On the Structure of $${L^\infty}$$ L ∞ -Entropy Solutions to Scalar Conservation Laws in One-Space Dimension

On the Structure of $${L^\infty}$$ L ∞ -Entropy Solutions to Scalar Conservation Laws in... We prove that if u is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. In particular, the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a C 0-sense up to the degeneracy due to the segments where $${f''=0}$$ f ′ ′ = 0 . We prove also that the initial data is taken in a suitably strong sense and we give some examples which show that these results are sharp. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Rational Mechanics and Analysis Springer Journals

On the Structure of $${L^\infty}$$ L ∞ -Entropy Solutions to Scalar Conservation Laws in One-Space Dimension

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Physics; Classical Mechanics; Physics, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Fluid- and Aerodynamics
ISSN
0003-9527
eISSN
1432-0673
D.O.I.
10.1007/s00205-017-1137-9
Publisher site
See Article on Publisher Site

Abstract

We prove that if u is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. In particular, the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a C 0-sense up to the degeneracy due to the segments where $${f''=0}$$ f ′ ′ = 0 . We prove also that the initial data is taken in a suitably strong sense and we give some examples which show that these results are sharp.

Journal

Archive for Rational Mechanics and AnalysisSpringer Journals

Published: Jun 7, 2017

References

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