Global ETD Search

1	Regularity for solutions of nonlocal fully nonlinear parabolic equations and free boundaries on two dimensional cones Chang Lara, Hector Andres 22 October 2013 (has links) On the first part, we consider nonlinear operators I depending on a family of nonlocal linear operators [mathematical equations]. We study the solutions of the Dirichlet initial and boundary value problems [mathematical equations]. We do not assume even symmetry for the kernels. The odd part bring some sort of nonlocal drift term, which in principle competes against the regularization of the solution. Existence and uniqueness is established for viscosity solutions. Several Hölder estimates are established for u and its derivatives under special assumptions. Moreover, the estimates remain uniform as the order of the equation approaches the second order case. This allows to consider our results as an extension of the classical theory of second order fully nonlinear equations. On the second part, we study two phase problems posed over a two dimensional cone generated by a smooth curve [mathematical symbol] on the unit sphere. We show that when [mathematical equation] the free boundary avoids the vertex of the cone. When [mathematical equation]we provide examples of minimizers such that the vertex belongs to the free boundary. / text Regularity theory Integro-differential equations Nonlocal equations Fully nonlinear equations Nonlocal drift Parabolic equations Free boundary problems Two phase problem
2	Systèmes de particules en interaction, approche par flot de gradient dans l'espace de Wasserstein / Interacting particles systems, Wasserstein gradient flow approach Laborde, Maxime 01 December 2016 (has links) Depuis l’article fondateur de Jordan, Kinderlehrer et Otto en 1998, il est bien connu qu’une large classe d’équations paraboliques peuvent être vues comme des flots de gradient dans l’espace de Wasserstein. Le but de cette thèse est d’étendre cette théorie à certaines équations et systèmes qui n’ont pas exactement une structure de flot de gradient. Les interactions étudiées sont de différentes natures. Le premier chapitre traite des systèmes avec des interactions non locales dans la dérive. Nous étudions ensuite des systèmes de diffusions croisées s’appliquant aux modèles de congestion pour plusieurs populations. Un autre modèle étudié est celui où le couplage se trouve dans le terme de réaction comme les systèmes proie-prédateur avec diffusion ou encore les modèles de croissance tumorale. Nous étudierons enfin des systèmes de type nouveau où l’interaction est donnée par un problème de transport multi-marges. Une grande partie de ces problèmes est illustrée de simulations numériques. / Since 1998 and the seminal work of Jordan, Kinderlehrer and Otto, it is well known that a large class of parabolic equations can be seen as gradient flows in the Wasserstein space. This thesis is devoted to extensions of this theory to equations and systems which do not have exactly a gradient flow structure. We study different kind of couplings. First, we treat the case of nonlocal interactions in the drift. Then, we study cross diffusion systems which model congestion for several species. We are also interested in reaction-diffusion systems as diffusive prey-predator systems or tumor growth models. Finally, we introduce a new class of systems where the interaction is given by a multi-marginal transport problem. In many cases, we give numerical simulations to illustrate our theorical results. Distance de Wasserstein Flots de gradient Schéma JKO Splitting Dérive non locale Diffusions non linéaires Diffusions croisées Systèmes de réaction-Diffusion Équations d'Hele-Shaw Transport optimal Transport multi-Marges Formule de Benamou-Brenier Lagrangien augmenté Mouvement de foules Espèces en interaction Wasserstein distance Gradient flows JKO scheme Splitting Nonlocal drift Nonlinear diffusions Cross-Diffusion Reaction-Diffusion systems Hele-Shaw equation Optimal transport Multi-Marginal transport Benamou-Brenier formula Augmented lagrangian Crowd motions Interacting species 519.2

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Regularity for solutions of nonlocal fully nonlinear parabolic equations and free boundaries on two dimensional cones

Systèmes de particules en interaction, approche par flot de gradient dans l'espace de Wasserstein / Interacting particles systems, Wasserstein gradient flow approach