Preprint 2003-032

Lyapunov functional for solutions of systems of conservation laws containing a strong rarefaction

Marta Lewicka

Abstract: We study the Cauchy problem for a strictly hyperbolic nn system of conservation laws in one space dimension

ut + f(u)x = 0,

The initial data u0 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 stronger L1 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 the L1 distance between the two solutions.

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Marta Lewicka, <>
Publishing information:
Revised version submitted February 20 2004.
Submitted by:
<> May 9 2003.

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