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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Solutions for a nonlocal conservation law with fading memory


Authors: Gui-Qiang Chen and Cleopatra Christoforou
Journal: Proc. Amer. Math. Soc. 135 (2007), 3905-3915
MSC (2000): Primary 35L65, 35L60, 35K40
Published electronically: September 7, 2007
MathSciNet review: 2341940
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Abstract: Global entropy solutions in $ BV$ for a scalar nonlocal conservation law with fading memory are constructed as the limits of vanishing viscosity approximate solutions. The uniqueness and stability of entropy solutions in $ BV$ are established, which also yield the existence of entropy solutions in $ L^\infty$ while the initial data is only in $ L^\infty$. Moreover, if the memory kernel depends on a relaxation parameter $ \de>0$ and tends to a delta measure weakly as measures when $ \de\to 0+$, then the global entropy solution sequence in $ BV$ converges to an admissible solution in $ BV$ for the corresponding local conservation law.


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Additional Information

Gui-Qiang Chen
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
Email: gqchen@math.northwestern.edu

Cleopatra Christoforou
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
Address at time of publication: Department of Mathematics, University of Houston, Texas 77204-3008
Email: cleo@math.northwestern.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08942-3
PII: S 0002-9939(07)08942-3
Keywords: Nonlocal conservation law, entropy solutions, vanishing viscosity, fading memory, existence, uniqueness, stability.
Received by editor(s): April 23, 2006
Received by editor(s) in revised form: September 26, 2006
Published electronically: September 7, 2007
Communicated by: Walter Craig
Article copyright: © Copyright 2007 American Mathematical Society