241422 Regularity theory for nonlinear, degenerate partial differential equations (BS) (WiSe 2017/2018)

In this lecture we will focus on the porous medium equation. The porous
medium equation arises, for example, as a model for the flow of an ideal gas
in a porous medium. The degenerate nature of the diffusivity of this PDE
leads to interesting effects such as limited regularity of solutions, finite
speed of propagation, open interfaces, waiting times etc. We will first
recall several applications of the porous medium equation, consider special
(self-similar) solutions and highlight particular properties of solutions to
degenerate PDE. Then, we will focus on the question of (optimal) regularity
of solutions to porous media equations, which relies on methods from
harmonic analysis (Fourier multipliers, interpolation theory, Besov spaces)
which will be recalled in the course of the lecture.