Lagrangian Solutions of Semigeostrophic Equations in Physical Space

The semigeostrophic equations are a simple model of large-scale atmosphere/ocean flows, where ’large-scale’ is defined to mean that the flow is rotation-dominated, [4]. They are also accurate in the case where one horizontal scale becomes small, allowing them to describe weather fronts and jet streams. Previous work by J.-D. Benamou and Y. Brenier, [2], and Cullen and Gangbo, [5], and Cullen and Maroofi, [6] proves that the semigeostrophic equations can be solved in the case respectively of 3-dimensional incompressible flow between rigid boundaries, vertically-averaged 3-d incompressible flow with a free surface, and fully compressible flow. However, all these results only prove the existence of weak solutions in ’dual’ variables, where the dual variables result from the change of variables introduced by [11]. 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 the results of [2] and [5] 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 [1] on transport equations and ODE for BV vector fields. It is not clear whether it is possible to prove existence of a weak solution to the Eulerian form of the equations. The transport theory of [1] gives uniqueness. However, the analysis of [2], [5], and [6] does not give uniqueness, because of the dependence of the transport velocity on the transported quantity. This question remains open. The remaining part of the paper is organized as following: in Section 2 we consider 3-dimensional incompressible semi-geostrophic system in a domain with rigid boundary, and in Section 3 we consider the semi-geostrophic shallow water model.