Topics in Analysis: Fluid Dynamics and Related Equations

The course studies the Muskat equation, which governs the evolution of the interface between two fluids in a porous medium, e.g. oil and water in sand. In two dimensions, when the two fluids have the same viscosity but different densities, the problem reduces to a contour dynamics equation. When the interface is the graph of a function, a lot is known; but in the "turnover regime" in which a vertical line may meet the interface in more than one point, hard problems remain. The lectures explain the state of the art as of a few years ago, then present recent work of Jia Shi.