### On the Convergence Rate of Vanishing Viscosity Approximations

Alberto Bressan and Tong Yang

Abstract: Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $\big\|u(t,\cdot)-u^\ve(t,\cdot)\big\|_{\L^1}= \O(1)(1+t)\cdot \sqrt\ve|\ln\ve|$ 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 $u*\varphi_{\strut \sqrt\ve}$ 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.

Paper:
Available as PostScript (512 Kbytes) or gzipped PostScript (200 Kbytes; uncompress using gunzip).
Author(s):
Alberto Bressan, <bressan@sissa.it>
Tong Yang, <matyang@cityu.edu.hk>
Publishing information:
To appear in Comm. Pure Appl. Math.
Comments:
Submitted by:
<matyang@cityu.edu.hk> June 17 2003.

[ 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | All Preprints | Preprint Server Homepage ]
© The copyright for the following documents lies with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use.

Conservation Laws Preprint Server <conservation@math.ntnu.no>
2003-10-24 14:20:03 UTC