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.
Conservation Laws Preprint Server <firstname.lastname@example.org> 2003-10-24 14:17:20 UTC