Convergence of approximation schemes for nonlocal front propagation equations

We provide a convergence result for numerical schemes approximating nonlocal front propagation equations. Our schemes are based on a recently investigated notion of weak solution for these equations. We also give examples of such schemes, for a dislocation dynamics equation, and for a Fitzhugh-Nagumo type system

Uniqueness Results for Nonlocal Hamilton-Jacobi Equations

We are interested in nonlocal Eikonal Equations describing the evolution of interfaces moving with a nonlocal, non monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh-Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts

Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations

In this article, we provide existence results for a general class of nonlocal and nonlinear second-order parabolic equations. The main motivation comes from front propagation theory in the cases when the normal velocity depends on the moving front in a nonlocal way. Among applications, we present level-set equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo systems

Minimizing movements for dislocation dynamics with a mean curvature term

International audienceWe prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough

Integral formulations of the geometric eikonal equation

Abstract We prove integral formulations of the eikonal equation ut = c(x, t)|Du|, equivalent to the notion of viscosity solution in the framework of the set-theoretic approach to front propagation problems. We apply these integral formulations to investigate the regularity of the front: we prove that under regularity assumptions on the velocity c, the front has locally finite perimeter in {c = 0}, and we give a time-integral estimate of its perimeter. Key words and phrases: Eikonal equation, viscosity solutions, set-theoretic approach, functions of bounded variation and sets of finite perimeter

Contribution à l'étude d'équations de propagations de fronts locales et non-locales

Ce travail porte sur l étude de propagations de fronts gouvernées par des lois locales et non-locales. Dans la méthode par lignes de niveau, le front est vu comme ligne de niveau 0 d une fonction auxiliaire. A la loi géométrique d évolution du front correspond alors une équation de Hamilton-Jacobi sur cette fonction, que nous envisageons dans le cadre des solutions de viscosité. Dans les modéles non-locaux, la difficulté principale pour prouver des résultats d existence ou d unicité est l absence de principe d inclusion entre les fronts. Dans la méthode par lignes de niveau, ceci correspond à une absence de principe de comparaison entre les fonctions, qui rend impossible l utilisation des techniques habituelles. L utilisation alternative de méthodes de point fixe associe à toute équation non-locale une famille d équations locales. La compréhension de la régularité des solutions des équations locales, et en particulier du périmètre de leurs lignes de niveau, apparaît alors cruciale dans les arguments de point fixe. Dans le chapitre 1, on prouve des formulations intégrales de l équation eikonale locale, dont on déduit des estimations sur le périmétre des lignes de niveau de ses solutions. Dans le reste des travaux, on s intéresse aux équations non-locales, et notamment à une notion de solution faible pour ces équations. Deux modèles non-locaux, la dynamique des dislocations et un système de type Fitzhugh-Nagumo, sont également étudiés en détails. On donne en particulier des résultats d existence, d unicité et d approximation numérique dé solutions faibles.The subject of this work is the study of front propagations governed by local and nonlocal Iaws. In the so-called level-set method, The fron is seen as the 0 level-set of an auxiliary function. In this context, the geometric evolution law of the front corresponds to a Hamilton-Jacob equation satisfied by this function; this equation is considered in the framework of viscosity solutions. In nonlocal models, the main obstacle to existence and uniqueness results is the absence of inclusion principle between fronts. In the level-set method, this corresponds to an absence of comparison principle between functions, which makes impossible the use of classical techniques. The alternative use of fixed point methods associates to any nonlocal equation a family of localequations. Understanding the regularity of the solutions of local equations, and in particular the perimeter of their level-sets, therefore appears crucial in fixed point type arguments. In Chapter 1, we prove integral formulations for the local eikonal equation, from which we deduce perimeter estimates on the level-sets of its solutions. In the rest of this work, we focus en nonlocal equations, and in particular on a notion of weak solution for these equations. Two nonlocal models, the dislocation dynamics equation and a Fitzhugh-Nagumo type system, are also studied in details. In particular, we give results on existence, uniqueness and numerical approximation of weak solutions.BREST-BU Droit-Sciences-Sports (290192103) / SudocSudocFranceF

Minimizing movements for dislocation dynamics with a mean curvature term

International audienceWe prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough

VISCOSITY SOLUTIONS FOR A POLYMER CRYSTAL GROWTH MODEL

Abstract. We prove existence of a solution for a polymer crystal growth model describing the movement of a front (Γ(t)) evolving with a nonlocal velocity. In this model the nonlocal velocity is linked to the solution of a heat equation with source δΓ. The proof relies on new regularity results for the eikonal equation, in which the velocity is positive but merely measurable in time and with Hölder bounds in space. From this result, we deduce a priori regularity for the front. On the other hand, under this regularity assumption