Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves

TL;DR

This paper investigates the stability of the Muskat problem, showing that the Rayleigh-Taylor condition can fail in finite time, leading to water waves turning, which has implications for fluid interface dynamics.

Contribution

It demonstrates that the Rayleigh-Taylor condition may initially hold but can break down in finite time within the Muskat problem, and proves the existence of water waves turning.

Findings

Rayleigh-Taylor condition can break down in finite time

Existence of water waves turning due to stability loss

Implications for fluid interface evolution

Abstract

The Muskat problem models the evolution of the interface given by two different fluids in porous media. The Rayleigh-Taylor condition is natural to reach the linear stability of the Muskat problem. We show that the Rayleigh-Taylor condition may hold initially but break down in finite time. As a consequence of the method used, we prove the existence of water waves turning.