Hyperbolic limit of the jin-xin relaxation model

We consider the special Jin-Xin relaxation model (0.1) ui+A(u)ux = ∈(uxx - utt). We assume that the initial data (u0,∈u0,t) are sufficiently smooth and close to (u,0) in L∞ and have small total variation. Then we prove that there exists a solution (u∈(t), ∈u∈t(t)) with uniformly small total variation for all t ≥ 0, and this solution depends Lipschitz-continuously in the L1 norm with respect to time and the initial data. Letting ∈ → 0, the solution u∈ converges to a unique limit, providing a relaxation limit solution to the quasi-linear, nonconservative system (0.2) ut + A(u)ux = 0. These limit solutions generate a Lipschitz semigroup S on a domain D containing the functions with small total variation and close to u. This is precisely the Riemann semigroup determined by the unique Riemann solver compatible with (0.1).