Semi-Discrete Optimal Transport Techniques for the Compressible Semi-Geostrophic Equations

We solve the compressible semi-geostrophic (SG) equations, a system of equations modelling large-scale atmospheric flows, by discretising the equations in space using semi-discrete optimal transport techniques. We show that, given suitable initial data, the discretised equations admit a unique, twice continuously differentiable, energy-conserving and global-in-time solution. We then show, by an appropriate limiting procedure, that for any compactly supported initial measure there exist global-in-time solutions of the compressible SG equations that are Lipschitz in time. This significantly generalises the original results due to Cullen and Maroofi, and it sets the theoretical foundation for solving the compressible SG equations numerically.

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