Preprint 2002-031

Stability of Multidimensional Undercompressive Shock Waves

Jean-Francois Coulombel

Abstract: This paper is devoted to the study of linear and nonlinear stability of undercompressive shock waves for first order systems of hyperbolic conservation laws in a multidimensional space. We first recall the framework proposed by Freist\"uhler to extend Majda's work on classical shock waves to undercompressive shock waves. Then we show how the so-called uniform stability condition yields a linear stability result in terms of a maximal $L^2$ estimate. We follow Majda's strategy on shock waves with several improvements and modifications inspired from M\'etivier's work. The linearized problems are solved by duality and the nonlinear equations by means of a Newton type iteration scheme. Finally, we show how this work applies to phase transitions in an isothermal van der Waals fluid.

Available as PostScript (328 Kbytes) or gzipped PostScript (128 Kbytes; uncompress using gunzip).
Jean-Francois Coulombel, <>
Publishing information:
Submitted by:
<> June 10 2002.

[ 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 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 <>
Last modified: Tue Jun 25 10:54:00 MEST 2002