Descrizione

In this book recent developments in the theory of optimal transportation and some of its applications to fluid dynamics are described. New variants of the original problem are explored and some common (and sometimes unexpected) features in this emerging variety of problems are figured out. In Chapter 1 the optimal transportation problem on manifolds with geometric costs coming from Tonelli Lagrangians is studied, while in Chapter 2 a generalization of the classical transportation problem called the optimal irrigation problem is considered. Then, Chapter 3 is about the Brenier variational theory of incompressible flows, which concerns a weak formulation of the Euler equations viewed as a geodesic equation in the space of measure-preserving diffeomorphism. Chapter 4 is devoted to the study of regularity and uniqueness of solutions of Hamilton-Jacobi equations applying the Aubry-Mather theory. Finally, the last chapter deals with a DiPerna-Lions theory for martingale solutions of stochastic differential equations.