On large periodic traveling wave solutions to the free boundary Stokes and Navier-Stokes equations

TL;DR

This paper establishes the existence and stability of large amplitude periodic traveling wave solutions in free boundary Stokes and Navier-Stokes equations with surface tension and external stress, using advanced operator analysis.

Contribution

It proves the existence and asymptotic stability of large amplitude periodic traveling wave solutions for free boundary viscous flows with arbitrary stress tensors.

Findings

Existence of large amplitude periodic traveling waves.

Asymptotic stability of these solutions.

Application of nonlocal operator analysis in Sobolev domains.

Abstract

We study the free boundary problem for a finite-depth layer of viscous incompressible fluid in arbitrary dimension, modeled by the Stokes or Navier-Stokes equations. In addition to the gravitational field acting in the bulk, the free boundary is acted upon by surface tension and an external stress tensor posited to be in traveling wave form. We prove that for any isotropic stress tensor with periodic profile, there exists a locally unique periodic traveling wave solution, which can have large amplitude. Moreover, we prove that the constructed traveling wave solutions are asymptotically stable for the dynamic free boundary Stokes equations. Our proofs rest on the analysis of the nonlocal normal-stress to normal-Dirichlet operators for the Stokes and Navier-Stokes equations in domains of Sobolev regularity.