Mixing solutions for the Muskat problem

Abstract

We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type $H^5$ initial data in the fully unstable regime. The proof combines convex integration, contour dynamics and a basic calculus for non smooth semiclassical type pseudodifferential operators which is developed.

73 pages, 2 figures. This version includes the case of variable opening of the mixing zone and emphasizes the semiclassical analysis viewpoint