The Stability Principle and global weak solutions of the free surface semi-geostrophic equations in geostrophic coordinates

The semi-geostrophic equations are used widely in the modelling of large-scale atmospheric flows. In this note, we prove the global existence of weak solutions of the incompressible semi-geostrophic equations, in geostrophic coordinates, in a three-dimensional domain with a free upper boundary. The proof, based on an energy minimization argument originally inspired by the Stability Principle as studied by Cullen, Purser and others, uses optimal transport techniques as well as the analysis of Hamiltonian ODEs in spaces of probability measures as studied by Ambrosio and Gangbo. We also give a general formulation of the Stability Principle in a rigorous mathematical framework.