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On the Convergence Rate of Vanishing Viscosity Approximations

Alberto Bressan·Tong Yang
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Abstract

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $\big\|u(t,\cdot)-u^\ve(t,\cdot)\big\|_{Ł^1}= Ø(1)(1+t)\cdot \sqrt\ve|\ln\ve|$ on the distance between an exact BV solution $u$ and a viscous approximation $u^\ve$, letting the viscosity coefficient $\ve\to 0$. In the proof, starting from $u$ we construct an approximation of the viscous solution $u^\ve$ by taking a mollification $u*ϕ_{\strut \sqrt\ve}$ 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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Alberto Bressan, Tong Yang. 2003-07-10. On the Convergence Rate of Vanishing Viscosity Approximations. https://arxiv.org/abs/math/0307141

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