On the Rayleigh-Taylor Instability for the two-Phase Navier-Stokes Equations

TL;DR

This paper rigorously analyzes the Rayleigh-Taylor instability in a two-phase Navier-Stokes system with surface tension and gravity, demonstrating instability when a heavy fluid is above a lighter one in an $L_p$-framework.

Contribution

It provides a rigorous proof of Rayleigh-Taylor instability for the two-phase Navier-Stokes equations near equilibrium using an abstract instability approach.

Findings

Instability occurs when the heavy fluid is on top of the light fluid.

The boundary symbol admits zeros in the unstable half-plane.

The instability is established in an $L_p$-setting.

Abstract

The two-phase free boundary problem with surface tension and downforce gravity for the Navier-Stokes system is considered in a situation where the initial interface is close to equilibrium. The boundary symbol of this problem admits zeros in the unstable halfplane in case the heavy fluid is on top of the light one, which leads to the well-known Rayleigh-Taylor instability. Instability is proved rigorously in an $L_p$-setting by means of an abstract instability result due to Henry.