Selection problems of Z^2-periodic entropy solutions and viscosity solutions

TL;DR

This paper investigates the selection criteria for Z^2-periodic entropy and viscosity solutions of hyperbolic scalar conservation laws and Hamilton-Jacobi equations, comparing finite difference approximation with vanishing viscosity methods.

Contribution

It introduces a new selection criterion for periodic solutions in finite difference approximations, highlighting differences from the vanishing viscosity approach.

Findings

Finite difference approximation yields a different selection criterion.

The new criterion distinguishes between hyperbolic scaling and viscosity methods.

Differences in solution characteristics are demonstrated between the two approximation techniques.

Abstract

Z^2-periodic entropy solutions of hyperbolic scalar conservation laws and Z^2-periodic viscosity solutions of Hamilton-Jacobi equations are not unique in general. However, uniqueness holds for viscous scalar conservation laws and viscous Hamilton-Jacobi equations. Ugo Bessi ('03) investigated the convergence of approximate Z^2-periodic solutions to an exact one in the process of the vanishing viscosity method, and characterized this physically natural Z^2-periodic solution with the aid of Aubry-Mather theory. In this paper, a similar problem is considered in the process of the finite difference approximation under hyperbolic scaling. We present a selection criterion different from the one in the vanishing viscosity method, which exhibits difference in characteristics between the two approximation techniques.