Time-splitting approximation of the Cauchy problem for a stochastic conservation law

International audienceIn this paper, we present a time discretization of a first-order hyperbolic equation of nonlinear type set in Rd and perturbed by a multiplicative noise. Using an operator splitting method, we are able to show the existence of an approximate solution. Thanks to recent techniques of well-posedness theory on this kind of stochastic equations, we show the convergence of such an approximate solution towards the unique stochastic entropy solution of the problem, as the time step of the splitting scheme converges to zero

A degenerate parabolic-hyperbolic Cauchy problem with a stochastic force

International audienceIn this paper we are interested in the Cauchy problem for a nonlinear degenerate parabolic-hyperbolic problem with multiplicative stochastic forcing. Using an adapted entropy formulation a result of existence and uniqueness of a solution is proved

Convergence of a TPFA scheme for a diffusion-convection equation with a multiplicative stochastic noise

The aim of this paper is to address the convergence analysis of a finite-volume scheme for the approximation of a stochastic non-linear parabolic problem set in a bounded domain of $\mathbb{R}^2$ and under homogeneous Neumann boundary conditions. The considered discretization is semi-implicit in time and TPFA in space. By adapting well-known methods for the time-discretization of stochastic PDEs, one shows that the associated finite-volume approximation converges towards the unique variational solution of the continuous problem strongly in $L^2(\Omega; L^2(0,T;L^2(\Lambda)))$.Comment: arXiv admin note: text overlap with arXiv:2203.0985

Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with multiplicative noise

We study here the discretization by monotone finite volume schemes of multi-dimensional nonlinear scalar conservation laws forced by a multiplicative noise with a time and space dependent flux-function and a given initial data in $L^{2}(\R^d)$. After establishing the well-posedness theory for solutions of such kind of stochastic problems, we prove under a stability condition on the time step the convergence of the finite volume approximation towards the unique stochastic entropy solution of the equation

A Multi-Scale Model of Soft Imperfect Interface with Nonlocal Damage

International audienceThe aim of this paper is to propose a model of bonded interface including nonlocal damage and unilateral conditions. The model is derived from the problem of a composite structure made by two adherents and a thin adhesive. The adhesive is damaged at microscopic level and is subjected to two regimes, one in traction and one in compression. The model of interface is derived by matched asymptotic expansions. In this paper, two cases corresponding to the two regimes are discussed. Moreover, this model can be considered as a model of contact with adhesion and unilateral constraint. At the end of the paper, a simple numerical example is presented to show the evolution of the model

Well-posedness result for a system of random heat equation coupled with a multiplicative stochastic Barenblatt equation

In this paper, a stochastic nonlinear evolution system under Neumann boundary conditions is investigated. Precisely, we are interested in finding an existence and uniqueness result for a system of random heat equation coupled with a Barenblatt's type equation with a multiplicative stochastic force in the sense of Itô. To do so, we investigate in a first step the case of an additive noise through a semi-implicit in time discretization of the system. This allows us to show the well-posedness of the system in the additive case. In a second step, the derivation of continuous dependence estimates of the solution with respect to the data allows us to show the desired existence and uniqueness result for the multiplicative case

Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise

We study here the approximation by a finite-volume scheme of a heat equation forced by a Lipschitz continuous multiplicative noise in the sense of It\^o. More precisely, we consider a discretization which is semi-implicit in time and a two-point flux approximation scheme (TPFA) in space. We adapt the method based on the theorem of Prokhorov to obtain a convergence in distribution result, then Skorokhod's representation theorem yields the convergence of the scheme towards a martingale solution and the Gy\"{o}ngy-Krylov argument is used to prove convergence in probability of the scheme towards the unique variational solution of our parabolic problem

ON ABSTRACT BARENBLATT EQUATIONS

In this paper we are interested in abstract problems of Barenblatt's type. In a first part, we investigate the problem f (∂ t u) + Au = g where f and A are maximal monotone operators and by assuming that A derives from a potential J . With general assumptions on the operators, we prove the existence of a solution. In the second part of the paper, we examine a stochastic version of the above problem: f [∂ t (u − t 0 hdw)] + Au = 0 , with some restrictive assumptions on the data due principally to the framework of the Itô integral

Study on stochastic partial differential equations

Cette thèse s’inscrit dans le domaine mathématique de l’analyse des équations aux dérivées partielles (EDP) non-linéaires stochastiques. Nous nous intéressons à des EDP paraboliques et hyperboliques que l’on perturbe stochastiquement au sens d’Itô. Il s’agit d’introduire l’aléatoire via l’ajout d’une intégrale stochastique (intégrale d’Itô) qui peut dépendre ou non de la solution, on parle alors de bruit multiplicatif ou additif. La présence de la variable de probabilité ne nous permet pas d’utiliser tous les outils classiques de l’analyse des EDP. Notre but est d’adapter les techniques connues dans le cadre déterministe aux EDP non linéaires stochastiques en proposant des méthodes alternatives. Les résultats obtenus sont décrits dans les cinq chapitres de cette thèse : Dans le Chapitre I, nous étudions une perturbation stochastique des équations de Barenblatt. En utilisant une semi- discrétisation implicite en temps, nous établissons l’existence et l’unicité d’une solution dans le cas additif, et grâce aux propriétés de la solution nous sommes en mesure d’étendre ce résultat au cas multiplicatif à l’aide d’un théorème de point fixe. Dans le Chapitre II, nous considérons une classe d’équations de type Barenblatt stochastiques dans un cadre abstrait. Il s’agit là d’une généralisation des résultats du Chapitre I. Dans le Chapitre III, nous travaillons sur l’étude du problème de Cauchy pour une loi de conservation stochastique. Nous montrons l’existence d’une solution par une méthode de viscosité artificielle en utilisant des arguments de compacité donnés par la théorie des mesures de Young. L’unicité repose sur une adaptation de la méthode de dédoublement des variables de Kruzhkov.. Dans le Chapitre IV, nous nous intéressons au problème de Dirichlet pour la loi de conservation stochastique étudiée au Chapitre III. Le point remarquable de l’étude repose sur l’utilisation des semi-entropies de Kruzhkov pour montrer l’unicité. Dans le Chapitre V, nous introduisons une méthode de splitting pour proposer une approche numérique du problème étudié au Chapitre IV, suivie de quelques simulations de l’équation de Burgers stochastique dans le cas unidimensionnel.This thesis deals with the mathematical field of stochastic nonlinear partial differential equations’ analysis. We are interested in parabolic and hyperbolic PDE stochastically perturbed in the Itô sense. We introduce randomness by adding a stochastic integral (Itô integral), which can depend or not on the solution. We thus talk about a multiplicative noise or an additive one. The presence of the random variable does not allow us to apply systematically classical tools of PDE analysis. Our aim is to adapt known techniques of the deterministic setting to nonlinear stochastic PDE analysis by proposing alternative methods. Here are the obtained results : In Chapter I, we investigate on a stochastic perturbation of Barenblatt equations. By using an implicit time discretization, we establish the existence and uniqueness of the solution in the additive case. Thanks to the properties of such a solution, we are able to extend this result to the multiplicative noise using a fixed-point theorem. In Chapter II, we consider a class of stochastic equations of Barenblatt type but in an abstract frame. It is about a generalization of results from Chapter I. In Chapter III, we deal with the study of the Cauchy problem for a stochastic conservation law. We show existence of solution via an artificial viscosity method. The compactness arguments are based on Young measure theory. The uniqueness result is proved by an adaptation of the Kruzhkov doubling variables technique. In Chapter IV, we are interested in the Dirichlet problem for the stochastic conservation law studied in Chapter III. The remarkable point is the use of the Kruzhkov semi-entropies to show the uniqueness of the solution. In Chapter V, we introduce a splitting method to propose a numerical approach of the problem studied in Chapter IV. Then we finish by some simulations of the stochastic Burgers’ equation in the one dimensional case

EDP stochastiques, schémas volumes-finis et application à la mécanique

In this manuscript are presented the research activities I have carried out since my arrival in Marseille ten years ago following my PhD thesis. They are part of the mathematical field of theoretical analysis and numerical approxi- mation of stochastic partial differential equations (SPDEs). More precisely, they concern the study of non-linear problems perturbed by a stochastic integral of Itô’s type depending or not of the solution, commonly referred to as multiplicative (respectively additive) noise. The approach employed here is based on the use and combination of techniques from PDE’s analysis, probability theory and stochastic calculus. Two types of SPDEs were studied: on the one hand, scalar con- servation laws of hyperbolic and then parabolic type for which construction and convergence analysis of finite-volume schemes have been developed, and on the other hand parabolic equations related to solid mechanics modeling for which well-posedness issues have been investigated.Sont présentées dans ce manuscrit les activités de recherche que j’ai menées depuis mon arrivée à Marseille il y a dix ans suite à mon doctorat. Ces dernières s’inscrivent dans le domaine mathématique de l’analyse théorique et de l’approximation numérique d’EDP stochastiques. Plus précisément, elles concernent l’étude de problèmes non-linéaires perturbés par une intégrale stochastique d’Itô dépendant ou non de la solution, on parle alors de bruit multiplicatif ou additif. L’approche mathématique choisie dans mes recherches consiste à adapter et combiner les techniques connues en analyse des EDP avec des outils de théorie des probabilités et de calcul stochastique. Les EDP stochastiques que j’ai étudiées ont été de deux types : d’un côté des lois de conservation scalaires de type hyperbolique puis parabolique pour lesquelles je me suis attachée à la construction et l’analyse de convergence de schémas numériques de type volumes-finis, et de l’autre des équations paraboliques liées à la modélisation en mécanique du solide pour lesquelles j’ai regardé des questions d’existence et d’unicité de solution