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Entropy criteria and stability of extreme shocks: a remark on a paper of Leger and Vasseur
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by Benjamin Texier and Kevin Zumbrun PDF
Proc. Amer. Math. Soc. 143 (2015), 749-754 Request permission

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
  • MR Author ID: 330192
  • Email: kzumbrun@indiana.edu
  • 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
  • © Copyright 2014 American Mathematical Society
  • 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
  • MathSciNet review: 3283661