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On compactness estimates for hyperbolic systems of conservation laws

Nguyen, Khai T.; Glass, Olivier; Ancona, Fabio (2015), On compactness estimates for hyperbolic systems of conservation laws, Annales de l'Institut Henri Poincaré. Analyse non linéaire, 32, 6. http://dx.doi.org/10.1016/j.anihpc.2014.09.002

Type
Article accepté pour publication ou publié
External document link
http://fr.arxiv.org/pdf/1403.5070v1
Date
2015
Journal name
Annales de l'Institut Henri Poincaré. Analyse non linéaire
Volume
32
Number
6
Publisher
Elsevier
Publication identifier
http://dx.doi.org/10.1016/j.anihpc.2014.09.002
Metadata
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Author(s)
Nguyen, Khai T.

Glass, Olivier

Ancona, Fabio
Abstract (EN)
We study the compactness in $L^{1}_{loc}$ of the semigroup mapping $(S_t)_{t > 0}$ defining entropy weak solutions of general hyperbolic systems of conservation laws in one space dimension. We establish a lower estimate for the Kolmogorov $\varepsilon$-entropy of the image through the mapping $S_t$ of bounded sets in $L^{1}\cap L^\infty$, which is of the same order $1/\varepsilon$ as the ones established by the authors for scalar conservation laws. We also provide an upper estimate of order $1/\varepsilon$ for the Kolmogorov $\varepsilon$-entropy of such sets in the case of Temple systems with genuinely nonlinear characteristic families, that extends the same type of estimate derived by De Lellis and Golse for scalar conservation laws with convex flux. As suggested by Lax, these quantitative compactness estimates could provide a measure of the order of "resolution" of the numerical methods implemented for these equations.
Subjects / Keywords
Hyperbolic systems of conservation laws; Temple systems; Compactness estimates; Kolmogorov entropy

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