Optimal transport, Euler equations, Mather and DiPerna-Lions theories

Introduction
The aim of this note is to present a part of the research I have done during and after my Phd.
The central argument of my research concerns the optimal transport problem and its applications,
but I also worked on other subjects. Some of them, which I will describe here, are: the
study of variational models for the incompressible Euler equations, Mather’s theory, and some
generalization of the Diperna-Lions theory for ODEs with non-smooth vector fields. The note
is therefore structured in four independent parts.
In the first part, I will introduce the optimal transport problem, starting with some preliminaries.
In Sections 1.2 and 1.3 I will describe some recent results, which I studied in
[8, 10, 11, 17, 22, 24], concerning existence, uniqueness and properties of optimal transport
maps in a Riemannian and sub-Riemannian setting.
I will then focus on an important problem in this area, which consists in studying the
regularity of the optimal transport map. This is something I studied in [18, 25, 23]. In Section
1.4 I will state some of the obtained results. We will see in particular that there are some
unexpected connections between regularity properties of the transport map on Riemannian
manifolds, and the geometric structure of the manifold. As an example, as I showed with
Rifford in [23], studying the regularity of the optimal transport one can prove as a corollary a
convexity result on the cut-locus of the manifold.
We will then see some applications of the optimal transport, showing how one can apply
it to prove some refined version of functional inequalities: in Section 1.5 we will see that the
optimal transport allows to prove a sharpened isoperimetric inequality in R n , a result I did in
[19] with Maggi and Pratelli. Moreover, always using the optimal transport, me and Ge were
recently able to prove isoperimetric-type inequalities on manifolds with constant curvature [16].
Finally in Section 1.6 I will show a variant of the optimal transport that I studied in [15],
and I called the “optimal partial transport problem”.
The second part concerns some variational methods introduced by Brenier for the study
of the incopressible Euler equations. These methods are based on a relaxation of Arnold’s
problem, which consists in looking at the Euler equations as geodesics in the space of volume
preserving diffeomorphism. After introducing the models, in Section 2.2 I will describe some
of the results obtained with Bernot and Santambrogio in [7], where we studied some particular
generalized solutions in two dimensions. Then Section 2.3 is focused on giving sufficient and necessary
conditions for being a generalized solution, a problem investigated with Ambrosio in [3, 4].
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