Local-in-time Physical Solutions of the Incompressible Semi-Geostrophic Equations in Eulerian Coordinates

TL;DR

This paper proves the local-in-time existence of smooth, convex solutions to the 3D incompressible semi-geostrophic equations in bounded domains using div-curl systems, addressing an open problem in Eulerian coordinates.

Contribution

It is the first to construct bounded solutions of semi-geostrophic equations in Eulerian coordinates on bounded domains, solving an open problem in the field.

Findings

Existence of local-in-time smooth solutions in 3D bounded domains.

Solutions maintain convexity and adhere to Cullen's Stability Principle.

Methodology relies on div-curl systems and recent estimates on the ageostrophic velocity.

Abstract

We prove the existence of local-in-time smooth solutions of the incompressible semi-geostrophic equations expressed in Eulerian co-ordinates in 3-dimensional smooth bounded simply-connected domains. Our solutions adhere to Cullen's Stability Principle in that the geopotential is guaranteed to be a convex map for all times of its existence. We achieve our results by appealing to the theory of so-called div-curl systems (or Hodge systems), making use of recent results of Wang, which yield useful estimates on the ageostrophic velocity field. To our knowledge, this work constitutes the first time that any notion of bounded solution of the semi-geostrophic equations in Eulerian co-ordinates has been constructed on a bounded domain. Indeed, our work solves an open problem as highlighted by, among others, A. Figalli in his CIME lectures on the semi-geostrophic equations. Our methods are largely elementary. We discuss the application of the novel ideas in this work to the case of variable Coriolis force in the final section of the article.