Hyperbolic limit of the Jin‐Xin relaxation model

We consider the special Jin‐Xin relaxation model$(0.1) \;\;\;\;\;\;\;\;\;\; u_t+ A(u) u_x = \epsilon (u_{xx} - u_{tt}).$We assume that the initial data ( $u_0, \epsilon u_{0,t}$ ) are sufficiently smooth and close to ( $\bar{u},0$ ) in L ∞ and have small total variation. Then we prove that there exists a solution ( $u^\epsilon (t), \epsilon u^{\epsilon}_t (t)$ ) 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 $\epsilon \longrightarrow 0$ , the solution $u^\epsilon$ 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 $\cal{S}$ on a domain $\cal{D}$ containing the functions with small total variation and close to $\bar{u}$ . This is precisely the Riemann semigroup determined by the unique Riemann solver compatible with (0.1). © 2005 Wiley Periodicals, Inc.