Preprint 2001-050

Vanishing Viscosity Solutions of Nonlinear Hyperbolic Systems

Stefano Bianchini and Alberto Bressan

Abstract: We consider the Cauchy problem for a strictly hyperbolic, n×n system in one space dimension: ut+A(u)ux=0 assuming that the initial data has small total variation.

We show that the solutions of the viscous approximations ut+A(u)ux=epsilonˇuxx are defined globally in time and satisfy uniform BV estimates, independent of epsilon. Moreover, they depend continuously on the initial data in the L1 distance, with a Lipschitz constant independent of tˇepsilon. Letting epsilon->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:Rn->Rn, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws ut+A(u)ux=0.

Available as PostScript (1.3 Mbytes) or gzipped PostScript (415 Kbytes; uncompress using gunzip).
Stefano Bianchini, <>
Alberto Bressan, <>
Publishing information:
To appear in Annals of Mathematics.
Submitted by:
<> November 30 2001. Updated version submitted April 1 2003. New update submitted October 24 2003.

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