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

We consider the Cauchy problem for the (strictly hyperbolic, genuinely nonl inear) system of conservation laws with relaxation [GRAPHICS] Assume there exists an equilibrium curve A(u), such that r(u, A(u)) = 0. Un der some assumptions on sigma and r, we prove the existence of global tin t ime) solutions of bounded variation, u(epsilon), upsilon (epsilon), for a > 0 fixed. As epsilon --> 0, we prove the convergence of a subsequence of u(epsilon), upsilon (epsilon) to some u, upsilon that satisfy the equilibrium equations u(t) - A(u)(x) = 0, upsilon (t,.) = A(u(t,.)) For Allt greater than or equal to 0.