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