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Nonlinear asymptotic stability in L ∞ for Lipschitz solutions to scalar conservation laws
[Stabilité asymptotique non linéaire L ∞ pour les solutions lipschitziennes des lois de conservation scalaires]
William Golding1
1 Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 581-592.

Dans cette note, nous montrons la stabilité non linéaire L ∞ pour les solutions lipschitziennes des lois de conservation scalaires, multidimensionnelles et vraiment non linéaires. En tant qu’application, nous sommes capables de calculer des taux de décroissance algébriques explicites de la norme L ∞ des perturbations des solutions lipschitziennes globales en temps, y compris les perturbations des ondes de raréfaction planes. Notre analyse utilise la méthode de De Giorgi appliquée à la formulation cinétique et est une extension de la méthode récemment introduite par Silvestre dans [Comm. Pure Appl. Math, 72 (6) : 1321-1348, 2019].

In this note, we show nonlinear stability in L ∞ for Lipschitz solutions to genuinely nonlinear, multi-dimensional scalar conservation laws. As an application, we are able to compute algebraic decay rates of the L ∞ norm of perturbations of global-in-time Lipschitz solutions, including perturbations of planar rarefaction waves. Our analysis uses the De Giorgi method applied to the kinetic formulation and is an extension of the method introduced recently by Silvestre in [Comm. Pure Appl. Math, 72 (6): 1321-1348, 2019].

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DOI : 10.5802/crmath.553
Classification : 35B40, 35B35, 35B65, 35L65

William Golding 1

1 Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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William Golding. Nonlinear asymptotic stability in $L^\infty $ for Lipschitz solutions to scalar conservation laws. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 581-592. doi : 10.5802/crmath.553. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.553/

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