A note on stability shifting for the Muskat problem

TL;DR

This paper demonstrates that solutions to the Muskat problem can change stability regimes over time and provides numerical evidence of solutions developing singularities, highlighting complex dynamic behaviors.

Contribution

It introduces the phenomenon of stability shifting in the Muskat problem and offers numerical evidence of singularity formation for solutions with moderate initial derivatives.

Findings

Solutions can shift from unstable to stable and back to unstable.

Numerical evidence of turning singularities in solutions with medium initial derivative norms.

Stability regimes are not fixed but can evolve dynamically.

Abstract

In this note, we show that there exist solutions of the Muskat problem that shift stability regimes: they start unstable, then become stable, and finally return to the unstable regime. We also exhibit numerical evidence of solutions with medium-sized $L^{\infty}$ norm of the derivative of the initial condition that develop a turning singularity.