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 C0-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.
On the Structure of L∞ -Entropy Solutions to Scalar Conservation Laws in One-Space Dimension
Marconi E.
2017
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 C0-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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