Workshop on Nonlinear Parabolic PDEs

May 20 - May 24, 2024

Nonlinear evolutionary partial differential equations are of fundamental importance in mathematical analysis. They are used to model time-dependent phenomena such as flows in porous media, turbulent filtration processes, groundwater flows through gravel or fractured crystalline rocks, shallow water waves, turbulent polytropic filtration of gas or the moving boundary between two phases of a material undergoing a phase change (for instance the melting of ice to water, Stefan problem. Such phenomena or processes are typically modeled by parabolic PDE’s with singular and/ or degenerate.

In recent years, there have been quite a number of breakthroughs in the theory of singular and degenerate parabolic PDEs. For example, it was shown that weak solutions of the porous medium equation are higher integrable in the sense of Elcrat and Meyers. This result was extended to porous medium type systems and Trudinger’s equation. Moreover, several parameter ranges of doubly nonlinear parabolic equations are nowadays covered. Also, the Hölder continuity of weak solutions is widely understood for both non-negative and signed solutions. There has also been significant progress in regularity theory for the Stefan problem. On the other hand, many important problems remain unresolved. Even the uniqueness of weak solutions to doubly non-linear equations is not yet fully understood.

The proposed workshop will focus on the following themes related to (doubly) nonlinear parabolic PDEs

  • comparison principles and uniqueness of weak solutions, 
  • boundary regularity, 
  • regularity of the spatial gradient and  
  • behavior of solutions in the subcritical range.

As will become apparent, the novelties are not mutually exclusive and several themes can overlap in a specific research problem.