Well-posedness of the Euler equations in a stably stratified ocean in isopycnal coordinates

Abstract

This article is concerned with the well-posedness of the incompressible Euler equations describing a stably stratified ocean, reformulated in isopycnal coordinates. Our motivation for using this reformulation is twofold: first, its quasi-2D structure renders some parts of the analysis easier. Second, it closes a gap between the analysis performed in the paper by Bianchini and Duch{\^e}ne in 2022 in isopycnal coordinates, with shear velocity but with a regularizing term, and the analysis performed in the paper by Desjardins, Lannes, Saut in 2020 in Eulerian coordinates, without any regularizing term but without shear velocity. Our main result is a local well-posedness result in Sobolev spaces on the system in isopycnal coordinates, with shear velocity, without any regularizing term. The time of existence that we obtain is uniform with respect to the size $\epsilon$ of the perturbation, and boils down to the large time $1/\epsilon$ with the assumptions of the paper by Desjardins, Lannes, Saut in 2020. With additional assumptions, it is also uniform in the shallow-water parameter. The main difficulty consists in transposing to the isopycnal reformulation the symmetric structure of the system which is more straightforward in Eulerian coordinates.