Relaxation limit for piecewise smooth solutions to systems of conservation laws

In this paper we study the asymptotic equivalence of a general system of 1-D conservation laws and Ihc corresponding relaxation model proposed by S. Jin and Z. Xin ( 1995, Comm. Pure Appl. Math. 48, 235-277) in the limit of small relaxation rate. It is shown that if the relaxation system satisfies the subcharacteristic condition and the solution of the hyperbolic conservation laws is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition, then there exist solutions of the relaxation systems that converge to the solution of the original conservation laws ("equilibrium" system) at a rate of order epsilon as the rate of relaxation epsilon goes to zero. The proof uses a matched asymptotic analysis and an energy estimate related to the nonlinear stability theory for viscous shock. profiles. (C) 2000 Academic Press.