Analysis Seminar

The Muskat problem on the half-plane models motion of an interface between two fluids of distinct densities in a porous medium that sits atop an impermeable layer, such as oil and water in an aquifer above bedrock.  We develop a local well-posedness theory for this model in the stable regime (lighter fluid above the heavier one), which includes considerably more general fluid interface geometries than even existing whole plane results and allows the interface to touch the bottom.  The latter applies to the important scenario of the heavier fluid invading a region occupied by the lighter fluid along the impermeable layer.  We also show that finite time singularities do arise in this setting, including from arbitrarily small smooth initial data, by obtaining maximum principles for the height, slope, and potential energy of the fluid interface.