The viscous surface wave problem with generalized surface energies

TL;DR

This paper proves global well-posedness and exponential decay to equilibrium for a 3D viscous fluid with a free surface influenced by generalized surface energies, including bending and tension effects.

Contribution

It introduces a nonlinear energy method to handle the complex surface energy contributions in the viscous surface wave problem.

Findings

Solutions exist globally for initial data near equilibrium.

Solutions decay exponentially fast to equilibrium.

The method handles fully nonlinear surface energy terms.

Abstract

We study a three-dimensional incompressible viscous fluid in a horizontally periodic domain with finite depth whose free boundary is the graph of a function. The fluid is subject to gravity and generalized forces arising from a surface energy. The surface energy incorporates both bending and surface tension effects. We prove that for initial conditions sufficiently close to equilibrium the problem is globally well-posed and solutions decay to equilibrium exponentially fast, in an appropriate norm. Our proof is centered around a nonlinear energy method that is coupled to careful estimates of the fully nonlinear surface energy.