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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

$L_1$ stability for $2\times 2$ systems
of hyperbolic conservation laws


Authors: Tai-Ping Liu and Tong Yang
Journal: J. Amer. Math. Soc. 12 (1999), 729-774
MSC (1991): Primary 35L67, 76L05; Secondary 35L65, 35A05
Published electronically: April 13, 1999
MathSciNet review: 1646841
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the evolution of the $L_1$ distance of solutions for systems of $2\times 2$ hyperbolic conservation laws. For the approximate solutions constructed by Glimm's scheme with the aid of the wave tracing method, we introduce a nonlinear functional which is equivalent to the $L_1$ distance between solutions, nonincreasing in time, and expressed explicitly in terms of the wave patterns of the solutions. This functional reveals the nonlinear mechanism of wave interactions and coupling which affect the $L_1$ topology.


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Additional Information

Tai-Ping Liu
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305-2060
Email: liu@math.stanford.edu

Tong Yang
Affiliation: Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
Email: matyang@cityu.edu.hk

DOI: http://dx.doi.org/10.1090/S0894-0347-99-00292-1
PII: S 0894-0347(99)00292-1
Received by editor(s): March 8, 1998
Received by editor(s) in revised form: September 9, 1998
Published electronically: April 13, 1999
Additional Notes: The first author’s research was supported in part by NSF Grant DMS-9623025
The second author’s research was supported in part by the RGC Competitive Earmarked Research Grant 9040290
Article copyright: © Copyright 1999 American Mathematical Society