A Glimm type functional for a special Jin-Xin relaxation model

We consider a special case of the Jin-Xin relaxation systems u(t) + v(x) = 0, v(t) + lambda (2)u(x) = (F(u) - v)/epsilon. (*) We assume that the integral curves of the eigenvectors r(i) of DF(u) are straight lines. In this setting we prove that fur every initial data ii, v with sufficiently small total variation the solution (u(epsilon), v(epsilon)) of (*) is well defined for all t > 0, and its total variation satisfies a uniform bound, independent of t, epsilon. Moreover, as epsilon tends to 0(+), the solutions (u(epsilon), v(epsilon)) converge to a unique limit (u(t), v(t)): u(t) is thr: unique entropic solution of the corresponding hyperbolic system u(t) + F(u)(x) = 0 and v(t, x) = F (u(t, x)) for all t > 0, a.e. x is an element of R. The proofs rely on the introduction of a new functional for the solutions of (*), corresponding to the Glimm interaction potential for the approaching waves of different families. (C) 2001 Editions scientifiques et medicales Elsevier SAS. AMS classification: 35L