Stability of Navier-Stokes equations with a free surface

TL;DR

This paper establishes a global well-posedness and exponential decay to equilibrium for the Navier-Stokes equations with a free surface in a three-dimensional periodic domain, using advanced energy methods.

Contribution

It introduces a novel nonlinear energy framework and a tripled bootstrap argument to analyze the stability of free surface Navier-Stokes flows.

Findings

Global existence of solutions in low regularity Sobolev spaces

Exponential decay rate towards equilibrium

Development of new energy estimates and bootstrap techniques

Abstract

We consider the viscous incompressible fluids in a three-dimensional horizontally periodic domain bounded below by a fixed smooth boundary and above by a free moving surface. The fluid dynamics are governed by the Navier-Stokes equations with the effect of gravity and surface tension on the free surface. We develop a global well-posedness theory by a nonlinear energy method in low regular Sobolev spaces with several techniques, including: the horizontal energy-dissipation estimates, a new tripled bootstrap argument inspired by Guo and Tice [Arch. Ration. Mech. Anal.(2018)]. Moreover, the solution decays asymptotically to the equilibrium in an exponential rate.