Well-posedness of the traveling wave problem for the free boundary compressible Navier-Stokes equations

TL;DR

This paper proves the existence and uniqueness of traveling wave solutions in viscous compressible fluids with free boundaries, establishing a well-posedness theory for these complex fluid dynamics problems.

Contribution

It provides the first general construction of traveling wave solutions for free boundary compressible Navier-Stokes equations, including cases with surface tension.

Findings

Existence of traveling wave solutions for nontrivial wave speeds.

Unique solutions depend continuously on data and wave speed.

Results include cases with and without surface tension in three or more dimensions.

Abstract

We prove that traveling waves in viscous compressible liquids are a generic phenomenon. The setting for our result is a horizontally infinite, finite depth layer of compressible, barotropic, viscous fluid, modeled by the free boundary compressible Navier-Stokes equations in dimension $n \ge 2$. The bottom boundary of the fluid is flat and rigid, while the top is a moving free boundary. A constant gravitational field acts normal to the flat bottom. We allow external forces to act in the fluid's bulk and external stresses to act on its free surface. These are posited to be in traveling wave form, i.e. time-independent when viewed in a coordinate system moving at a constant, nontrivial velocity parallel to the lower rigid boundary. In the absence of such external sources of stress and force, the fluid system reverts to equilibrium, which corresponds to a flat, quiescent fluid layer with vertically stratified density. In contrast, when such sources of stress or force are present, the system admits traveling wave solutions. We establish a small data well-posedness theory for this problem by proving that for every nontrivial traveling wave speed there exists a nonempty open set of stress and forcing data that give rise to unique traveling wave solutions, and that these solutions depend continuously on the data and the wave speed. When $n \ge 3$ we prove this with surface tension accounted for at the free boundary, while in the case $n=2$ we prove this with or without surface tension. To the best of our knowledge, this result constitutes the first general construction of traveling wave solutions to any free boundary compressible fluid equations.