Continuous dependence on the density for stratified steady water waves

TL;DR

This paper proves that for small-amplitude stratified water waves, solutions depend continuously on the density profile, justifying the common approximation of smooth stratification by layered models.

Contribution

It establishes a Lipschitz continuous mapping from streamline density functions to wave solutions, including piecewise smooth densities with jumps.

Findings

Solutions exist for a neighborhood of density functions.

The dependence of solutions on density is Lipschitz continuous.

Includes densities with arbitrarily many jumps.

Abstract

There are two distinct regimes commonly used to model traveling waves in stratified water: continuous stratification, where the density is smooth throughout the fluid, and layer-wise continuous stratification, where the fluid consists of multiple immiscible strata. The former is the more physically accurate description, but the latter is frequently more amenable to analysis and computation. By the conservation of mass, the density is constant along the streamlines of the flow; the stratification can therefore be specified by prescribing the value of the density on each streamline. We call this the streamline density function. Our main result states that, for every smoothly stratified periodic traveling wave in a certain small-amplitude regime, there is an $L^\infty$ neighborhood of its streamline density function such that, for any piecewise smooth streamline density function in that neighborhood, there is a corresponding traveling wave solution. Moreover, the mapping from streamline density function to wave is Lipschitz continuous in a certain function space framework. As this neighborhood includes piecewise smooth densities with arbitrarily many jump discontinues, this theorem provides a rigorous justification for the ubiquitous practice of approximating a smoothly stratified wave by a layered one. We also discuss some applications of this result to the study of the qualitative features of such waves.