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 the 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 as the rate of relaxation goes to zero. The proof uses a matched asymptotic analysis and an energy estimate related to the nonlinear stability theory for viscous shock profiles