On the vanishing viscosity approximation in the vectorial case

We present here some preliminary results in the study of the large time behavior of vanishing viscosity approximations for systems of conservation laws. In the first part we adapt the approximation technique used in [BY1] to obtain sharper bounds on the convergence rate of the viscous approximations, u(epsilon), in the case the solution u of the hyperbolic system is self similar and contains exactly one interaction between shocks of different families. Then, in the second part, we present a new proof of the result by D. Serre, [S], on existence and uniqueness of solutions of parabolic systems of conservation laws defined globally in time, for -infinity < t < +infinity.