Konstantin Khanin: Lagrangian dynamics on shocks manifolds and the optimal transport problem

Viscosity solutions of the Burgers equation and of the more general Hamilton-Jacobi equation are closely related to the dynamical properties of the minimizers for the corresponding Lagrangian action. However, most of the characteristics are merging with shocks. In this talk we shall discuss how the dynamics can be naturally defined after such a merger.

In the one- dimensional case the problem is simple since the shocks are isolated points. On the contrary, in the multi-dimensional case the shocks form submanifolds of finite codimension, which allows for a rather non-trivial dynamics. Although the velocity field has jump discontinuities on shocks, one can still determine, essentially in a unique way, the effective velocity field on the shock manifold. The effective dynamics has interesting connection with the optimal transport problem.