On the zero relaxation limit for a system modeling the motions of a viscoelastic solid

We consider a simple model of the motions of a viscoelastic solid. The model consists of a two-by-two system of conservation laws including a strong relaxation term. We establish the existence of a BV-solution of this system for any positive value of the relaxation parameter. We also show that this solution is stable with respect to the perturbations of the initial data in L₁. By deriving the uniform bounds, with respect to the relaxation parameter, on the total variation of the solution, we obtain the convergence of the solutions of the relaxation system towards the solutions of a scalar conservation law as the relaxation parameter δ goes to zero. Due to the Lip† bound on the solutions of the relaxation system, an estimate on the rate of convergence towards equilibrium is derived. In particular, an script O sign(√δ) bound on the L¹-error is established.