Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues

Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem. We establish the complementarity relation and also a segregation result. Several major mathematical difficulties arise in the two species case. Firstly, the system structure makes comparison principles fail. Secondly, segregation and internal layers limit the regularity available on some quantities to BV. Thirdly, the Aronson-B{\'e}nilan estimates cannot be established in our context. We are lead, as it is classical, to add correction terms. This procedure requires technical manipulations based on BV estimates only valid in one space dimension. Another novelty is to establish an L1 version in place of the standard upper bound

Stability of flows associated to gradient vector fields and convergence of iterated transport maps

In this paper we address the problem of stability of flows associated to a sequence of vector fields under minimal regularity requirements on the limit vector field, that is supposed to be a gradient. We apply this stability result to show the convergence of iterated compositions of optimal transport maps arising in the implicit time discretization (with respect to the Wasserstein distance) of nonlinear evolution equations of a diffusion type. Finally, we use these convergence results to study the gradient flow of a particular class of polyconvex functionals recently considered by Gangbo, Evans ans Savin. We solve some open problems raised in their paper and obtain existence and uniqueness of solutions under weaker regularity requirements and with no upper bound on the jacobian determinant of the initial datum

Fast Diffusion Equations with Caffarelle-Kohn-Nirenberg Weights: Regularity and Asymptotics.

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 25-02-202

Modèles de diffusion non linéaire issus de la théorie de la fusion par confinement inertiel

Cette thèse est consacrée à l'étude mathématique de deux modèles de réaction-diffusion qui interviennent en Fusion par Confinement Inertiel. Dans un premier chapitre, nous proposons un nouveau modèle thermodiffusif décrivant un effet de stabilisation par ablation transverse aux petites longueurs d'onde. Cette approche a été suggérée par [Clavin,Masse 2004], suite a une étude linéaire auto-consistante du modèle thermo-hydrodynamique complet [Sanz,Masse,Clavin 2006] dans laquelle une relation de dispersion heuristique a été établie. Une première étude [Clavin,Masse,Roquejoffre 2011] a permis, pour un modèle approché, d'obtenir rigoureusement une relation de dispersion très proche. Nous prouvons, dans le cadre d'une approximation d'écoulement longitudinal, qu'on retrouve bien la relation de dispersion auto-consistante. Un deuxième chapitre est consacré à l'existence de solutions d'onde pour un modèle de flamme non-linéaire en écoulement cisaillé et avec croissance linéaire à l'infini dans la direction de propagation. Nous montrons que cette solution existe pour des vitesses de propagation plus rapides qu'une certaine vitesse critique, explicitement calculée en fonction de l'écoulement prescrit. Cette solution, qui possède une interface libre, est tout à fait analogue à la solution d'onde plane de l'Equation des Milieux Poreux ; la nouveauté réside ici en la présence d'un écoulement cisaillé longitudinal. Dans un dernier chapitre nous étudions numériquement la frontière libre, pour laquelle nos simulations semblent indiquer la présence de coins. Par une étude semi-heuristique, nous donnons un scénario possible permettant d'étudier la régularité et la description géométrique de la frontière libre.This PhD thesis is devoted to the study of two reaction-diffusion models arising in Inertial Confinement Fusion. In chapter 1 we derive a new thermodiffusive model, describing a stabilization at short wave-lengths by transversal mass ablation. A self-consistent analysis [SMC06] of the full thermo-hydrodynamical model yielded a heuristic dispersion relation. It was suggested in [Masse,Clavin 2004] that the stabilization can be investigated looking at a much simpler model, namely the linear relaxation of wrinkled fronts. A first rigorous analysis was performed for an approximated model in [Clavin,Masse,Roquejoffre 2011], where a very similar dispersion relation was obtained. We prove here that, in the context of a longitudinal flow approximation, the dispersion relation obtained in our model is exactly the self-consistent one. In chapter 2, we establish an existence result for traveling wave solutions in some non-linear flame model with a shear flow and growth condition at infinity in the propagation direction. We show that this solutions exists for propagation speeds larger than some critical speed explicitly computed in terms of the flow. This solution, which has a free boundary, is very similar to the planar traveling wave existing for the Porous Media Equation. The main novelty is here the presence of a prescribed longitudinal shear flow. In the last chapter we use numerical simulations to investigate the free boundary, in which corners seem to appear. We give a semi-heuristic argument, which may allow one to study the free boundary regularity and its geometrical description