Porous media flow as the limit of a nonstrictly hyperbolic system of conservation laws

We show here that the weak solutions of the quasilinear hyperbolic system {epsilon u(t)(epsilon) + (epsilon(n + 1/2)(u(epsilon))(2) + f(upsilon(epsilon)))(x) = -u(epsilon) upsilon(t)(epsilon) + (u(epsilon)upsilon(epsilon))(x) = 0 converge, as epsilon tends to zero, to the solutions of the reduced problem (u + f(upsilon)(x) = 0 upsilon(t) + (u upsilon)(x) = 0, so that upsilon satisfies the nonlinear parabolic equation upsilon(t) - (f(upsilon)(x) upsilon)(x) = 0. The limiting procedure is carried out by using the theory of compensated compactness. Finally we obtain the existence of Lyapounov functionals for the limit parabolic equation as weak limit of the convex entropies as epsilon tends to zero for the corresponding hyperbolic system.