CAUCHY-PROBLEM FOR HYPERBOLIC CONSERVATION-LAWS WITH A RELAXATION TERM

This paper considers the Cauchy problem for hyperbolic conservation la ws arising in chromatography: (u + v)(t) + f(u)(x) = 0, v(t) = A(u) - v/g(delta, u, v)' with bounded measurable initial data, where the rela xation term g(delta, u, rr) converges to zero as the parameter delta > 0 tends to zero. We show that a solution of the equilibrium equation (u + A(u))(t) + f(u)(x) = 0 is given by the limit of the solutions of the viscous approximation (u + v)(t) + f(u)(x) = epsilon(u + v)(xx), v (t) = epsilon v(xx) + A(u) - v/g(delta, u, v)' of the original system as the dissipation epsilon and the relaxation delta go to zero related by delta = O(epsilon). The proof of convergence is obtained by a simp lified method of compensated compactness [2], avoiding Young measures by using the weak continuity theorem (3.3) of two by two determinants.