AbstractGiven a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ‖u(t, ·) −uϵ(t, ·)‖=O(1)(1 +t) ·$\sqrt{\varepsilon}$|ln ϵ| on the distance between an exact BV solutionuand a viscous approximationuϵ, letting the viscosity coefficient ϵ → 0. In the proof, starting fromuwe construct an approximation of the viscous solutionuϵby taking a mollificationu*$\varphi_{\sqrt{\varepsilon}}$and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed ϵ. Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. © 2004 Wiley Periodicals, Inc.