On the Convergence Rate of Vanishing Viscosity Approximations

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $\big|u(t,\cdot)-u^\ve(t,\cdot)\big|{\L^1}= \O(1)(1+t)\cdot \sqrt\ve|\ln\ve|$onthedistancebetweenanexactBVsolution$u$andaviscousapproximation$u^\ve$,lettingtheviscositycoefficient$\ve\to 0$\mathrm{.}Intheproof,startingfrom$u$ we construct an approximation of the viscous solution $u^{\text{ve}}$ by taking a mollification $u*\phi{\strut \sqrt\ve}$ and inserting viscous shock profiles at the locations of finitely many large shocks, for each fixed $\text{ve}$. 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.