A Fully Nonlinear Version of the Incompressible Euler Equations: The Semigeostrophic System

The semi-geostrophic equations are used in meteorology. They appear as a variant of the two-dimensional Euler incompressible equations in vorticity form, where the Poisson equation that relates the stream function and the vorticity field is just replaced by the fully non linear elliptic Monge-Ampère equation. This work gathers new results concerning the semi-geostrophic equations: Existence and stability of measure valued solutions, existence and uniqueness of solutions under certain continuity conditions for the density, convergence to the incompressible Euler equations.