On the validity of Chapman-Enskog expansions for shock waves with small strength

TL;DR

This paper validates the Chapman-Enskog expansion for shock waves with small strength by providing uniform estimates comparing relaxation model waves and diffusive equations.

Contribution

It rigorously justifies the Chapman-Enskog expansion for discontinuous shock solutions, establishing precise pointwise estimates.

Findings

Uniform estimates for wave differences

Validation of Chapman-Enskog expansion for shocks

Comparison between relaxation and diffusive models

Abstract

We justify a Chapman-Enskog expansion for discontinuous solutions of hyperbolic conservation laws containing shock waves with small strength. Precisely, we establish pointwise uniform estimates for the difference between the traveling waves of a relaxation model and the traveling waves of the corresponding diffusive equations determined by a Chapman-Enskog expansion procedure to first- or second-order.