Preprint 2003-032

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

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

u_t + f(u)_x = 0,
u(0,x)=u₀(x).

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.

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.