We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy’s law. The free boundary is given by the discontinuity among the densities and viscosities of the fluids. This physical scenario is known as the two dimensional Muskat problem or the two-phase Hele-Shaw flow. We prove local-existence in Sobolev spaces when, initially, ...

We prove that the Cauchy problem for Muskat equation is well-posed locally in time any initial data critical space of Lipschitz functions with three-half derivative $$L^2$$ . Moreover, we solution exists globally under a smallness assumption.

We exhibit a new decomposition of the nonlinearity for Muskat equation and use it to commute Fourier multipliers with equation. This allows study solutions critical regularity. As corollary, we obtain first well-posedness result arbitrary large data in space H?2(R2)?W1,?(R2). Moreover, prove existence initial which are not Lipschitz.