The compressible viscous surface-internal wave problem: stability and vanishing surface tension limit

TL;DR

This paper studies the stability of two-layer compressible viscous fluids with free boundaries, analyzing the effects of surface tension and gravity, and establishes conditions for stability and the zero surface tension limit.

Contribution

It provides a sharp nonlinear stability criterion for the free boundary problem and characterizes the influence of surface tension on stability, including the zero surface tension limit.

Findings

Large surface tension can prevent Rayleigh-Taylor instability.

The equilibrium is stable for all non-negative surface tensions when the lower fluid is heavier.

Explicit decay rates to equilibrium are established.

Abstract

This paper concerns the dynamics of two layers of compressible, barotropic, viscous fluid lying atop one another. The lower fluid is bounded below by a rigid bottom, and the upper fluid is bounded above by a trivial fluid of constant pressure. This is a free boundary problem: the interfaces between the fluids and above the upper fluid are free to move. The fluids are acted on by gravity in the bulk, and at the free interfaces we consider both the case of surface tension and the case of no surface forces. We establish a sharp nonlinear global-in-time stability criterion and give the explicit decay rates to the equilibrium. When the upper fluid is heavier than the lower fluid along the equilibrium interface, we characterize the set of surface tension values in which the equilibrium is nonlinearly stable. Remarkably, this set is non-empty, i.e. sufficiently large surface tension can prevent the onset of the Rayleigh-Taylor instability. When the lower fluid is heavier than the upper fluid, we show that the equilibrium is stable for all non-negative surface tensions and we establish the zero surface tension limit.