Degraded mixing solutions for the Muskat problem

TL;DR

This paper demonstrates the existence of numerous mixing solutions for the Muskat problem in unstable regimes, showing degraded macroscopic behavior and providing a quantitative framework applicable to various evolution equations.

Contribution

It introduces a refined convex integration approach to construct mixing solutions with degraded behavior and establishes a quantitative h-principle for a broad class of evolution equations.

Findings

Existence of infinitely many mixing solutions in the unstable Muskat regime

Quantitative estimates of fluid volume proportions in the mixing zone

Application of the framework to vortex sheets in Euler equations

Abstract

We prove the existence of infinitely many mixing solutions for the Muskat problem in the fully unstable regime displaying a linearly degraded macroscopic behaviour inside the mixing zone. In fact, we estimate the volume proportion of each fluid in every rectangle of the mixing zone. The proof is a refined version of the convex integration scheme presented in [DS10, Sze12] applied to the subsolution in [CCF16]. More generally, we obtain a quantitative h-principle for a class of evolution equations which shows that, in terms of weak*-continuous quantities, a generic solution in a suitable metric space essentially behaves like the subsolution. This applies of course to linear quantities, and in the case of IPM to the power balance $\mathbf{P}$ (14) which is quadratic. As further applications of such quantitative h-principle we discuss the case of vortex sheet for the incompressible Euler equations.