Abstract:We consider the special Jin–Xin relaxation model

u_{t}+A(u)u_{x}= ε(u_{xx}−u_{tt}).We assume that the initial data (

u_{0}, εu_{0,t}) are sufficiently smooth and close to (u, 0) inL^{∞}and have small total variation. Then we prove that there exists a solution (u^{ε(t)}, εu^{εt(t)}) with uniformly small total variation for allt≥ 0, and this solution depends Lipschitz continuously in theL^{1}norm w.r.t. the initial data and time.We then take the limit ε→0, and show that

u^{ε(t)}tends to a unique Lipschitz continuous semigroup \mathcal{S} on a domain \mathcal{D} containing the functions with small total variation and close tou. The semigroup \mathcal{S} defines a semigroup ofrelaxation limiting solutions} to the quasilinear non conservative systemMoreover this semigroup coincides with the trajectory of a Riemann Semigroup, which is determined by the unique Riemann solver compatible with \eqref{E:speJX01}.

u_{t}+A(u)u_{x}= 0.

**Paper:**- Available as PDF (400 Kbytes), Postscript (2 Mbytes) or gzipped PostScript (496 Kbytes; uncompress using gunzip).
**Author(s):**- Stefano Bianchini, <bianchini@iac.cnr.it>
**Publishing information:****Comments:****Submitted by:**- <bianchini@iac.cnr.it> November 5 2004.

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Conservation Laws Preprint Server <conservation@math.ntnu.no> 2004-11-05 14:14:52 UTC