Asymptotic stability of shear-flow solutions to incompressible viscous free boundary problems with and without surface tension

TL;DR

This paper analyzes the stability of shear-flow solutions in viscous free boundary problems, demonstrating exponential decay to equilibrium with surface tension and almost exponential decay without, including a vanishing surface tension limit.

Contribution

It provides the first rigorous proof of asymptotic stability for shear flows in viscous free boundary problems with and without surface tension, and establishes a vanishing surface tension limit.

Findings

Exponential decay of perturbations with surface tension.

Almost exponential decay without surface tension.

Validation of the vanishing surface tension limit.

Abstract

This paper concerns the dynamics of a layer of incompressible viscous fluid lying above a rigid plane and with an upper boundary given by a free surface. The fluid is subject to a constant external force with a horizontal component, which arises in modeling the motion of such a fluid down an inclined plane, after a coordinate change. We consider the problem both with and without surface tension for horizontally periodic flows. This problem gives rise to shear-flow equilibrium solutions, and the main thrust of this paper is to study the asymptotic stability of the equilibria in certain parameter regimes. We prove that there exists a parameter regime in which sufficiently small perturbations of the equilibrium at time $t=0$ give rise to global-in-time solutions that return to equilibrium exponentially in the case with surface tension and almost exponentially in the case without surface tension. We also establish a vanishing surface tension limit, which connects the solutions with and without surface tension.