Turning waves and breakdown for incompressible flows

TL;DR

This paper investigates the finite-time breakdown of interfaces in incompressible fluid flows, specifically Muskat and water wave problems, showing that solutions can become singular and lose their graphical structure.

Contribution

It demonstrates that interfaces in these fluid models can reach a blow-up regime in finite time, including unstable regimes where the Rayleigh-Taylor condition fails.

Findings

Interface reaches a non-graph regime in finite time

Solution blows up with infinite derivative norm

Unstable regimes with sign change in Rayleigh-Taylor condition

Abstract

We consider the evolution of an interface generated between two immiscible incompressible and irrotational fluids. Specifically we study the Muskat and water wave problems. We show that starting with a family of initial data given by $(\al,f_0(\al))$, the interface reaches a regime in finite time in which is no longer a graph. Therefore there exists a time $t^*$ where the solution of the free boundary problem parameterized as $(\al,f(\al,t))$ blows-up: $\|\da f\|_{L^\infty}(t^*)=\infty$. In particular, for the Muskat problem, this result allows us to reach an unstable regime, for which the Rayleigh-Taylor condition changes sign and the solution breaks down.