Local existence of smooth solutions for the semigeostrophic equations on curved domains

TL;DR

This paper establishes local-in-time existence of smooth solutions to the semigeostrophic equations on curved, simply connected domains with arbitrary conformally flat metrics, using a novel Eulerian coordinate approach.

Contribution

It introduces a new Eulerian coordinate method for solving semigeostrophic equations on curved domains, extending previous flat domain results.

Findings

Proves local existence of smooth solutions in curved domains.

Develops a construction avoiding dual variables used in flat cases.

Handles general conformally flat metrics with non-zero Coriolis force.

Abstract

We prove local-in-time existence of smooth solutions to the semigeostrophic equations in the general setting of smooth, bounded and simply connected domains of $\mathbb{R}^2$ endowed with an arbitrary conformally flat metric and non-vanishing Coriolis term. We present a construction taking place in Eulerian coordinates, avoiding the classical reformulation in dual variables, used in the flat case with constant Coriolis force, but lacking in this general framework.