# ${L}^{2}$-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

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topSerre, 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 $p>1$. 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 $p>1$. 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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