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We study the asymptotic equivalence of the Jin-Xin relaxation model to its formal limit of genuinely nonlinear 2 by 2 conservation laws (isentropic Euler equation in Lagrangian coordinate). We consider the case where the initial data are allowed to have jump discontinuities such that the corresponding solutions to the Euler equation contain centered rarefaction waves. In particular, Riemann data connected by rarefaction curves are included. We show that, as long as the initial data is a small perturbation of a non-vacuum constant state, the solution for the relaxation system exists globally in time and converges, as e —> 0, to the solution of the corresponding Euler equation uniformly except for an initial layer whose width is essentially of order O(e).

Asymptotic towards rarefaction wave of the Jin–Xin relaxation model for the $p$ system