Well-posedness of the Muskat problem with H2 initial data

We study the dynamics of the interface between two incompressible fluids in a two-dimensional porous medium whose flow is modeled by the Muskat equations. For the two-phase Muskat problem, we establish global well-posedness and decay to equilibrium for small H2 perturbations of the rest state. For the one-phase Muskat problem, we prove local well-posedness for H2 initial data of arbitrary size. Finally, we show that solutions to the Muskat equations instantaneously become infinitely smooth.