Time--dependent equations on networks

TL;DR

This paper investigates the well-posedness of time-dependent Hamilton-Jacobi equations on networks, introducing a novel comparison approach linked to semidiscrete problems that simplifies analysis and avoids complex test functions.

Contribution

It presents a new method for proving comparison results for Hamilton-Jacobi equations on networks, bypassing traditional doubling variable techniques.

Findings

Existence and uniqueness of solutions established.

Stability properties demonstrated.

Comparison results linked to semidiscrete problems.

Abstract

We study well posedness of time--dependent Hamilton--Jacobi equations on a network, coupled with a continuous initial datum and a flux limiter. We show existence and uniqueness of solutions as well as stability properties. The novelty of our approach is that comparison results are proved linking the equation to a suitable semidiscrete problem, bypassing doubling variable method. Further, we do not need special test functions, and perform tests relative to the equations on different arcs separately.