Traveling wave solutions to the inclined or periodic free boundary incompressible Navier-Stokes equations

TL;DR

This paper constructs traveling wave solutions for the free boundary incompressible Navier-Stokes equations in inclined or periodic settings, introducing new functional analytic tools for anisotropic Sobolev spaces.

Contribution

It develops novel analytical methods for anisotropic Sobolev spaces and constructs traveling wave solutions in inclined or periodic fluid domains.

Findings

Existence of traveling wave solutions in inclined/periodic domains

Development of new properties of anisotropic Sobolev spaces

Solutions as perturbations of shear flows

Abstract

This paper concerns the construction of traveling wave solutions to the free boundary incompressible Navier-Stokes system. We study a single layer of viscous fluid in a strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. The fluid is acted upon by a bulk force and a surface stress that are stationary in a coordinate system moving parallel to the fluid bottom. We also assume that the fluid is subject to a uniform gravitational force that can be resolved into a sum of a vertical component and a component lying in the direction of the traveling wave velocity. This configuration arises, for instance, in the modeling of fluid flow down an inclined plane. We also study the effect of periodicity by allowing the fluid cross section to be periodic in various directions. The horizontal component of the gravitational field gives rise to stationary solutions that are pure shear flows, and we construct our solutions as perturbations of these by means of an implicit function argument. An essential component of our analysis is the development of some new functional analytic properties of a scale of anisotropic Sobolev spaces, including that these spaces are an algebra in the supercritical regime, which may be of independent interest.