Selection problems of Z^2-periodic entropy solutions and viscosity solutions
This work provides theoretical insight into solution selection for periodic PDEs, relevant to mathematicians studying conservation laws and Hamilton-Jacobi equations.
The paper addresses the non-uniqueness of Z^2-periodic entropy solutions for hyperbolic conservation laws and viscosity solutions for Hamilton-Jacobi equations, showing that the selection criterion for finite difference approximations differs from that of the vanishing viscosity method.
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.