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

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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 \({f''=0}\). 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.

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Authors and Affiliations

  1. SISSA, Via Bonomea 265, 34136, Trieste, Italy

    S. Bianchini & E. Marconi

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  1. S. Bianchini
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  2. E. Marconi
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Correspondence to S. Bianchini.

Additional information

Communicated by C. De Lellis

The authors thank the CMSA at Harvard where part of this work has been written. This research has been partially supported by MIUR PRIN Project No. 2012L5WXHJ.

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Bianchini, S., Marconi, E. On the Structure of \({L^\infty}\)-Entropy Solutions to Scalar Conservation Laws in One-Space Dimension. Arch Rational Mech Anal 226, 441–493 (2017). https://doi.org/10.1007/s00205-017-1137-9

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  • DOI: https://doi.org/10.1007/s00205-017-1137-9

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