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Entropy criteria and stability of extreme shocks: a remark on a paper of Leger and Vasseur


Authors: Benjamin Texier and Kevin Zumbrun
Journal: Proc. Amer. Math. Soc. 143 (2015), 749-754
MSC (2010): Primary 35L65, 35L67, 35B35
DOI: https://doi.org/10.1090/S0002-9939-2014-12426-9
Published electronically: October 10, 2014
MathSciNet review: 3283661
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Abstract: We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $ L^2$ dependence on initial data of Lax $ 1$- or $ n$-shock solutions of an $ n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freistühler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $ BV$ or $ H^s$ perturbations.


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Additional Information

Benjamin Texier
Affiliation: Université Paris-Diderot, Institut de Mathématiques de Jussieu - Paris Rive Gauche, UMR CNRS 7586, 75252 Paris Cedex 05, France–and–Ecole Normale Supérieure, Département de Mathématiques et Applications, UMR CNRS 8553, 75005 Paris, France
Email: texier@math.jussieu.fr

Kevin Zumbrun
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: kzumbrun@indiana.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12426-9
Received by editor(s): May 20, 2013
Published electronically: October 10, 2014
Additional Notes: The first author’s research was partially supported by the Project “Instabilities in Hydrodynamics” funded by the Mairie de Paris (under the “Emergences” program) and the Fondation Sciences Mathématiques de Paris.
The second author was partially supported under NSF grants no. DMS-0300487 and DMS-0801745
Communicated by: Walter Craig
Article copyright: © Copyright 2014 American Mathematical Society