# 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

, Volume 226 (1) – Jun 7, 2017
53 pages

/lp/springer_journal/on-the-structure-of-l-infty-l-entropy-solutions-to-scalar-conservation-p97IAswh7G
Publisher
Springer 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

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