Regularity of solutions to the Muskat equation

TL;DR

This paper proves that smooth solutions to the Muskat problem with different densities are analytic except at points where fluid overturn occurs, providing insight into the regularity and singularity formation in fluid interface problems.

Contribution

It establishes the analyticity of smooth solutions to the Muskat problem away from overturn points, advancing understanding of solution regularity in fluid interface dynamics.

Findings

Solutions are analytic except at overturn points.

Regularity results depend on fluid density differences.

Provides conditions under which solutions maintain smoothness.

Abstract

In this paper, we show that if a solution to the Muskat problem in the case of different densities and the same viscosity is sufficiently smooth, then it must be analytic except at the points where a turnover of the fluids happens.