Vanishing Viscosity Solutions of Nonlinear Hyperbolic Systems

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 $\text{ve}$. Moreover, they depend continuously on the initial data in the ${\text{L}}^1$ distance, with a Lipschitz constant independent of $t,\text{ve}$. Letting $\text{ve}\to 0$, 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 $f:{\mathbb{R}}^n\mapsto {\mathbb{R}}^n$, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws $u_t+f(u{)}_x=0$.