Vanishing Viscosity Solutions of Nonlinear Hyperbolic Systems

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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= uxx are defined globally in time and satisfy uniform BV estimates, independent of. Moreover, they depend continuously on the initial data in the Ł¹ distance, with a Lipschitz constant independent of t,. Letting → 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:ⁿⁿ, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws uₜ+f(u)ₓ=0.