Solutions of the Fully Compressible Semi-Geostrophic System

TL;DR

This paper rigorously proves the existence of weak solutions for the fully compressible semi-geostrophic system in both physical and geostrophic coordinates, using optimal transport theory and Wasserstein space analysis.

Contribution

It provides the first rigorous proof of weak Lagrangian solutions in physical coordinates and offers an alternative proof in dual coordinates, advancing mathematical understanding of atmospheric flow models.

Findings

Existence of weak Lagrangian solutions in physical coordinates

Alternative proof of weak solutions in geostrophic coordinates

Application of optimal transport and Wasserstein space methods

Abstract

The fully compressible semi-geostrophic system is widely used in the modelling of large-scale atmospheric flows. In this paper, we prove rigorously the existence of weak Lagrangian solutions of this system, formulated in the original physical coordinates. In addition, we provide an alternative proof of the earlier result on the existence of weak solutions of this system expressed in the so-called geostrophic, or dual, coordinates. The proofs are based on the optimal transport formulation of the problem and on recent general results concerning transport problems posed in the Wasserstein space of probability measures.