Introduction

Many relevant phenomena, not only in the natural sciences but also in engineering and economics, are modeled by (systems of) partial differential equations (PDEs) that exhibit some sort of degeneracy or singularity. Examples include the motion of multi-phase fluids in porous media, the melting of crushed ice (and phase transitions, in general), the behavior of composite materials or the pricing of assets in financial markets. Because of its significance in terms of the applications, but also due to the novel analytical techniques that it generates, the class of degenerate and singular parabolic equations is an important branch in the contemporary analysis of partial differential equations.

The purpose of these lecture notes is to describe intrinsic scaling, a method for obtaining continuity results for the weak solutions of degenerate and singular parabolic equations, and to convince the reader of the strength of this approach to regularity, by giving evidence of its wide applicability in different situations. To understand what is at stake, let us start by placing the problem in its historical context. As in many other mathematical journeys, it all started with one of Hilbert's problems.