Vanishing viscosity solutions of nonlinear hyperbolic systems

We consider the Cauchy problem for a strictly hyperbolic, n × n system in one-space dimension: u t + A(u)u x = 0, assuming that the initial data have small total variation.We show that the solutions of the viscous approximations u t + A(u)u x = εu xx are defined globally in time and satisfy uniform BV estimates, independent of ε.Moreover, they depend continuously on the initial data in the L 1 distance, with a Lipschitz constant independent of t, ε.Letting ε → 0, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data.In the conservative case where A = Df is the Jacobian of some flux function f : R n → R n , the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws u t + f (u) x = 0.