Well-Posedness of the Free Boundary Incompressible Porous Media Equation

We consider the free boundary incompressible porous media equation which describes the dynamics of a density transported by a Darcy flow in the field of gravity, with a free boundary between the fluid region and the dry region above it. For any stratified density state, we identify a stability condition for the initial free boundary. Under this condition, we prove that small localized perturbations of the stratified density lead to unique local-in-time solutions in Sobolev spaces. Our proof involves analytic ingredients that are of independent interest, including tame fractional Sobolev estimates for operators that map the Dirichlet boundary function and the forcing function of Poisson's equation to its solution in domains of Sobolev regularity.