Hyperbolic Limit of the Jin-xin Relaxation Model

We consider the special Jin-Xin relaxation model (0.1) ut + A(u)ux = 2(uxx − utt), We assume that the initial data (u0, 2u0,t) are sufficiently smooth and close to (ū, 0) in L∞ and have small total variation. Then we prove that there exists a solution (u2(t), 2ut(t)) with uniformly small total variation for all t ≥ 0, and this solution depends Lipschitz continuously in the L1 norm w.r.t. the initial data and time. We then take the limit 2 → 0, and show that u2(t) tends to a unique Lipschitz continuous semigroup S on a domain D containing the functions with small total variation and close to ū. The semigroup S defines a semigroup of relaxation limiting solutions to the quasilinear non conservative system (0.2) ut + A(u)ux = 0. Moreover this semigroup coincides with the trajectory of a Riemann Semigroup, which is determined by the unique Riemann solver compatible with (0.1).