Global-in-time dynamics for Muskat and two-phase Stokes gravity waves

In this talk we consider the evolution of an interface evolving by an incompressible flow. On the one hand, we study the one-phase Muskat problem, where the fluid is filtered in a porous medium. In the gravity-stable case, we show that initial Lipschitz graphs of arbitrary size provide global-in-time well-posedness. On the other hand, we study the interface dynamics given by two fluids of different densities evolving by the linear Stokes law. We show stability to the flat stable case and exponential growth in the unstable regime.