L 2 -type contraction for systems of conservation laws

Denis Serre[1]; Alexis F. Vasseur[2]

  • [1] UMPA, ENS-Lyon 46 allée d’Italie, 69364 Lyon Cedex 07, France
  • [2] University of Texas at Austin 1 University Station C1200, Austin, TX 78712-0257, USA

Journal de l’École polytechnique — Mathématiques (2014)

  • Volume: 1, page 1-28
  • ISSN: 2270-518X

Abstract

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The semi-group associated with the Cauchy problem for a scalar conservation law is known to be a contraction in L 1 . However it is not a contraction in L p for any p > 1 . Leger showed in [20] that for a convex flux, it is however a contraction in L 2 up to a suitable shift. We investigate in this paper whether such a contraction may happen for systems. The method is based on the relative entropy method. Our general analysis leads us to the new geometrical notion of Genuinely non-Temple systems. We treat in details two examples: – the Keyfitz–Kranzer system with rotationally invariant flux, for which the L 2 contraction holds true, – the Euler system of gas dynamics, for which it does not.

How to cite

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Serre, Denis, and Vasseur, Alexis F.. "$L^2$-type contraction for systems of conservation laws." Journal de l’École polytechnique — Mathématiques 1 (2014): 1-28. <http://eudml.org/doc/275608>.

@article{Serre2014,
abstract = {The semi-group associated with the Cauchy problem for a scalar conservation law is known to be a contraction in $L^1$. However it is not a contraction in $L^p$ for any $p&gt;1$. Leger showed in [20] that for a convex flux, it is however a contraction in $L^2$ up to a suitable shift. We investigate in this paper whether such a contraction may happen for systems. The method is based on the relative entropy method. Our general analysis leads us to the new geometrical notion of Genuinely non-Temple systems. We treat in details two examples: – the Keyfitz–Kranzer system with rotationally invariant flux, for which the $L^2$ contraction holds true, – the Euler system of gas dynamics, for which it does not.},
affiliation = {UMPA, ENS-Lyon 46 allée d’Italie, 69364 Lyon Cedex 07, France; University of Texas at Austin 1 University Station C1200, Austin, TX 78712-0257, USA},
author = {Serre, Denis, Vasseur, Alexis F.},
journal = {Journal de l’École polytechnique — Mathématiques},
keywords = {Conservation laws; relative entropy; shock stability; Temple systems; functional regression; sparse recovery; LASSO; oracle inequality; infinite dictionaries},
language = {eng},
pages = {1-28},
publisher = {École polytechnique},
title = {$L^2$-type contraction for systems of conservation laws},
url = {http://eudml.org/doc/275608},
volume = {1},
year = {2014},
}

TY - JOUR
AU - Serre, Denis
AU - Vasseur, Alexis F.
TI - $L^2$-type contraction for systems of conservation laws
JO - Journal de l’École polytechnique — Mathématiques
PY - 2014
PB - École polytechnique
VL - 1
SP - 1
EP - 28
AB - The semi-group associated with the Cauchy problem for a scalar conservation law is known to be a contraction in $L^1$. However it is not a contraction in $L^p$ for any $p&gt;1$. Leger showed in [20] that for a convex flux, it is however a contraction in $L^2$ up to a suitable shift. We investigate in this paper whether such a contraction may happen for systems. The method is based on the relative entropy method. Our general analysis leads us to the new geometrical notion of Genuinely non-Temple systems. We treat in details two examples: – the Keyfitz–Kranzer system with rotationally invariant flux, for which the $L^2$ contraction holds true, – the Euler system of gas dynamics, for which it does not.
LA - eng
KW - Conservation laws; relative entropy; shock stability; Temple systems; functional regression; sparse recovery; LASSO; oracle inequality; infinite dictionaries
UR - http://eudml.org/doc/275608
ER -

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