Conservation Laws with Coinciding Smooth Solutions but Different Conserved Variable

TL;DR

This paper investigates hyperbolic conservation laws with identical eigenstructure but different conserved variables, showing solutions with same initial data differ only at third order, and applies results to fluid dynamics and traffic models.

Contribution

It proves that solutions to such systems differ at third order in initial total variation, providing refined estimates and applications to Euler equations, scalar cases, and traffic models.

Findings

Solutions differ at third order in initial total variation.

Improved estimates for the distance between solutions of Euler equations.

Precise bounds for scalar conservation laws and traffic models.

Abstract

Consider two hyperbolic systems of conservation laws in one space dimension with the same eigenvalues and (right) eigenvectors. We prove that solutions to Cauchy problems with the same initial data differ at third order in the total variation of the initial datum. As a first application, relying on the classical Glimm-Lax result, we obtain estimates improving those in by Saint Raymond on the distance between solutions to the isentropic and non-isentropic inviscid compressible Euler equations, under general equations of state. Further applications are to the general scalar case, where rather precise estimates are obtained, to an approximation by Di Perna of the p-system and to a traffic model.