Localized mixing zone for Muskat bubbles and turned interfaces

TL;DR

This paper develops localized mixing solutions for the Muskat problem, demonstrating the continuation of interface evolution beyond classical instabilities by combining classical analysis with convex integration techniques.

Contribution

It introduces a novel approach to construct mixing solutions with localized zones, extending the understanding of interface evolution in unstable porous media flows.

Findings

Existence of mixing solutions beyond Rayleigh-Taylor instability

Localized mixing zones near unstable interfaces

Compatibility of Muskat problem with convex integration methods

Abstract

We construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh-Taylor and smoothness breakdown exhibited in [18,17]. At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat problem and the convex integration method.