2003/07/10 by Alberto Bressan, Tong Yang, Bressan, Alberto +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP
paper · pdf · doi:10.48550/arxiv.math/0307141
34 pages, 5 figures
arxiv created 2003/07/10 · arxiv updated 2009/12/01
Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ‖u(t,⋅)-u^\ve(t,⋅)‖Ł1= Ø(1)(1+t)⋅ √\ve|ln\ve| on the distance between an exact BV solution u and a viscous approximation u^\ve, letting the viscosity coefficient \ve→ 0. In the proof, starting from u we construct an approximation of the viscous solution u^\ve by taking a mollification u*ϕ\strut √\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.