Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients

14/02/2020, 14:00 - 15:00 in MAP/0G/018

Meszaros, Alpar Richard (Durham University)


Abstract:

Motivated by some physical and biological models, in this talk we consider a class of degenerate parabolic equations. Our analysis is based on gradient flows in the space of probability measures equipped with the distance arising in the Monge-Kantorovich optimal transport problem. The associated internal energy functionals in general fail to be differentiable, therefore classical results do not apply in our setting. We study the combination of both linear and porous medium type diffusions and we show the existence and uniqueness of the solutions in the sense of distributions in suitable Sobolev spaces. Our notion of solution allows us to give a fine characterization of the emerging ‘critical regions’, observed previously in numerical experiments. A link to a three phase free boundary problem is also pointed out. The talk is based on a joint work with Dohyun Kwon (UCLA). The talk is aimed to a general audience in mathematical analysis and no prior knowledge on the theory of optimal transport is assumed.

Mathematical Sciences Research Centre