Particle dynamics inside shocks in Hamilton-Jacobi equations

TL;DR

This paper introduces a unique, global, nonsmooth flow for Hamilton-Jacobi equations that extends particle trajectories into shocks, providing a variational description of the dynamics within discontinuities.

Contribution

It establishes the existence and uniqueness of a canonical flow inside shocks for any convex Hamiltonian, extending classical particle trajectories beyond smooth regions.

Findings

Existence of a unique global nonsmooth flow inside shocks.

Variational characterization of the effective velocity field.

Discussion of relation to dissipative anomalies in vanishing viscosity limit.

Abstract

Characteristics of a Hamilton-Jacobi equation can be seen as action minimizing trajectories of fluid particles. For nonsmooth "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 nonsmooth coalescing flow that extends particle trajectories and determines dynamics inside the shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss relation to the "dissipative anomaly" in the limit of vanishing viscosity.