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

Abstract. We prove existence of weak solutions of the 3D compressible semi-geostrophic (SG) equations with compactly supported measure-valued initial data. These equations model large-scale atmospheric flows. Our proof uses a particle discretization 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, energy-conserving, and global-in-time solution, generated by particle trajectories that are twice continuously differentiable. In general, by discretizing 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 generalizes the original results due to Cullen and Maroofi [ Arch. Ration. Mech. Anal., 167 (2003), pp. 309–336], and it provides the theoretical foundation for the design of numerical schemes using semi-discrete optimal transport to solve the 3D compressible SG equations.