Breakdown of smoothness for the Muskat problem

TL;DR

This paper demonstrates that even stable initial conditions in the Muskat problem can evolve into unstable regimes and experience breakdown, highlighting the complex dynamics and potential for singularity formation.

Contribution

It reveals the existence of analytic initial data leading to regime transition and breakdown in the Muskat problem, a novel insight into its stability properties.

Findings

Existence of initial data causing regime transition.

Solutions can lose regularity and break down.

Highlights complex dynamics in the Muskat problem.

Abstract

In this paper we show that there exist analytic initial data in the stable regime for the Muskat problem such that the solution turns to the unstable regime and later breaks down i.e. no longer belongs to $C^4$.