Abstract:We consider the Cauchy problem for a strictly hyperbolic,n×nsystem 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}=epsilonˇu_{xx}are defined globally in time and satisfy uniform BV estimates, independent ofepsilon. Moreover, they depend continuously on the initial data in theL^{1}distance, with a Lipschitz constant independent oftˇepsilon. Lettingepsilon->0, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case whereA=Dfis the Jacobian of some flux functionf:R^{n}->R^{n}, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation lawsu_{t}+A(u)u_{x}=0.

**Paper:**- Available as PostScript (1.3 Mbytes) or gzipped PostScript (415 Kbytes; uncompress using gunzip).
**Author(s):**- Stefano Bianchini, <bianchin@iac.rm.cnr.it>
- Alberto Bressan, <bressan@sissa.it>
**Publishing information:**- To appear in Annals of Mathematics.
**Comments:****Submitted by:**- <bressan@sissa.it> November 30 2001. Updated version submitted April 1 2003. New update submitted October 24 2003.

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Conservation Laws Preprint Server <conservation@math.ntnu.no> 2003-10-24 14:17:20 UTC