Well-posedness of the Muskat problem with $H^2$ 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 $H^2$ perturbations of the rest state. For the one-phase Muskat problem, we prove local well-posedness for $H^2$ initial data of arbitrary size. Finally, we show that solutions to the Muskat equations instantaneously become infinitely smooth. 
C.H. Arthur Cheng, Rafael Granero-Belinch\'on, and Steve Shkoller
math.AP