Abstract:We study the Cauchy problem for a strictly hyperbolicn×nsystem of conservation laws in one space dimension

u_{t}+f(u)_{x}= 0,

u(0,x)=u_{0}(x).The initial data

u_{0}is a small BV perturbation of a single rarefaction wave with an arbitrary strength. All characteristic fields are assumed to be genuinely nonlinear or linearly degenerate in the vicinity of the reference rarefaction curve. We prove that a suitable BV stability condition yields uniform bounds on the total variation of perturbation, thus implying the existence of a global admissible solution. On the other hand, a strongerL^{1}stability condition guarantees the existence of the Lipschitz continuous flow of solutions. Our proof relies on the construction of a Lyapunov functional which is almost decreasing in time and which is equivalent to theL^{1}distance between the two solutions.

**Paper:**- Available as PostScript (408 Kbytes) or gzipped PostScript (152 Kbytes; uncompress using gunzip).
**Author(s):**- Marta Lewicka, <lewicka@math.uchicago.edu>
**Publishing information:****Comments:**- Revised version submitted February 20 2004.
**Submitted by:**- <lewicka@math.uchicago.edu> May 9 2003.

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Conservation Laws Preprint Server <conservation@math.ntnu.no> 2004-02-21 19:56:01 UTC