On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations

TL;DR

This paper explores how particle trajectories evolve in Hamilton-Jacobi equations under regularizations like viscosity and noise, especially when shocks form, providing a unified framework for understanding dynamics in nonsmooth solutions.

Contribution

It introduces a regularized flow extending trajectories inside shocks, defining an effective velocity field for convex Hamiltonians, and compares viscous and noise-based regularizations.

Findings

Viscous regularization yields a unique, discontinuous effective velocity field.

Weak noise limit results in a non-unique effective velocity field.

Both regularizations satisfy a self-consistency condition for particle dynamics.

Abstract

Characteristic curves of a Hamilton-Jacobi equation can be seen as action minimizing trajectories of fluid particles. However this description is valid only for smooth solutions. For nonsmooth "viscosity" solutions, which give rise to discontinuous velocity fields, this picture holds only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we discuss two physically meaningful regularisation procedures, one corresponding to vanishing viscosity and another to weak noise limit. We show that for any convex Hamiltonian, a viscous regularization allows to construct a nonsmooth flow that extends particle trajectories and determines dynamics inside the shock manifolds. This flow consists of integral curves of a particular "effective" velocity field, which is uniquely defined everywhere in the flow domain and is discontinuous on shock manifolds. The effective velocity field arising in the weak noise limit is generally non-unique and different from the viscous one, but in both cases there is a fundamental self-consistency condition constraining the dynamics.