Stratified periodic water waves with singular density gradients

TL;DR

This paper develops a mathematical framework for analyzing symmetric periodic water waves with stratified density, including singular gradients, using weak formulations and nonlinear analysis.

Contribution

It establishes three equivalent classical formulations for stratified water waves with singular density gradients and constructs symmetric periodic traveling wave solutions.

Findings

Existence of symmetric periodic traveling waves with stratified density

Wave solutions can have singular density gradients

Framework applies to waves with strong solutions and singularities

Abstract

We consider Euler's equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. For this problem we establish three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients. Based upon this equivalence we then construct two-dimensional symmetric periodic traveling waves that are monotone between each crest and trough. Our analysis uses, to a large extent, the availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis.