Interface evolution: the Hele-Shaw and Muskat problems

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Abstract
The following article appeared in Annals of Mathematics 173.1 (2011):477-542 and may be found at http://annals.math.princeton.edu/2011/173-1/p10We 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, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition.The first named author was supported in part by grant MTM2005-04730 of the MEC (Spain). The other two authors were partially supported by grant MTM2005-05980 of the MEC (Spain) and grant StG-203138CDSIF of the ERC