Mixing solutions for the Muskat problem

TL;DR

This paper proves the existence of mixing solutions for the Muskat problem in the unstable regime using advanced mathematical techniques, expanding understanding of fluid mixing in porous media.

Contribution

It introduces a novel combination of convex integration, contour dynamics, and semiclassical pseudodifferential calculus to establish existence results for the Muskat problem.

Findings

Existence of mixing solutions for all $H^5$ initial data in the unstable regime.

Development of a calculus for non-smooth semiclassical pseudodifferential operators.

Application of convex integration to porous media equations.

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.