Global bv solutions and relaxation limit for a system of conservation laws

We consider the Cauchy problem for the (strictly hyperbolic, genuinely nonlinear) system of conservation laws with relaxation ut - vx = 0, vt - s(u)x = 1/h r(u,v). Assume there exists an equilibrium curve A(u), such that r(u,A(u))=0. Under some assumptions on s and r, we prove the existence of global (in time) solutions of bounded variation, uh, vh, for h > 0 fixed. As h -> 0, we prove the convergence of a subsequence of uh, vh to some u, v that satisfy the equilibrium equations ut - A(u)x = 0, v(t, · ) = A(u(t, · )) \forall t \geq 0.