, r) for all (t, x, r)

∈ [0, T ) × R × R d ,
where (λ i ) ⋆ and (λ i ) ⋆ are respectively the upper and lower semi-continuous envelopes of λ i .

Proof of Lemma 5.4.5

We only show the proof of the rst inequality, the second is proved similarly. Indeed, we know that there exists a sequence (ε n , t εn , x εn , r εn ) → (0, t, x, r), as n goes to +∞, such that where we have used the fact that ρ d+2 εn 0 and ∫ R d+2 ρ d+2 εn = 1. Thanks to the convergence, as n → +∞, of (ε n , t εn , x εn , r εn ) to (0, t, x, r), we can deduce, as in (5.49), that for every α > 0 there exists n α > 0, such that, for all n n α , we have It is easy to verify that v i εm is non-decreasing in the case of non-decreasing solution since the convolution preserves the monotonicity and therefore using assumption (5.3), we obtain

λ i (t
∂ t φ(t ε i m , x ε i m )-λ i (t ε i m , x ε i m )) ∂ x φ(t ε i m , x ε i m ) 0.
(5.54)

Now, we proceed as in Theorem 5.1.3, passing to the limit, m → +∞, rstly and then α → 0 we get ∂ t ϕ(t 0 , x 0 ) -λ i (t 0 , x 0 , r 1 , • • • , r i , • • • , r d ) |∂ x ϕ(t 0 , x 0 )| 0 with r i = u i (t 0 , x 0 ).

Which proves using Lemma 5.4.5 that min { ∂ t ϕ(t 0 , x 0 ) -(λ i ) ⋆ (t 0 , x 0 , r) |∂ x ϕ(t 0 , x 0 )| : r ∈ U (t 0 , x 0 ), r i = u i (t 0 , x 0 ) } 0.

and therefore u = (u 1 , • • • , u d ) is viscosity sub-solution of (5.1). In the same way, we prove that u = (u 1 , • • • , u d ) is viscosity super-solution of (5.1), under assumption (5.3). However, if this monotonicity condition is not satised, we can show by repeating the same process and using equality (5.39) that u i and u i verify both (5.40) and (5.41) except for x 0 ∈ D i where D i is a countable set in R. 2

It is worth mentioning that the hypothesis of monotonicity assumed on the speed, which plays a crucial role throughout the above proof, is not needed in the case of dislocation dynamics and gas dynamics.

Global existence of discontinuous solution for gas dynamics

In this section we study the global existence of the isentropic gas dynamics system, dened as First of all, we remark that system (5.55) is a diagonalizable hyperbolic system. Indeed, in the case where ρ > 0 and (ρ, u) is a smooth solution, we can check easily that the following two variables r 1 = u + ρ θ and r 2 = u -ρ θ with θ = (γ-1) 2 ,

satisfy the following diagonal system Let T > 0 and ρ 1 η be the mollier dened in (5.8). Assume that (r 1 0 , r 2 0 ) ∈ (W 1,∞ (R)) 2 , the functions r i 0 are non-decreasing for i = 1, 2 and r 1 0,η = ρ 1 η ⋆ r 1 0 , r 2 0,η = ρ 1 η ⋆ r 2 0 .

           ∂ t r 1 + λ1 (r 1 , r 2 )∂ x r 1 = 0 in (0, T ) × R, ∂ t r 2 + λ2 (r 1 , r 2 )∂ x r 2 = 0 in (0, T ) × R,
Then we have i) Existence, uniqueness and bounds System (5.57) has a unique solution (r ∥∂ x r i 0 ∥ L ∞ (R) for i = 1, 2, and 0 < t < T.

1 η , r 2 η ) ∈ ( W 2,∞ ([0, T ) × R) ) 2 ∩ (C ∞ ([0, T ) × R))
(5.59) Moreover, the functions r i η (t, •) are non-decreasing for i = 1, 2 and 0 t < T . ii) Convergence Up to extract a subsequence, (r 1 η , r 2 η ) converges locally uniformly, as η goes to zero, to a function

(r 1 , r 2 ) ∈ ( W 1,∞ ([0, T ) × R)
) 2 , where (r 1 , r 2 ) is a continuous viscosity solution of (5.56), Moreover for all 0 t < T the functions r i (t, •) are non-decreasing for i = 1, 2 and satisfy estimates (5.58) and (5.59).

The next subsection is reserved for the proof of Theorem 5.1.5.

5.5.1 Proof of Theorem 5.1.5

We divide the proof in four steps.

Step 1. (Lipschitz continuous solution for diagonal system) For 0 < ε 1, we consider system (5.56) with the following initial data

r 1 0,ε = ρ 1 ε ⋆ (u 0 + ρ θ 0 ) + ε, r 2 0,ε = ρ 1 ε ⋆ (u 0 -ρ θ 0 ) -ε,
where θ = γ-1 2 and ρ 1 ε is the mollier dened in (5.8). Under assumption (5.20) we can check that these initial data satisfy the conditions of Theorem 5.5.1. This implies that there exists a continuous viscosity solution (r 1 ε , r 2 ε ) ∈ ( W 1,∞ ([0, T ) × R) ) 2 of (5.56), satisfying following L ∞ estimate ∥r i ε (t, •)∥ L ∞ (R) ∥r i 0,ε ∥ L ∞ (R) for i = 1, 2, and 0 < t < T.

Since γ > 1 and ρ 0 0, we deduce that ∥r i ε (t, •)∥ L ∞ (R) ∥u 0 ∥ L ∞ (R) + (∥ρ 0 ∥ L ∞ (R) ) θ + 1 for i = 1, 2, and 0 < t < T.

(5.60)

Moreover, we know that the functions r i ε (t, •) are non-decreasing for i = 1, 2 and t > 0. Therefore

∥∂ x r i ε (t, •)∥ L 1 (R) 2∥r i ε (t, •)∥ L ∞ (R) 2 ( ∥u 0 ∥ L ∞ (R) + (∥ρ 0 ∥ L ∞ (R) ) θ + 1
) for i = 1, 2, and 0 < t < T. (5.61) Since (r 1 ε , r 2 ε ) verify system (5.56) almost everywhere, we can also see that there exists a positive constant µ independent of ε, such that ∥∂ t r i ε ∥ L ∞ ((0,T );L 1 (R)) µ for i = 1, 2.

(5.62)

Now, we will prove that r 1 ε -r 2 ε > 0. To this end, we recall, from Theorem 5.5.1, that r 1 ε = lim η→0 r 1 ε,η and r 2 ε = lim η→0 r 2 ε,η , where (r 1 ε,η , r 2 ε,η ) is the smooth solution of the following regularized parabolic system ∂ t r i ε,η + λi (r 1 ε,η , r 2 ε,η )∂ x r i ε,η , = η∂ xx r i ε,η , for i = 1, 2, with regular initial data r 1 0,ε,η , r 2 0,ε,η (see Theorem 5.5.1). To simplify, we set r ε,η = r 1 ε,η -r 2 ε,η and we can check that r ε,η satises the following equation

∂ t r ε,η = - ( r 1 ε,η + r 2 ε,η 2 ) ∂ x r ε,η - γ -1 4 r ε,η ∂ x (r 1 ε,η + r 2 ε,η ) + ε∂ xx r ε,η .
Using the maximum principle theorem for parabolic equations (see Lieberman [START_REF] Lieberman | Second order parabolic dierential equations[END_REF]Th 2.10]), the η-uniform estimate (5.59) and the fact that r 1 0,ε,η -r 2 0,ε,η 2ε > 0, we get r 1 ε,η (t, •) -r 1 ε,η (t, •) 2εe -αt > 0, for all 0 t < T , with α = γ -1 2 max i=1,2

||∂ x r i 0,ε || L ∞ (R) . We pass to the limit η → 0 and obtain r 1 ε (t, •) -r 1 ε (t, •) 2εe -αt > 0.

(5.63)

Step 2. (From (r 1 ε , r 2 ε ) toward (ρ ε , u ε )) Let (r 1 ε , r 2 ε ) the Lipschitz continuous solution of system (5.56), constructed in Step 1. For θ = γ-1 2 , we use the following variable change

u ε = r 1 ε + r 2 ε 2 and ρ θ ε = r 1 ε -r 2 ε 2 ,
(5.64) that is possible since r 1 ε -r 2 ε > 0. With a simple computation we can check that these new variables solve (almost everywhere) the following system

   ∂ t (ρ θ ε ) + u ε ∂ x (ρ θ ε ) + θρ θ ε ∂ x u ε = 0, ∂ t u ε + u ε ∂ x u ε + θρ θ ε ∂ x (ρ θ ε ) = 0.
According to (5.63), we know that ρ ε β ε > 0 (for some positive constant β ε ), in addition to the fact that (r 1 ε , r 2 ε ) ∈ (W 1,∞ ([0, T ) × R) 2 , we conclude that the functions u ε and ρ ε dened above belong to W 1,∞ ([0, T ) × R), and moreover solve the following system   

∂ t ρ ε + ∂ x (ρ ε u ε ) = 0, ∂ t (ρ ε u ε ) + ∂ x (ρ ε u 2 ε + p(ρ ε )) = 0, (5.65)
with the following initial data u ε (0, x) = ρ 1 ε ⋆ u 0 (x), and ρ θ ε (0, x) = ρ 1 ε ⋆ ρ θ 0 (x) + ε, for x ∈ R.

Step 3. (Convergence from (ρ ε , u ε ) toward (ρ, u)) In order to fulll the proof of Theorem 5.1.5, we still have to pass to the limit in (5.65) as ε → 0. Indeed, from ε-uniform estimates (5.66)

As the proof of Theorem 5.1.1 (ii) (see Section 5.3), we use Simon's lemma (cf. Lemma 5.3.1) in the particular case X = BV (K), B = Y = L 1 (K) associated to the following compact embedding BV (K) → L p (K) for 1 p < +∞ and for all compact K ⊂ R, we will be able to extract a subsequence (u εn , ρ θ εn ) that converges towards a limit (u, ρθ ) strongly in C([0, T ); L p (K)), for all 1 p < +∞, T > 0 and compact K ⊂ R. Thanks to estimates (5.66) we can extract a subsequence, still denoted by (u εn , ρ θ εn ) , satisfying the following convergence estimates (u εn , ρ θ εn ) -→ (u, ρθ ) strongly in C([0, T ); L p (K)), for all compact K ⊂ R, (u εn , ρ θ εn ) -→ (u, ρθ ) weakly-⋆ in L ∞ ((0, T ) × R),

(u εn , ρ θ εn ) -→ (u, ρθ ) weakly-⋆ in L ∞ ((0, T ); BV (R)).

Using the fact that ρ εn 0, the rst estimate in (5.66) shows that ρ εn is also uniformly bounded with respect to ε n and then, up to extract a subsequence, we can verify that ρ εn -→ ρ weakly-⋆ in L ∞ ((0, T ) × R),

for some function ρ ∈ L ∞ ((0, T ) × R). It remains to verify that (u, ρ) is solution of (5.55) in the distributional sense (i.e. in D ′ ((0, T ) × R)). Indeed, for all compact K ⊂ R, we have on one hand that ρ εn converges to ρ weakly-⋆ in L ∞ ((0, T ) × K) and on the other hand u εn converges to u strongly L 1 ((0, T ) × K). This gives us a weak-⋆ convergence in L ∞ ((0, T ) × K) times a strong convergence L 1 ((0, T ) × K) in the term ρ εn u εn . Hence the product ρ εn u εn converges weakly in L 1 ((0, T ) × K) to ρu and then in the distributional sense. Similarly, we can prove that ρ εn u 2 εn converges to ρu 2 in the distributional sense, using the weak-⋆ convergence in L ∞ ((0, T ) × K) of ρ εn toward ρ and the strong convergence in L 2 ((0, T ) × K) of u εn toward u, in other words the strong convergence in L 1 ((0, T ) × K) of u 2 εn toward u 2 .

Moreover, since ρ θ εn converges to ρθ strongly in L 1 ((0, T ) × K), then up to extract a subsequence ρ θ εn converges to ρθ almost everywhere in (0, T ) × K. Therefore ρ εn converges to (ρ θ )

1 θ
almost everywhere in (0, T ) × K. This implies, using the dominated convergence theorem and the fact that ρ εn is uniformly bounded with respect to ε n , that ρ εn converges to (ρ θ ) 1 θ strongly in L 1 ((0, T ) × K) and in particular in D ′ ((0, T ) × R). However, we know that ρ εn converges also to ρ in D ′ ((0, T ) × R), thanks to the weak-⋆ convergence. Thus, ρθ = ρ θ in D ′ ((0, T ) × R). In the same way, we can prove that ρ γ εn converges to ρ γ strongly in L 1 ((0, T ) × K), for all T > 0 and compact K ⊂ R.

This shows that (u, ρ) is solution in D ′ ((0, T ) × R), to the following system    ∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 + p(ρ)) = 0.

Taking the lim inf in estimates (5.66) and using the lower semi-continuity of ∥ • ∥ L ∞ ((0,T )×R) and ∥ • ∥ L ∞ ((0,T );BV (R)) with respect to weak-⋆ topology, we can prove that u and ρ θ satisfy (5.21).

Step 4. (Recovering the initial data) Now we prove that the initial conditions (u 0 , ρ 0 ) coincide with (u(0, •), ρ(0, •)). Indeed, by the ε-uniformly estimate given in (5.66) we can prove easily that

∥u ε (t, •) -ρ 1 ε ⋆ u 0 ∥ L 1 (R) + ∥ρ θ ε (t, •) -ρ 1 ε ⋆ ρ θ 0 -ε∥ L 1 (R) C 3 t.

General Conclusion and Perspectives

The aim of this thesis is to study particular models applied on dislocation dynamics. We rst prove the existence and the uniqueness of a coupled singular parabolic system that was initially proposed by Groma, Czikor and Zaiser. After that, focusing on Groma and Balogh model, we establish the global existence of a BV solution of a non-local coupled system of Hamilton-Jacobi equations in the case where d = 2. Then, a generalization of this work to the case of d-equations is presented where we obtain the global existence of a BV solution which is not necessarily monotone and with no sign restrictions on the velocity. This work was complicated due to several factors as the presence of the absolute value in the system, the strong coupling of the equations and the BV space.

Lastly, as prospects, we are willing to prove the global in time existence and uniqueness of our GCZ model. For GB model, the uniqueness of the two studied models, stands as an open problem since the fact that our models are non-linear and non-monotone, bans the possibility of applying the comparison principle and proving the uniqueness of the solution via standard viscosity solutions methods.

As a future numerical work, we are proposing a scheme to approximate, on the discrete level, our theoretical results presented in Chapters 4 and 5 of this thesis starting from BV initial data. We are willing to prove the convergence of our scheme and then to conrm our results by presenting an error estimation between the theoretical solution and the obtained numerical one.

After that, some numerical simulations will be presented in this work.

I

  would like also to express my sincere recognition to the Rectors of the Lebanese University Fouad AYOUB and University of Technology of Compiègne Philippe COURTIER as well as the Deans for the Doctoral School of Sciences and Technology Fawaz EL OMAR and the Doctoral School of Sciences for Engineering Christine PRELLE. Special thanks to the Lebanese University for funding my Ph.D. studies. To my friends Hawraa NABULSI, Fatima ABBASS, Saly MALAK, Aya OUSAILY, Bouthayna NEMOUCHI, Mariam BACHA, Hanin ALJEBAWY and Iman KLEILAT, I am thankful for that bond which made me feel that I have real sisters who always care for me. iii I would immensely thank my uncle Doctor Hassan MAHDI who is my unfailing support and continuous motivation. Your presence in my life is a gift from God making me reassured that, whatever happens, someone is always there for me. Finally, I must express my very profound gratitude to the persons with the greatest indirect contribution to this work, my mother Dalal MAHDI, my father Hussein RIZIK and my siblings Mohamad, Manar and Hadi. Their trust, prayers and constant encouragement gave me the ability to tackle challenges and sustained me thus far. They are the main reason behind every success I achieve. d'une solution régulière en utilisant le principe de comparison et un argument de point xe. Ensuite, nous démontrons un résultat d'existence global en temps pour le modèle de GB, en nous basant sur la notion des solutions de viscosité discontinues et sur une nouvelle estimation sur la variation totale de la solution, ainsi que sur la propagation à vitesse nie des équations régissantes. Ce résultat est étendu aussi au cas des systèmes d'équations d'Hamilton-Jacobi général. v vi Notations • u ⋆ : Upper semi-continuous envelope • u ⋆ : Lower semi-continuous envelope • u: Upper relaxed semi-limit • u: Lower relaxed semi-limit • U : graph closure of u • T V (u): Total variation of u • BV (E): Set of functions with bounded variations • U SC(E): Set of upper semi-continuous functions • LSC(E): Set of lower semi-continuous functions

Figure 1 . 1 :

 11 Figure 1.1: Microscopic observation of dislocations in N i 3 Al.

Figure 1 . 2 :

 12 Figure 1.2: Dislocations of positive and negative types.

Figure 1 . 3 :

 13 Figure 1.3: Edge and screw dislocations.

Figure 1 . 4 :

 14 Figure 1.4: Gas dynamics.

  .22) are, respectively, discontinuous viscosity sub-solution and super-solution of (1.18) in the sense of the following denition Denition 1.3.2. (Discontinuous viscosity sub-solution, super-solution and solution)

Theorem 1 .

 1 3.4. (Global existence result for non-decreasing solutions)

(1. 45 )

 45 are, respectively, discontinuous viscosity sub-solution and super-solution of system(1.36) in the sense of the following denition of discontinuous viscosity solutions introduced by Ishii in [58, Denition 2.1] for the Hamilton-Jacobi system Denition 1.4.1. (Discontinuous viscosity sub-solution, super-solution and solution)

Chapter 2 Modelization 2 . 1

 221 Groma-Czikor-Zaiser model Motivated by the dynamics of dislocation densities in a bounded crystal, Groma, Czikor and Zaiser [48] have proposed a 2-dimensional model describing the dynamics of parallel edge dislocation densities. Establishing this model aims to describe the possible accumulation of dislocations on the boundary layer of the material by shedding light on the evolution of the dislocation densities taking into consideration the short range dislocationdislocation interactions. These interactions represent a regime in which the dislocation separations are small. They are illustrated by the physical terms τ b and τ f which are presented later (see (2.4) and (2.5)). The two types of dislocation densities are a consequence of the two types of edge dislocations that result from the positive or negative direction of the Burgers vector.

Figure 2 . 1 :

 21 Figure 2.1: The model of Groma, Czikor and Zaiser.

Figure 2 . 2 :

 22 Figure 2.2: Dislocation points in a cross sectional surface.

Figure 2 . 3 :

 23 Figure 2.3: Passage from 2D to 1D.

  To simplify, we consider the homogeneous case σ = 2µε e (U ) + λtr(ε e (U ))I d , (2.25) where µ > 0 and λ + µ > 0 (consequence of (2.21)) are the Lamé coecients and I d is the identity matrix. The following lemma holds Lemma 2.2.1. (Equivalence between 2-D and 1-D models) Assume that the Burgers vector ⃗ b = (1, 0) and the unknowns of the system depend only on x = x 1 +x 2 (as shown in Figure 2.3), then the 2-D problem (2.23) is equivalent to the following 1-D problem

Chapter 3 Formal

 3 derivation and existence result of an approximate model on dislocation densities This chapter is an exposure of the article Formal derivation and existence result of an approximate model on dislocation densities, ZAMM Z. Angew. Math. Mech. 98, no. 6, 1015-1032, which was written in collaboration with Hassan Ibrahim and Zynab Salloum. This article is concerned with the theoretical study of a mathematical model arising from dislocation dynamics. Our model is a coupled singular system of nonlinear parabolic equations which was proposed by Groma, Czikor and Zaiser to describe the possible accumulation of dislocations on the boundary layer of a material. So, in our work, we focus on the short range dislocation-dislocation interactions neglecting the long ones. This result is the rst result dealing with GCZ model with the real physical exterior stress, unlike previous results which were done by considering a constant or bounded space-time dependent exterior stress eld. By means of comparison principle and xed point argument, we were able to prove the short time existence and uniqueness of a continuous solution in some Hölder space. ZAMM Z. Angew. Math. Mech. Volume 98, No. 6 (2018), 1015-1032 Formal derivation and existence result of an approximate model on dislocation densities 3.1 Introduction Dislocations are line defects, or microscopic irregularity inside materials where the atoms are out of position in the crystalline structure. They represent a non-stationary phenom-

3. 3 .

 3 A comparison principle, on the gradient of the solution, is given in Section 3.4. In Section 3.5, we use a xed point argument on convenient spaces to establish the short time existence and uniqueness.3.2 The modeling3.2.1 Physical motivation and setting of the problemGroma, Czikor and Zaiser[START_REF] Groma | Spatial correlations and higher-order gradient terms in a continuum description of dislocations dynamics[END_REF] have proposed a model to described the possible accumulation of dislocations on the boundary layer of a material. They have presented a two-dimensional model describing the dynamics of parallel edge dislocations densities in a bounded three-dimensional crystal. To more explain the physical model, we can take a channel (refer to Figure2.1) that contains a certain number of parallel edge dislocations and bounded by walls that are impenetrable by dislocations. This channel has a nite width in the x-direction and an innite extension in the y-direction. The dislocation lines

Section 3. 3 .

 3 Now, consider the domain I T = I × (0, T ), I = (0, 1), T > 0.

40 3. 3 .

 403 Notations and the main resultsRegularizing equation (3.5) by adding εθ ± xx and writing down the equations in ρ and κ, we get (for

( 3 .

 3 7)-(3.9). Theorem 3.3.1. (Short time existence and uniqueness) Let ε > 0 a xed constant.

(3. 22 )

 22 Using (3.20) and (3.22), direct and initial computations lead

Lemma 3 . 4 . 3 .

 343 Let θ ∈ R and M (x, t) = cosh(θx)M (x, t). Under the condition of Lemma 3.4.2, the function M can not have a positive minimum on∂I.

  [START_REF] Groma | Link between the microscopic and mesoscopic length-scale description of the collective behavior of dislocations[END_REF] is obtained in two steps. In a rst step, while having the initial and boundary conditions (3.8) and (3.9), we nd a solution ρ ∈ Y of the second equation of (3.45), then we plug it into the rst equation to get a solution κ ∈ Y . Here, the existence and uniqueness of both solutions are guaranteed by[START_REF] Ladyzenskaja | Linear and quasilinear equations of parabolic type, Translated from the Russian[END_REF] Theorem 9.1]. It is worth mentioning that Theorem 9.1 in[START_REF] Ladyzenskaja | Linear and quasilinear equations of parabolic type, Translated from the Russian[END_REF] requires a compatibility condition of order 0 (3.11) on the initial and boundary data. Those conditions are satised by our boundary assumptions on ρ 0 and κ 0 (see Proposition 3.5.1).

. 48 )

 48 Using the second equation of(3.46) and (3.48), we nally get ∥ρ -r∥ Y cT p-3 2p ∥κ -k∥ Y .

( 3 . 54 )

 354 Using (3.51),(3.52),(3.53) and (3.54), we nally get∥F ∥ L p (I T ) cT p-3 2p ( ∥κ -k∥ Y + ∥ρ -r∥ Y) . (3.55) Using the rst equation of (3.46) and (3.55), we nally get ∥κ -k∥ Y cT p-3 2p ( ∥κ -k∥ Y + ∥ρ -r∥ Y) .

(3. 56 ) 2 Remark 3 . 5 . 2 .

 562352 Equations(3.49) and(3.56) show that, for T > 0 suciently small, the application Ξ is a contraction, and eventually it has a unique xed point (ρ, κ) solution of system (3.44), (3.8) and (3.9). Lastly, to get rid of the articial truncations I and J in(3.44), and to show (3.43), we use the embedding(3.38) and the initial conditions(3.42). This is again with the possibility of reducing the time T . The embedding(3.38) infers that κ x ∈ C α,α/2 (I T ) then, since γ 0 2 κ x 2M 0 , we easily deduce

. 59 )

 59 By introducing ρ = ρϕ δ and κ = κϕ δ ,(3.60)

2 and the proof follows. 2 4 Global

 24 Chapter BV solution for a non-local coupled system modeling the dynamics of dislocation densities This chapter is an exposure of the article Global BV solution for a non-local coupled system modeling the dynamics of dislocation densities, J. Dierential Equations 264, no. 3, 1750-1785 that was written in collaboration with Ahmad El Hajj and Hassan Ibrahim. After working on Groma, Czikor and Zaiser model in the regular case, we step in this article to work with a Groma-Balogh model in BV space. This model is a coupling of two non-local Hamilton-Jacobi equations that arises from dislocation dynamics. By considering a particular periodic condition, we were able to study the long range dislocation-dislocation interactions focusing on what is going inside the material away from its boundary. For this system, the global existence of discontinuous viscosity solutions is attained relying on a fundamental BV estimate and a nite speed propagation property. The uniqueness of the solution of this system stands as an open problem since the non-linearity and non-monotonicity, ban the possibility of applying the comparison principle and proving the uniqueness of the solution via standard viscosity solutions methods.

  Figure 4.1. In this case, the dislocation points are moving along with the Burgers vectors ± ⃗ b, hence it is convenient to distinguish two types of dislocations, the positive (following + ⃗ b) and the negative (following -⃗ b) dislocations.

Figure 4 . 1 :

 41 Figure 4.1: The cross section of the dislocation lines.

Theorem 4 .

 4 1.1. (Global weak existence result for a local problem)

Theorem 4 .

 4 1.2. (Global existence result for non-decreasing solutions)Assume that the assumptions (4.3), (4.4) and (4.7) hold. If the initial data ρ ± 0 are nondecreasing, then the system (4.6) admits a discontinuous non-decreasing viscosity solution ρ ± satisfying (4.15), (4.16), (4.17), (4.18) and (4.19).

Remark 4 . 1 . 5 .

 415 The uniqueness of the solution obtained in Theorem 4.1.4 remains an open question in the case of BV initial data.

1 ,

 1 has been proved by Serre [80, Vol I, Th 3.6.1]. This result is only local in time even in dimension N = 1.

(4. 24 )

 24 For the regularized equation (4.24), we have the following result.

Theorem 4 . 2 . 1 .

 421 (Existence and uniqueness of Lipschitz continuous solution of (4.21))

(4. 35 )Step 3 (

 353 Fixed point argument)The previous estimates (4.34) and (4.35) and the classical xed point argument result in the existence of a time T * > 0 depending on η, ∥L∥ L ∞ (0,T ) and ∥v 0 ∥ (L ∞ ((0,T );W 1,∞ (T))) 2 , for which system (4.31) admits a solution v ±,δ η ∈ L ∞ ((0, T * ); W 1,∞ (T)). For the Long time existence, refer to the iteration process done in Chapter 5 Section 5.2.

2 4. 3 A

 23 priori uniform estimates on the solution of(4.21) In this section, we show two η-uniform estimates on the solution of equation (4.21) obtained in Theorem 4.2.1 (ii). The rst one concerns the BV estimate of the equation and is a key result. Lemma 4.3.1. (BV estimate) Assume that (4.22) holds. Then, for all 0 < η 1, the solution v η of (4.21), given by Theorem 4.2.1 (ii), satises (4.28), (4.29) and the following estimate ∫ dx, for all t 0.

  Denition 4.4.1. (Continuous viscosity sub-solution, super-solution and solution) A function v ± ∈ C(R × [0, T )) is a viscosity sub-solution of (4.21) if and only if

Theorem 4 . 4 . 2 .

 442 (Existence and uniqueness of Lipschitz continuous viscosity solution)

Lemma 4 .

 4 4.3. (Stability of continuous viscosity solutions, Barles [10, Th 2.3])

Lemma 4 .

 4 4.4. (Simon's Lemma [81, Corollary 4]) Let X, B and Y be three Banach spaces, where X → B with compact embedding and B → Y with continuous embedding. If

2 4. 5

 25 Existence of sub and super discontinuous viscosity solutions This section is devoted to the proof of Theorem 4.1.1

(4. 57 )

 57 First, let us recall the denition of discontinuous viscosity solutions of the local system (4.6). Denition 4.5.1. (Discontinuous viscosity sub-solution, super-solution and solution)

Theorem 4 .

 4 5.2. (A stability result for discontinuous viscosity solution, [9, Th 4.1])

2 4. 5 . 2

 252 Proof of Theorem 4.1.

2 4. 6

 26 Link between the sub and the super discontinuous viscosity solutionsIn this section, we will prove Theorem 4.1.1 (iii), and Theorem 4.1.2. First, we show in Subsection 4.6.1, some preliminary results that we use in Subsection 4.6.2, in order to complete the proof of Theorem 4.1.2. Finally, in Subsection 4.6.3, we present the proof of Theorem 4.1.1 (iii).

4. 6 . 2

 62 Proof of Theorem 4.1.2 Proof of Theorem 4.1.2

2 4. 6 . 3

 263 Proof of Theorem 4.1.1 (iii) Proof of Theorem 4.1.

( 4 .

 4 90)This equality links the sub-solution ρ ± and the super-solution ρ ± . It remains to show that ρ ± (•, τ ) = ρ ± (•, τ ) except at most on a countable set in R, for all τ ∈ [0, T ].

0 are 1 -

 1 periodic functions; L 0 and α are given positive constants. By applying Theorems 4.1.1 and 4.1.2, we prove a global existence result of this system (announced in Theorem 4.1.4) with a BV initial data. To do this, we divide this section into two subsections. In the rst one, after regularizing the initial data, we prove the existence of a Lipschitz continuous viscosity solution of the regularized problem. Then, in the second subsection, we use equality (4.20) and the stability result of discontinuous viscosity solutions, in order to pass to the limit and prove the result. 4.7.1 Existence, uniqueness and a priori estimates of the regularized problem

4. 7 . 2

 72 Passage to the limit and proof of Theorem 4.1.4 Proof of Theorem 4.1.4

(5. 23 )

 23 Before showing this result let us recall the denition of the continuous viscosity solution for (5.22). Denition 5.2.1. (Continuous viscosity sub-solution, super-solution and solution) A function v ∈ C([0, T ) × R) is a viscosity sub-solution of (5.22) if and only if v(0, x) v 0 (x) and for any

( 2 )

 2 -(i) in Denition 5.4.1.

2 5. 4 . 3

 243 Proofs of the main results Theorem 5.1.1 (iii) and Theorem 5.1.3 After giving meaning to the initial data in the previous sub-section, it remains to show that the sub-solution and super-solution introduced in Theorem 5.1.1 (iii) and Theorem 5.1.3, satisfy the points (1)-(ii) and (2)-(ii) in Denition 5.4.1, respectively. To do that, we need the following lemma. Lemma 5.4.5. (Envelopes of the velocity)

  λ i εn (t εn , x εn , r n ) sup |y-x| 2α,|τ -t| 2α |w-r| 2α λ i (τ, y, w).

  x) = ρ 1 ε ⋆ u j ε (t, •)(x) for j = 1, • • • , d.

     ∂ t ρ + ∂ x (ρu) = 0 in (0, T ) × R, ∂ t (ρu) + ∂ x ( ρu 2 + p(ρ) ) = 0, with p(ρ) = (γ-1) 2 4γ ρ γ in (0, T ) × R, u(0, x) = u 0 (x) and ρ(0, x) = ρ 0 (x) 0 for x ∈ R,(5.55)where T > 0, γ > 1 and respecting the usual notation for the physical quantities, ρ represents the density of the uid, u is the velocity of the uid and p the pressure. In what follows, we propose a fairly simple proof to show the existence of discontinuous solutions of system (5.55), using an existence result of Lipschitz continuous solution developed in[START_REF] Hajj | Some niqueness results for diagonal hyperbolic systems equation with large and monotone data[END_REF] Theorem 1.3] and some compactness argument.

r 1 1 -+ r 2 2 - γ -1 4 (r 1 -∂ t r 1 η

 112411 (0, x) = r 1 0 (x), r 1 (0, x) = r 2 0 (x)for x ∈ R,(5.56)where λ1 and λ2 are dened as follows r 2 ) = u + θρ θ λ2 (r 1 , r 2 ) = r 1 r 2 ) = u -θρ θ .To prove Theorem 5.1.5, we need to recall the existence result proved by El Hajj et al. in[START_REF] Hajj | Some niqueness results for diagonal hyperbolic systems equation with large and monotone data[END_REF] Theorem 1.3] for the following parabolic regularization of system (5.56), dened for 0< η 1 + λ1 (r 1 η , r 2 η )∂ x r 1 η -η∂ xx r 1 η = 0 in (0, T ) × R, ∂ t r 2 η + λ2 (r 1 η , r 2 η )∂ x r 2 η -η∂ xx r 2 η = 0 in (0, T ) × R, r 1 η (0, x) = r 1 0,η(x), r 1 η (0, x) = r 2 0,η (x) for x ∈ R.

(5. 57 )

 57 Theorem 5.5.1. (Lipschitz continuous solution for diagonal system)

( 5 .

 5 [START_REF] Ishii | Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games[END_REF],(5.61),(5.62) and according to (5.64) we deduce that there exists three positive constantsC 1 , C 2 and C 3 independent of ε, such that            ∥u ε ∥ L ∞ ((0,T )×R) + ∥ρ θ ε ∥ L ∞ ((0,T )×R) C 1 , ∥∂ x u ε ∥ L ∞ ((0,T );L 1 (R)) + ∥∂ x ρ θ ε ∥ L ∞ ((0,T );L 1 (R)) C 2 , ∥∂ t u ε ∥ L ∞ ((0,T );L 1 (R)) + ∥∂ t ρ θ ε ∥ L ∞ ((0,T );L 1 (R)) C 3 .

  1.3. Global BV solution for a non-local coupled system modeling the dynamics of dislocation densities as a way of regarding what is going on in the interior of a material away from its boundary.Our mathematical result, concerning Groma, Czikor and Zaiser model, has been established in the framework of Hölder and Sobolev spaces, however, for the analytical study of Groma and Balogh model, we are going to work in BV space dened as follows

  (1.56) is automatically satised since if ρ 0 ≡ 0 on a subset ω ⊂ R, then ρ ≡ 0 in [0, T ) × ω t where ω t ⊂ ω and the function u can be chosen locally arbitrarily in [0, T ) × ω t . So, in vacuum state (ρ ≡ 0), we do not have uniqueness of the solution.

	1.58)
	Moreover, the functions u(t, •) ± ρ θ (t, •) are non-decreasing for all 0 t < T .
	Remark 1.5.2. (Vacuum case)

System

  x ∥ L p (I T ) λ where λ > 0 is a xed suciently large constant. For this reason, dene the spaces Y ρ and Y κ as follows

.44) with the same initial and boundary conditions (3.8) and (3.9). It is worth noticing that when (3.43) is satised then (3.44) coincides with (3.7). On the other hand, condition (3.43) also suggests that we consider functions u ∈ Y of bounded gradients, i.e.

∥u

  .58) Remark 3.5.4. The assumption(3.14) on the initial data, together with the constant boundary values, dene a compatibility condition of order 1 which represents the continuity on the parabolic boundary (see(3.11) and (3.12) for more details). In other words,

	we get

  This boundary condition let us conclude that (see[START_REF] Ladyzenskaja | Linear and quasilinear equations of parabolic type, Translated from the Russian[END_REF] Theorem 5.2]), if the source terms (see equation(3.40)) in (3.7) is of class C β,β/2 (I T ) for 0 < β < 2, then the solution (ρ, κ) will be of class C 2+β, 2+β 2 (I T ).

  Proof of Theorem 4.4.2 Proof of i): Let v ± η be the solution of (4.21) constructed in Theorem 4.2.1 (ii). First, by Lemma 4.4.3 (i), we know that v ± η is a continuous viscosity solution of (4.21). Let K be a compact subset in R. Using estimates (4.28), (4.29), (4.40) and applying Lemma 4.4.4 in the particular case where

  dx for all t 0. , since v ± are Lipschitz continuous, we know that the L 1 -norm of ∂v ± Remark 4.4.5. (L 1 -continuity in time) Under assumption (4.22), we can remark, using estimate (4.47), that the solution v ± of (4.43) satises the following L 1 -Lipschitz estimate in time

	Moreover∂x coincides with the total variation of v ± , which proves (4.48).
	The last estimate (4.47) directly comes from (4.44), (4.45) and (4.48), by using the Hölder's
	inequality and the fact that the Lipschitz continuous viscosity solution v ± satises the system
	(4.43) almost everywhere.

2

  there exists n ζ > 0 such that, for all n n ζ , we have ϵ n ζ, |x ϵn -x| ζ and t ϵn ζ.

	Thus, for every ζ > 0 and n n ζ , we have
	ρ ± ϵn (x ϵn , t ϵn )	sup
		|z-x| ζ(2+C)

  We begin by showing the far right inequality in (4.78), namely,ρ ± (x, t + h) 2h + ρ ± 1 (x + hγ c , t) -ρ ± 2 (x -hγ c , t). (x ϵm , t h ϵm ) → (x, t + h) when m → +∞, we can see that there exists m h > 0, such that, for all m m h , |x ϵm -x| < h and |t h ϵm -t -h| < h.Since, for all t ∈ [0, T ), the sequence ρ ± ϵm (•, t) is uniformly bounded in L ∞ (-2, 2) ∩ BV (-2, 2) and strongly convergent in L 1 (-2, 2), we can deduce, from Lemma 4.6.3, that there exists a

				(4.80)
	Indeed, consider h > 0 satisfying (4.79). From the denition of ρ ± , we know that there exists a
	sequence (ϵ m , x ϵm , t h ϵm ) → (0, x, t + h), when m → +∞, such that
		ρ ± (x, t + h) = lim m-→+∞	ρ ± ϵm (x ϵm , t h ϵm ).
	Using Lemma 4.5.4, we deduce that, for all m m h ,
		ρ ± ϵm (x ϵm , t h ϵm )	sup |y-xϵ m | C(t h ϵm -t)	ρ ± ϵm (y, t)
			sup |y-xϵ m | 2hC	ρ ± ϵm (y, t)	(4.81)
			sup	ρ ± ϵm (y, t).
			|y-x| h(2C+1)
	subsequence ρ ± ϵn (•, t) and a positive constant n h t , such that, for all n n h t ,
	ρ ± ϵn (y, t) 2h + ρ ± 1 (y + h, t) -ρ ± 2 (y -h, t), ∀y ∈ [-1, 1],	(4.82)
	where ρ ± 1 and ρ ± 2 are two bounded, right-continuous and non-decreasing functions (with respect to x) satisfying ρ ± = ρ ± 1 -ρ ± 2 . Collecting (4.81) and (4.82), we obtain that, for all h > 0 satisfying (4.79) and for all n n h t ,
	ρ ± ϵn (x ϵn , t h ϵn
	h	1 2(2C + 1)		(4.79)

1

2 ,

1

2 ], t ∈ [0, T ), and for all h > 0 verifying and t + h < T.

As

  As in the proof of Step 2 in Subsection 4.5.2, we can see that a ⋆ a and a a ⋆ . Moreover,

	from equality (4.20), we know that
	∫ 1	∫ 1	∫ 1	∫ 1
	ρ -(y, t)dy =	ρ -(y, t)dy and	ρ + (y, t)dy =	ρ + (y, t)dy.
	0	0	0	0

  Lemma 5.3.1.(Simon's Lemma[START_REF] Simon | Compacts sets in the space L p (0; T ; B)[END_REF] Corollary 4]) Let X, B and Y be three Banach spaces, where X → B with compact embedding and B → Y with continuous embedding. If (θ n ) n is a sequence uniformly bounded in L ∞ ((0, T ); X) and (∂ t θ n ) n is uniformly bounded in L r ((0, T ); Y ) where r > 1, then, (θ n ) n is relatively compact in C((0, T ); B).

  is uniformly Lemma 5.4.2. (Finite speed propagation property) Under assumptions (5.2) and (5.5), ifu ε = (u i ε ) i=1,••• ,dis the unique continuous viscosity solution of (5.6), given by Theorem 5.1.1 (i), then u i ε satises, for all h 0, the following estimate

	inf
	|y-x| tΛ

  Now, we show a local estimate valid on sequences of non-decreasing functions converging locally and strongly in L 1 (R).

			.44)
	Furthermore, if we note
	α(t, x) = sup	u i ε (h, y),
		|y-x| tΛ
	which implies (5.43). The same proof is done for the inequality
	inf |y-x| tΛ	u i ε (h, y) u i ε (t + h, x),
	Lemma 5.4.3. (Sequences of non-decreasing functions)

then, by Lax-Oleinik formula (See [10, Lemma 2.1]), we know that α is the unique continuous viscosity solution of

(5.44)

. Using the comparison principle (see

[17,

Th 1.1]), we deduce that u i ε (t + h, x) α(t, x), on (0, T ) × R, by considering the following equation ∂ t w = -Λ|∂ x w|. 2

  εn , x εn ). Lemma 5.4.2 with h = 0 and t = t εn , we get u i εn (x εn , t εn ) sup |y-xε n | tε n Λ

	n-→+∞ εn (t Using u i u i εn (0, y)
	(∫	)
	sup	u i 0 (z)ρ 1 εn (y -z)dz
	|y-xε n | tε n Λ	R
	(
	sup	sup
	|y-xε n | tε n Λ	|z-y| εn

  , x, r) = lim n-→+∞ λ i εn (t εn , x εn , r εn ).

	From (5.7), we can see that
	λ i εn (t

εn , x εn , r n ) = ∫ (0,T )×R×R d λ i (τ,

y, w)ρ d+2 εn (t εn -τ, x εn -y, r εn -w) dydτ dw sup |y-xε n | εn,|τ -tε n | εn |w-rε n | εn λ i (τ, y, w),

  ∥∂ x r i η (t, •)∥ L ∞ (R) max

	max i=1,2	i=1,2
		(5.58)

2 , satisfying the following L ∞ estimates

∥r i η (t, •)∥ L ∞ (R) ∥r i 0 ∥ L ∞ (R) for i = 1

, 2, and 0 < t < T,

Chapter 2. Modelization

∥f ∥ L ∞ (R) for all t > 0, where γ 0 is a positive constant depending only on η.

Now, we pass to the limit, n → +∞, rstly and then α → 0 we get λ i (t, x, r) (λ i ) ⋆ (t, x, r).

ρ n 0, and ∫ R n ρ n = 1.

(5.8)

Our result lies initially in proving the global in time existence of the solution of the above regularized system, using the xed point argument and the comparison principle of the associated linear problem obtained by freezing u in the velocity. Afterwards, to pass from the solution of the regularized system (5.6) to that of system (5.1), we will show that the upper and lower relaxed semi-limits, of Barles and Perthame [START_REF] Barles | Exit time problems in optimal control and vanishing viscosity method[END_REF][START_REF] Barles | Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations[END_REF], which are dened as follows u i (t, x) = lim sup ⋆ u i ε (t, x) = lim sup ε-→0 (s,y)-→(t,x) u i ε (s, y),

(5.9) and u i (t, x) = lim inf ⋆ u i ε (t, x) = lim inf ε-→0 (s,y)-→(t,x) u i ε (s, y), (5.10) are, respectively, discontinuous viscosity sub-solution and super-solution of system (5.1) in the sense of discontinuous viscosity solutions introduced by Ishii in [58, Denition 2.1] for the Hamilton-Jacobi system and recalled below in Denition 5.4.1. Lastly, leaning on some εindependent a priori estimates, essentially on new BV uniform bound, we come to prove an almost everywhere equality between u i and u i in R, for all t > 0. This shows the existence of a function u = (u i ) i=1,••• ,d , where u i is dened as a strong limit of u i ε in C([0, T ); L 1 loc (R)) and u is almost everywhere discontinuous viscosity solution of (5.1).

Main results

In this subsection, we rst present, in Theorem 5.1.1, a global existence result of a special discontinuous viscosity solution of (5.1). After that, we show in Theorem 5.1.3, as a consequence of this result, the existence of a classical discontinuous viscosity solution of (5.1) by taking nondecreasing initial data. Finally, a global existence result to 1D isentropic gas dynamics system will be presented in Theorem 5.1.5. Now we will rst show that system (5.1) admits a BV discontinuous viscosity solution, in some weak sense.

Existence result for Eikonal system

(5.11)

(5.12) (5.13) where

. ii) Convergence Assume that u i ε , satises (5.11), (5.12) and (5.13) for i = 1, • • • , d. Then, up to extract a subsequence, the function u i ε converges, as ε goes to zero, to a function

) , (5.14) strongly in C

Moreover, u i satises, for all T > 0 and for i = 1, • • • , d, the following inequalities

(5.15)

(5.16)

)

|t -s|, for all s, t ∈ [0, T ), (5.17) and the following equality

), except at most on a countable set in R, for all t ∈ [0, T ), (5.18) where u i and u i are, respectively, the upper relaxed semi-limit and the lower relaxed semi-limit dened in (5.9) and (5.10).

iii) Global existence of weak discontinuous viscosity solution Let u ε be the solution of (5.6), constructed in (i). Suppose that assumption (5.3) is satised.

, are respectively discontinuous viscosity sub-solution and super-solution of system (5.1) (in the sense of Denition 5.4.1). Generally, (without assumption (5.3)) u and u, are both almost everywhere in space discontinuous viscosity sub-solution and super-solution of system (5.1) and moreover verify equality (5.18).

The key point to establish this theorem is the uniform BV estimate on u i ε (5.12). We rst consider the regularized problem of (5.1) and we show that the smooth solution admits the L ∞ bound (5.11) and the fundamental BV estimate (5.12). These estimates will allow us to pass to the limit when the regularization vanishes. Then we will show, from the classical stability properties of viscosity solutions, that the relaxed semi-limits u and u are, respectively, sub-and super-solution of (5.1) . These estimates also imply that the set of the discontinuous points, with respect to x, of the solution u is at most countable. Taking into account the nite speed propagation property of (5.1) and the time continuous estimate (5.13), it is then possible to show this property uniformly in time, which proves in particular (5.18). Remark 5.1.2. (Weak discontinuous viscosity solution) We note that, the solution constructed in the previous theorem, can be seen as a discontinuous viscosity solution but in some weak sense, since it veries only an almost everywhere equality in space between u i and u i , which is reected by (5.18). Such a solution will be called a weak discontinuous viscosity solution of (5.1).

Recall that in the framework of non-decreasing solutions, the Hamilton Jacobi system (5.1) becomes a classical transport system. For such a system, Bianchini and Bressan proved in [START_REF] Bianchini | Vanishing viscosity solutions of nonlinear hyperbolic systems[END_REF] a striking global existence and uniqueness result for general non-conservative (d × d) strictly hyperbolic systems, including diagonal systems like system (5.1). The key step in their proof was an a priori estimate on the total variation of the approximate solutions proved relying on small total variation of initial data. Their existence result is a generalization of Glimm's result [START_REF] Glimm | Solutions in the large for nonlinear hyperbolic systems of equations[END_REF], proved in the case of conservation laws. Let us mention that an existence result has also been obtained by LeFloch and Liu [START_REF] Lefloch | Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form[END_REF][START_REF] Lefloch | Existence theory for nonlinear hyperbolic systems in nonconservative form[END_REF] in the non-conservative case. After that, El Hajj and Monneau proved in [START_REF] Hajj | Some niqueness results for diagonal hyperbolic systems equation with large and monotone data[END_REF] a global existence and uniqueness result for strictly hyperbolic diagonal systems, with the assumption

, by widening the regularity assumption and considering continuous solutions. That was a generalization to the work done by Lax [START_REF] Lax | Hyperbolic systems of conservation laws and the mathematical theory of shock waves[END_REF], in the case of (2 × 2) strictly hyperbolic systems, where the existence of Lipschitz solutions was proved. Now, we can state the following theorem as a consequence of Theorem 5.1.1 in the case of non-decreasing solutions. This result remains valid even in the case where system (5.1) is not necessarily strictly hyperbolic and the initial data in BV norm are large.

Theorem 5.1.3. (Global existence of non-decreasing discontinuous viscosity solution) Assume that (5.2) and (5.3) are satised. Suppose that u i 0 ∈ L ∞ (R) and the function (5.15), (5.16) and (5.17 1. In the case of non-decreasing and bounded initial data assumption (5.5) automatically holds, namely

Let us mention that, in the absence of the assumption (5.3) it is always possible to prove (as in Theorem 5.1.1 (iii)) the existence of a function u that is discontinuous viscosity solution almost everywhere in space.

3. Let us also mention that, if we consider the following diagonal hyperbolic systems

for T > 0 and i = 1, ..., d, as part of solutions that are not necessarily non-decreasing, it is always possible to prove a similar global existence result, of weak discontinuous viscosity solution, to that announced in Theorem 5.1.1. Indeed, with a slight modication of the proof, we can verify that the a priori estimates (5.11), (5.12) and (5. Existence result for 1D gas dynamics Now, we present, relying on an existence result done in [START_REF] Hajj | Some niqueness results for diagonal hyperbolic systems equation with large and monotone data[END_REF] and considering some monotonicity assumptions on the initial data, the global existence of a discontinuous solution of the following 1D isentropic gas dynamics system bounded in L ∞ ((0, T ); L 1 (K 0 )). Using Simon's lemma in the particular case X = BV (K 0 ), B = Y = L 1 (K 0 ) and the following compact embedding BV (K 0 ) → L 1 (K 0 ), we can extract a subsequence, denoted by (u i ε n,K 0

) ε n,K 0 , that converges strongly in L ∞ ((0, T ); L 1 (K 0 )) to some limit u i , as n → +∞. By a standard diagonalization procedure, we can extract a subsequence (u i εn ) εn (independent on i and K) that converges to the limit u i strongly in C([0, T ); L 1 (K)) for all compact K ⊂ R. Now, thanks to estimates (5.11) and (5.12) we can extract a subsequence, still denoted by (u i εn ) εn , satisfying the following convergence estimates

(5.38)

Taking the lim inf in estimates (5.11), (5.12) and using the lower semi-continuity of ∥ • ∥ L ∞ (R) and | • | BV (R) , we can prove that u i satises (5.14), (5.15) and (5.16). Moreover, using estimate (5.13), we can remark that u i εn satises the following L 1 -Lipschitz estimate in time

)

|t -s| for all t, s ∈ [0, T ).

Which implies, according to the strong convergence in C([0, T ); L 1 loc (R)), that u i satises also the L 1 -Lipschitz estimate in time (5.17).

Step 2. (Set of discontinuity points) It remains to show equality (5.18). Indeed, since u i (t, •) ∈ L ∞ (R) ∩ BV (R), we know that this function coincides with a right-continuous function, almost everywhere in R and consequently in L 1 loc (R). This is a direct consequence of the fact that the BV -functions are continuous except at most on a countable set. Now, we can prove as in [START_REF] Boudjerada | Global existence results for eikonal equation with BV initial data[END_REF]Section 6.3] that there exists a countable set D i (independent on t) such that (5.39) where u i and u i are dened in (5.9) and (5.10), respectively. This result is obtained based on estimates (5.11), (5.12), (5.13) and on the nite speed propagation property of the equation satised by u i εn , proved below in Lemma 5.4.2.

For a vector

It should be remarked that U (t, x) is closed; i.e., for all sequence {r m } ⊂ R d , if r m ∈ U (t, x) and r m -→ m→+∞ r for some r ∈ R d , then r ∈ U (t, x). Assume that λ i is locally bounded on (0,

(1) (Discontinuous viscosity sub-solution)

We call u a discontinuous viscosity sub-solution of (5.1) if it satises

(5.40)

(2) (Discontinuous viscosity super-solution)

Similarly, we call u a discontinuous viscosity super-solution of (5.1) if it satises

(5.41)

(3) (Discontinuous viscosity solution)

Finally, we call u a discontinuous viscosity solution of (5.1) if it is both a discontinuous viscosity sub-solution and super-solution of (5.1)

Noting that the minimum and the maximum in (5.40) and (5.41) are attained, since the sets

} are non-empty and compact and moreover (λ i ) ⋆ and (λ i ) ⋆ are upper and lower semi-continuous, respectively.

To prove Theorem 5.1.1 (iii) and Theorem 5.1.3, we need also to establish in the following subsection some preliminary results.

Preliminary results

We start this subsection by showing the following nite speed propagation property, valid for continuous viscosity solutions of system (5.6).

For the proof of this lemma see [START_REF] Boudjerada | Global existence results for eikonal equation with BV initial data[END_REF]Lemma 6.2].

We end by proving the following technical lemma. Lemma 5.4.4. (Envelopes in non-decreasing case) Assume (5.2) is satised. Suppose that u i 0 ∈ L ∞ (R) and the function

is the unique continuous viscosity solution of (5.6) given by Theorem 5.1.1 (i). Then, for i = 1, • • • , d, we have

for all (t, x) ∈ [0, T ) × R, (5.45) where u i and u i are dened in (5.9) and (5.10), respectively. Proof of Lemma 5.4.4

We will only show the proof of the rst equality, the second can be proved in a similar way.

Step 1. We will prove the following inequality for i

(5.46) Let a > 0, x ∈ [-a 2 , a 2 ] and t ∈ [0, T ). In fact, by the denition of u i , we know that there exists a sequence (ε m , t εm , x εm ) → (0, t, x), when m → +∞, such that u i (t, x) = lim m-→+∞ u i εm (t εm , x εm ).

For all α > 0, we can state that, there exists m α > 0, such that, for all m m α , we have |x εm -x| α and |t εm -t| α. u i εm (h α , y).

Moreover, from the maximum principle of (5.6) and since the initial data is non-decreasing, we know that u i εm is non-decreasing (with respect to x) and therefore, for all m m α , u i εm (t εm , x εm ) u i εm (h α , x + α(2Λ + 1)).

Now, as we have indicated in Section 5.3, since u i εm satises estimates (5.11), (5.12), (5.13) we can extract a subsequence, still denoted by (u i εm ) εm , that converges in the sense of (5.38) Then, for all compact K ⊂ R, we get ∥u(t, •) -u 0 ∥ L 1 (K) + ∥ρ θ (t, •) -ρ θ 0 ∥ L 1 (K) C 3 t.

where we have used the strongly convergence in C([0, T ); L 1 (K)), This proves that u(0, •) = u 0 (•) and ρ(0, •) = ρ 0 (•) in D ′ (R). 2