On Hamilton Jacobi equations with time measurable Hamiltonians posed on a 1-dimensional junction

TL;DR

This paper develops a framework for solving time-measurable Hamilton-Jacobi equations on a simple network with a junction, introducing flux-limited viscosity solutions and proving comparison and existence results.

Contribution

It introduces a novel notion of flux-limited viscosity solutions for time-measurable Hamiltonians on networks, extending the theory to discontinuous time dependencies.

Findings

Established a comparison principle for convex Hamiltonians.

Proved existence of solutions via optimal control construction.

Discussed extensions to nonconvex Hamiltonians and complex networks.

Abstract

In this paper, we study evolutive Hamilton Jacobi equations with Hamiltonians that are discontinuous in time, posed on a simple network consisting of two edges on the real line connected at a single junction. We introduce a notion of (flux-limited) viscosity solution for Hamiltonians H=H(t,x,p) that are assumed to be only measurable in time t. The flux limiter, A=A(t), acting at the junction, is not required to be continuous but only in L infinity. In the case of convex Hamiltonians, we prove a comparison principle and establish an existence result via the construction of an optimal control problem. Generalisations to the nonconvex case and to more general networks are also discussed.