Lagrangian Solutions of Semigeostrophic Equations in Physical Space

The semigeostrophic equations are a simple model of large-scale atmosphere/ocean flows. Previous work by J.-D. Benamou and Y. Brenier, M. Cullen and W. Gangbo, and M. Cullen and H. Maroofi proves that the semigeostrophic equations can be solved in the cases, respectively, of 3-dimensional (3-d) incompressible flow between rigid boundaries, vertically averaged 3-d incompressible flow with a free surface, and fully compressible flow. However, all these results prove only the existence of weak solutions in "dual" variables, where the dual variables result from a change of variables introduced by Hoskins. This makes it difficult to relate the solutions to the full Euler or Navier--Stokes equations, or to those of other simple atmosphere/ocean models. We therefore seek to extend these results to prove existence of a solution in physical variables. We do this using the Lagrangian form of the equations in physical space. The proof is based on the recent results of L. Ambrosio on transport equations and ODE for B...