Free-boundary problems are partial differential equations in which the unknown function and its domain must be simultaneously determined. They arise ubiquitously as mathematical models for phenomena in many fields, most notably in physics, biology and finance. Free boundary problems are typically highly nonlinear and nonlocal in nature, making their analysis challenging. I will discuss two fundamental mathematical themes – regularity and stability – in the context of two important problems in fluid mechanics, the water waves and the Muskat problems. These problems fall into the class of hyperbolic-dispersive and degenerate parabolic PDE, respectively. A synthesis of tools from harmonic analysis, microlocal analysis, functional analysis, and spectral theory is employed to tackle these problems.

https://ncsu.zoom.us/j/98092581180?pwd=bXZ3cEkvTVlMSUJlMGIyTVl0WGppdz09