Diffusive relaxation limit of the multi-dimensional hyperbolic Jin-Xin system

TL;DR

This paper investigates how the multi-dimensional Jin-Xin hyperbolic system approaches viscous conservation laws through a diffusive relaxation process, establishing global solutions and explicit convergence rates.

Contribution

It extends existing techniques to handle additional nonlinear terms and proves the global well-posedness and convergence rate in the multi-dimensional setting.

Findings

Proved global well-posedness of strong solutions.

Established uniform estimates with respect to the relaxation parameter.

Derived an explicit convergence rate for the relaxation limit.

Abstract

We study the diffusive relaxation limit of the Jin-Xin system toward viscous conservation laws in the multi-dimensional setting. For initial data being small perturbations of a constant state in suitable homogeneous Besov norms, we prove the global well-posedness of strong solutions satisfying uniform estimates with respect to the relaxation parameter. Then, we justify the strong relaxation limit and exhibit an explicit convergence rate of the process. Our proof is based on an adaptation of the techniques developed by Crin-Barat and Danchin to be able to deal with additional low-order nonlinear terms.