Preprint 1999-014

Periodic Solutions of Conservation Laws Constructed Through Glimm Scheme

Hermano Frid

Abstract: In this note we present a periodic version of an adaptation of the Glimm scheme introduced by Nishida (1968) and further extended by Bakhvalov (1970) and DiPerna (1973). For special classes of $2\X2$ systems of conservation laws these adaptions of the Glimm scheme give global existence of solutions of the Cauchy problem with large initial data in $L^\infty\cap BV_{loc}(\R)$, for Nishida-Bakhvalov's class, and in $L^\infty\cap BV(\R)$, in the case of DiPerna's class. For periodic initial data, our periodic formulation establishes that the periodic solutions so constructed, $u(\cdot ,t)$, are uniformly bounded in $L^\infty\cap BV([0,\ell])$, for all $t>0$, where $\ell$ is the period. We then obtain the asymptotic decay of these solutions by applying a theorem of Chen-Frid in \cite{CF} combined with a compactness theorem of DiPerna in \cite{Di83}. The question about the decay of Nishida's solution was proposed by Glimm-Lax \cite{GL} and remained open since then. The classes considered include systems motivated by isentropic gas dynamics.

Available as PostScript.
Hermano Frid, <>
Publishing information:
New version, submitted July 22.
Submitted by:
<> April 20 1999.

[ 1996 | 1997 | 1998 | 1999 | 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: Fri Jul 23 10:13:22 1999