  We consider the Cauchy problem for a strictly hyperbolic, $n\times n$ system
in one space dimension: $u_t+A(u)u_x=0$, assuming that the initial data has
small total variation.
  We show that the solutions of the viscous approximations $u_t+A(u)u_x=\ve
u_{xx}$ are defined globally in time and satisfy uniform BV estimates,
independent of $\ve$. Moreover, they depend continuously on the initial data in
the $\L^1$ distance, with a Lipschitz constant independent of $t,\ve$. Letting
$\ve\to 0$, 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 $f:\R^n\mapsto\R^n$, the vanishing
viscosity limits are precisely the unique entropy weak solutions to the system
of conservation laws $u_t+f(u)_x=0$.
