The dynamics of a gravity-driven Stokes free boundary

The dynamics of viscous waves is a very active subject in PDEs. When inertial forces are negligible compared to viscous forces, the fluids flow in the Stokes regime. In this talk, we will describe the dynamics of two incompressible, viscous and immiscible fluids filling a horizontally periodic strip. We consider the fluids to have different densities and to be subject to gravity forces. This induces the dynamics of the free boundary between the fluids. We will study the derivation of this model through a contour dynamics approach and we will address fundamental questions such as the well-posedness and stability of the free boundary.