Zero Relaxation Limit for Piecewise Smooth Solutions to a Rate-Type Viscoelastic System in the Presence of Shocks

We study a rate-type viscoelastic system proposed in I. Suliciu Ž Int. J. Engng. Sci. 28Ž1990., 827 841., which is a 3 3 hyperbolic system with relaxation. As the relaxation time tends to zero, this system converges to the well-known p-system formally. In the case that the solutions of the p-system are piecewise smooth, including finitely many noninteracting shock waves, we show that there exist smooth solutions for Suliciu’s model which converge to those of the p-system strongly as the relaxation time goes to zero. The method used here is the so-called matched asymptotic analysis suggested in J. Goodman and Z. P. Xin Ž Arch. Ration. Mech. Anal. 121Ž1992., 235 265., which includes two parts: the matched asymptotic expansion and stability analysis