In my research, I am investigating structural properties of non-linear partial differential equations (PDEs) from the point of view of (non-)uniqueness and (ir)regularity. The goal is to understand the behaviour of solutions to PDEs that are motivated by physics or geometry.
The main tool in this analysis is convex integration, a technique originally developed by Gromov in the study of differential inclusions.
It provides a structured yet flexible framework to construct highly irregular weak solutions, which usually goes hand in hand with severe non-uniqueness of such solutions. Conceptually, the goal is to develop a bottom-up approach (in terms of regularity) to uniqueness questions in non-linear PDEs, which is complementary to the usual top-down approach based on energy methods or weak-strong uniqueness.
Problems I am working on are related to
- Isometric embeddings
- Hall-magnetohydrodynamics
- Harmonic maps