The viscous surface-internal wave problem: global well-posedness and decay

TL;DR

This paper proves the global existence and decay rates of solutions for a viscous two-fluid free boundary problem, showing surface tension stabilizes the interface and identifying critical thresholds for stability.

Contribution

It establishes the global well-posedness and decay rates for viscous surface-internal wave problems with and without surface tension, including stability criteria for Rayleigh-Taylor instability.

Findings

Solutions decay exponentially with surface tension

Without surface tension, decay is almost exponential

Critical surface tension stabilizes the heavier-over-lighter fluid configuration

Abstract

We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a three-dimensional horizontally periodic setting. We establish the global well-posedness of the problem both with and without surface tension. We prove that without surface tension the solution decays to the equilibrium state at an almost exponential rate; with surface tension, we show that the solution decays at an exponential rate. Our results include the case in which a heavier fluid lies above a lighter one, provided that the surface tension at the free internal interface is above a critical value, which we identify. This means that sufficiently large surface tension stabilizes the Rayleigh-Taylor instability in the nonlinear setting. As a part of our analysis, we establish elliptic estimates for the two-phase stationary Stokes problem.