Hyperbolic limit of the Jin-Xin relaxation model

We consider the special Jin-Xin relaxation model (0.1) u 1 + A(u)u x = ε (u xx - u tt). We assume that the initial data (u 0, εu 0,t) are sufficiently smooth and close to (ū, 0) in L ∞ and have small total variation. Then we prove that there exists a solution (u ε (t), εu t ε(t)) with uniformly small total variation for all t ≥ 0, and this solution depends Lipschitz-continuously in the L 1 norm with respect to time and the initial data. Letting ε → 0, the solution u ε converges to a unique limit, providing a relaxation limit solution to the quasi-linear, nonconservative system (0.2) u t + A(u)u x= 0. These limit solutions generate a Lipschitz semigroup S on a domain D containing the functions with small total variation and close to ū. This is precisely the Riemann semigroup determined by the unique Riemann solver compatible with (0.1).