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Structure of entropy solutions for multi-dimensional scalar conservation laws


De Lellis, C; Otto, F; Westdickenberg, M (2003). Structure of entropy solutions for multi-dimensional scalar conservation laws. Archiv for Rational Mechanics and Analysis, 170(2):137-184.

Abstract

An entropy solution u of a multi-dimensional scalar conservation law is not necessarily in BV, even if the conservation law is genuinely nonlinear. We show that u nevertheless has the structure of a BV function in the sense that the shock location is codimension-one rectifiable. This result highlights the regularizing effect of genuine nonlinearity in a qualitative way; it is based on the locally finite rate of entropy dissipation. The proof relies on the geometric classification of blow-ups in the framework of the kinetic formulation.

An entropy solution u of a multi-dimensional scalar conservation law is not necessarily in BV, even if the conservation law is genuinely nonlinear. We show that u nevertheless has the structure of a BV function in the sense that the shock location is codimension-one rectifiable. This result highlights the regularizing effect of genuine nonlinearity in a qualitative way; it is based on the locally finite rate of entropy dissipation. The proof relies on the geometric classification of blow-ups in the framework of the kinetic formulation.

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30 citations in Web of Science®
27 citations in Scopus®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:finite entropy dissipation; geometric classification of blow-ups; entropy-entropy flux pair; BV functions; shock wave
Language:English
Date:2003
Deposited On:04 Nov 2009 11:00
Last Modified:05 Apr 2016 13:25
Publisher:Springer
ISSN:0003-9527
Additional Information:The original publication is available at www.springerlink.com
Publisher DOI:https://doi.org/10.1007/s00205-003-0270-9
Related URLs:http://www.ams.org/mathscinet-getitem?mr=2017887
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1036.35127

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