We will begin by reviewing classical results in the dynamics of Lagrangian flows, primarily within the framework of Aubry-Mather theory and weak KAM theory. From the perspective of transport, the regular Lagrangian flow determined by these theories establishes a connection between viscosity solutions of the Hamilton-Jacobi equation and optimal transport. Over the past decade, we and our collaborators have developed an intrinsic approach to study the singularities and their evolution in the Hamilton-Jacobi equation. In particular, the theory of Hamiltonian generalized gradient flows that we have developed in recent years has preliminarily established a transport theory for the corresponding potential functionals in the context of irregular Lagrangian flows. We have reason to believe that we can further explore transport problems related to entropy functionals and free energy functionals, as well as issues ranging from the structure of the cut locus in deterministic systems, the existence of invariant measures beyond Mather, to vanishing viscosity and zero-temperature limits. This is a relatively open field, and we will discuss both existing achievements and future expectations.
NB:This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006