Well-posedness of the stationary and slowly traveling wave problems for the free boundary incompressible Navier-Stokes equations

TL;DR

This paper proves the existence, uniqueness, and continuous dependence of stationary and traveling wave solutions in a three-dimensional viscous incompressible fluid with free boundary, gravity, surface tension, and applied forces, using advanced harmonic analysis and implicit function theorem techniques.

Contribution

It is the first to establish well-posedness of the solitary stationary wave problem and to embed it continuously into the traveling wave framework.

Findings

Existence and uniqueness of stationary and traveling wave solutions.

Continuous dependence of solutions on physical parameters and data.

Development of a novel analytical approach combining harmonic analysis and implicit function theorem.

Abstract

We establish that solitary stationary waves in three dimensional viscous incompressible fluids are a generic phenomenon and that every such solution is a vanishing wave-speed limit along a one parameter family of traveling waves. The setting of our result is a horizontally-infinite fluid of finite depth with a flat, rigid bottom and a free boundary top. A constant gravitational field acts normal to bottom, and the free boundary experiences surface tension. In addition to these gravity-capillary effects, we allow for applied stress tensors to act on the free surface region and applied forces to act in the bulk. These are posited to be in either stationary or traveling form. In the absence of any applied stress or force, the system reverts to a quiescent equilibrium; in contrast, when such sources of stress or force are present, stationary or traveling waves are generated. We develop a small data well-posedness theory for this problem by proving that there exists a neighborhood of the origin in stress, force, and wave speed data-space in which we obtain the existence and uniqueness of stationary and traveling wave solutions that depend continuously on the stress-force data, wave speed, and other physical parameters. To the best of our knowledge, this is the first proof of well-posedness of the solitary stationary wave problem and the first continuous embedding of the stationary wave problem into the traveling wave problem. Our techniques are based on vector-valued harmonic analysis, a novel method of indirect symbol calculus, and the implicit function theorem.