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) that includes considerably more general fluid interface geometries than even prior whole plane results, and crucially allows the interface to touch the bottom. The latter can be used to model physically relevant scenarios where the heavier fluid invades 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.
By Matthew Juniper
By Sergey Gavrilyuk
By David Lannes
