Free-surface Euler equations with density variations, and shallow-water limit

TL;DR

This paper extends the analysis of free-surface Euler equations to include density variations and vorticity, providing well-posedness results and justifying the shallow-water limit with a focus on long-time behavior.

Contribution

It introduces a framework for the well-posedness of Euler equations with density variations and establishes the convergence to shallow-water equations, including long-time existence results.

Findings

Established well-posedness in Sobolev spaces for the extended system.

Proved convergence from free-surface Euler to shallow-water equations.

Demonstrated existence of solutions on a logarithmic time-scale.

Abstract

In this paper we study the well-posedness in Sobolev spaces of the incompressible Euler equations in an infinite strip delimited from below by a non-flat bottom and from above by a free-surface. We allow the presence of vorticity and density variations, and in these regards the present system is an extension of the well-studied water waves equations. When the bottom is flat and with no density variations (but when the flow is not necessarily irrotational), our study provides an alternative proof of the already known large-time well-posedness results for the water waves equations. Our main contribution is that we allow for the presence of density variations, while also keeping track of the dependency in the shallow water parameter. This allows us to justify the convergence from the free-surface Euler equations towards the non-linear shallow water equations in this setting. Using an already established large time existence result for these latter equations, we also prove the existence of the free-surface Euler equations on a logarithmic time-scale, in a suitable regime.