The fully compressible semi-geostrophic system from meteorology

The semi-geostrophic equations are an approximation to the 3-D Euler equations for an atmosphere where the effects of rotation dominate. They are used by meteorologists to model the formation of fronts. Mathematically rigorous results were obtained by Benamou and Brenier and by Cullen and Gangbo in the incompressible case. In this paper we extend the results of Benamou and Brenier to the fully compressible case and show the existence of weak solutions to a reformulation of these equations in so-called dual variables. The Monge-Kantorovich theory appears as a material tool in our study.