Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $big|u(t,cdot)-u^ve(t,cdot)big|_{L^1}= O(1)(1+t)cdot sqrtve|lnve|$ on the distance between an exact BV solution $u$ and a viscous approximation $u^ve$, letting the viscosity coefficient $veto 0$. In the proof, starting from $u$ we construct an approximation of the viscous solution $u^ve$ by taking a mollification $u*phi_{strut sqrtve}$ and inserting viscous shock profiles at the locations of finitely many large shocks, for each fixed $ve$. Error estimates are then obtained by introducing new Lyapunov functionals which control shock interactions, interactions between waves of different families and by using sharp decay estimates for positive nonlinear waves.
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