title: Particle dynamics inside shocks in Hamilton-Jacobi equations.
abstract: The characteristic curves of a Hamilton-Jacobi equation can be seen as action-minimizing trajectories of fluid particles. For non-smooth 'viscosity' solutions, which give rise to discontinuous velocity fields, this description is usually pursued only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that, for any convex Hamiltonian, there exists a uniquely defined canonical global non-smooth coalescing flow that extends particle trajectories and determines the dynamics inside shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss the relation to the 'dissipative anomaly' in the limit of vanishing viscosity.
authors.affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada. khanin@math.toronto.edu
authors.forename: Konstantin
authors.initials: K
authors.lastname: Khanin
authors.forename: Andrei
authors.initials: A
authors.lastname: Sobolevski