The vanishing viscosity limit for Hamilton-Jacobi equations on Networks

TL;DR

This paper studies the vanishing viscosity approximation for Hamilton-Jacobi equations on networks, proving convergence of the elliptic solutions to the original problem's unique solution as viscosity approaches zero.

Contribution

It introduces a vanishing viscosity method for Hamilton-Jacobi equations on networks and establishes convergence to the unique solution, including the handling of Kirchhoff-type conditions.

Findings

Existence and uniqueness of the elliptic approximation solution.

Convergence of the approximation to the original solution as viscosity vanishes.

Handling of Kirchhoff-type conditions at network vertices.

Abstract

For a Hamilton-Jacobi equation defined on a network, we introduce its vanishing viscosity approximation. The elliptic equation is given on the edges and coupled with Kirchhoff-type conditions at the transition vertices. We prove that there exists exactly one solution of this elliptic approximation and mainly that, as the viscosity vanishes, it converges to the unique solution of the original problem.