Vanishing viscosity solutions of nonlinear hyperbolic systems

We consider the Cauchy problem for a strictly hyperbolic, n x 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) = epsilon u(xx) are defined globally in time and satisfy uniform BV estimates, independent of e. Moreover, they depend continuously on the initial data in the L-1 distance, with a Lipschitz constant independent of t, epsilon. Letting epsilon -> 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.