Reaction-Diffusion porous medium equations and systems

par FILIPPO SANTAMBROGIO (Institut Camille Jordan, UCBL)

mercredi 13 nov. 2024, 10:00 → 11:30 Europe/Paris

Fokko du Cloux (Braconnier)

The talk will focus on various estimates related to porous medium equations, with possible drifts or reaction terms, and possibly coupling equations for multi-population models.

In some  papers with various collaborators, Noemi David showed estimates on quantities of the form \int p|D²p|² and \int |Dp|⁴ , where p is the pressure. Some of these esimate also survive in the incompressible limit when the exponent in the porous medium diffusion tends to infinity. In his paper "Lagrangian solutions to the Porous Media Equation and Reaction Diffusion Systems" Matt Jacobs uses the first of these estimates (together with an estimate on how much p is close to 0) to prove that the vector field -Dp is almost Sobolev and define a unique Lagrangian flow map. He applies this to the construction of a segregated solution for a system of cross-reaction diffusion equations.

The talk will be devoted to presenting the main ideas of these estimates and their application. I will also explain how to find a solution to the same class of cross-reaction-diffusion systems via EDI methods (Energy Dissipation Inequality).