A unified approach towards the impossibility of finite time vanishing depth for incompressible free boundary flows

TL;DR

This paper proves that under certain smoothness conditions, the depth of internal water waves and porous medium flows cannot reach zero in finite time, regardless of physical effects like gravity or stratification.

Contribution

It establishes a general impossibility result for finite-time vanishing depth in free boundary flows, extending previous understanding to various physical scenarios.

Findings

Depth cannot vanish in finite time under smoothness assumptions

Results hold regardless of gravity and surface tension effects

Applicable to both water waves and porous media flows

Abstract

In this paper we study the motion of an internal water wave and an internal wave in a porous medium. For these problems we establish that, if the free boundary and, in the case of the Euler equations, also the tangential velocity at the interface are sufficiently smooth, the depth cannot vanish in finite time. This results holds regardless of gravity and surface tension effects or, if applicable, the stratification in multiphase flows.