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