Vanishing viscosity solutions of nonlinear hyperbolic systems

Abstract

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 have small total variation.

We show that the solutions of the viscous approximations $u_t+A(u)u_x=\varepsilon u_{xx}$ are defined globally in time and satisfy uniform BV estimates, independent of $\varepsilon$. Moreover, they depend continuously on the initial data in the ${\rm L}^1$ distance, with a Lipschitz constant independent of $t,\varepsilon$. Letting $\varepsilon\to 0$, 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 $u_t+f(u)_x=0$.

Authors

Stefano Bianchini

Functional Analysis and Applications
Scuola Internazionale Superiore di Studi Avanzati
Via Beirut 4
34014 Trieste
Italy

Alberto Bressan

Department of Mathematics
The Pennsylvania State University
University Park, PA 16082
United States