Nonlinear asymptotic stability in $L^\infty $ for Lipschitz solutions to scalar conservation laws
In this note, we show nonlinear stability in $L^\infty $ 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^\infty $ norm of perturbations of global-in-time Lipschitz solutions, includi...
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Académie des sciences
2024-07-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.553/ |
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| author | Golding, William |
| author_facet | Golding, William |
| author_sort | Golding, William |
| collection | DOAJ |
| description | In this note, we show nonlinear stability in $L^\infty $ 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^\infty $ 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, $\mathbf{72}$ (6): 1321-1348, 2019]. |
| format | Article |
| id | doaj-art-647fca2b4cd74515a15396e6447efe48 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-07-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-647fca2b4cd74515a15396e6447efe482025-02-07T11:21:51ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-07-01362G658159210.5802/crmath.55310.5802/crmath.553Nonlinear asymptotic stability in $L^\infty $ for Lipschitz solutions to scalar conservation lawsGolding, William0Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USAIn this note, we show nonlinear stability in $L^\infty $ 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^\infty $ 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, $\mathbf{72}$ (6): 1321-1348, 2019].https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.553/ |
| spellingShingle | Golding, William Nonlinear asymptotic stability in $L^\infty $ for Lipschitz solutions to scalar conservation laws Comptes Rendus. Mathématique |
| title | Nonlinear asymptotic stability in $L^\infty $ for Lipschitz solutions to scalar conservation laws |
| title_full | Nonlinear asymptotic stability in $L^\infty $ for Lipschitz solutions to scalar conservation laws |
| title_fullStr | Nonlinear asymptotic stability in $L^\infty $ for Lipschitz solutions to scalar conservation laws |
| title_full_unstemmed | Nonlinear asymptotic stability in $L^\infty $ for Lipschitz solutions to scalar conservation laws |
| title_short | Nonlinear asymptotic stability in $L^\infty $ for Lipschitz solutions to scalar conservation laws |
| title_sort | nonlinear asymptotic stability in l infty for lipschitz solutions to scalar conservation laws |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.553/ |
| work_keys_str_mv | AT goldingwilliam nonlinearasymptoticstabilityinlinftyforlipschitzsolutionstoscalarconservationlaws |