Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks

TL;DR

This paper develops a comprehensive theory for Hamilton-Jacobi equations on networks with quasi-convex, possibly discontinuous Hamiltonians, introducing a flux limiter concept and a vertex test function method to establish comparison principles, existence, and uniqueness.

Contribution

It introduces a novel flux limiter framework and a vertex test function method for Hamilton-Jacobi equations on networks with quasi-convex Hamiltonians, enabling general comparison, existence, and uniqueness results.

Findings

Equivalence of general and specific vertex conditions via flux limiter

Development of a vertex test function for comparison principles

Establishment of existence and uniqueness results for the equations

Abstract

We study Hamilton-Jacobi equations on networks in the case where Hamiltonians are quasi-convex with respect to the gradient variable and can be discontinuous with respect to the space variable at vertices. First, we prove that imposing a general vertex condition is equivalent to imposing a specific one which only depends on Hamiltonians and an additional free parameter, the flux limiter. Second, a general method for proving comparison principles is introduced. This method consists in constructing a vertex test function to be used in the doubling variable approach. With such a theory and such a method in hand, we present various applications, among which a very general existence and uniqueness result for quasi-convex Hamilton-Jacobi equations on networks.