Preprint

Inserted: 10 apr 2025
Last Updated: 10 apr 2025

Year: 2025

Abstract:

We prove existence of weak solutions of the 3D compressible semi-geostrophic (SG) equations with measure-valued initial data. These equations model large-scale atmospheric flows. Our proof uses a particle discretisation and semi-discrete optimal transport techniques. We show that, if the initial data is a discrete measure, then the compressible SG equations admit a unique, twice continuously differentiable, energy-conserving and global-in-time solution. In general, by discretising the initial measure by particles and sending the number of particles to infinity, we show that for any compactly supported initial measure there exists a global-in-time solution of the compressible SG equations that is Lipschitz in time. This significantly generalises the original results due to Cullen and Maroofi (2003), and it sets the theoretical foundation for solving the compressible SG equations numerically.

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