Mathematical analysis of partial differential equations in the semi-geostrophic theory

The semi-geostrophic equations constitute a mathematical model used to study the evolution of rotation-dominated atmospheric and oceanic flows on large horizontal scales. Although it was introduced in the late 1950s, the system attracted the attention of the mathematical community only in the late 1990s, when its connection with optimal transport theory was discovered. In this thesis, we present an overview of the main formulations and the main results in the literature of the analysis of the semi-geostrophic equations. It follows an account of the author's contribution to this theory. Firstly, we discuss the analysis of the surface semi-geostrophic equations, which model a semi-geostrophic flow in regime of constant potential vorticity. The system consists of an active scalar equation, whose activity is determined by way of a Neumann-to-Dirichlet map associated to a fully nonlinear second-order Neumann boundary value problem on the infinite strip R 2 × (0, 1). We present the results on the local-in-time existence and uniqueness of classical solutions of this active scalar equation in Holder spaces. Secondly, we consider a class of steady solutions of the semi-geostrophic equations on the whole space and derive the linearised dynamics around such solutions. The linearised equation consists of a transport equation featuring a pseudo-differential operator of order 0. We study well-posedness of this equation in L 2 (R 3 , R 3 ) introducing a representation formula for the solutions, and extend the result to the space of tempered distributions on R 3 . We investigate stability of the steady solutions of the semi-geostrophic equations by looking at plane-wave solutions of the associ ated linearised problem, and discuss differences in the case of the quasi-geostrophic equations. We conclude with an overview of the main open questions that arise from the theory presented in this thesis.