Title:

Stable regime singularity for the Muskat problem

Abstract:

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 show that unlike on the whole plane, finite time singularities do arise in the stable regime (lighter fluid above the heavier one) in this setting, including from arbitrarily small smooth initial data.  We achieve this by developing a local well-posedness theory for this model as well as obtaining maximum principles for the height, slope, and potential energy of the fluid interface.  The former allows the interface to touch the bottom, which applies to the important scenario of the heavier fluid invading a region occupied by the lighter fluid along the impermeable layer, and includes considerably more general fluid interface geometries than even previous whole plane results.