Norm inflation for the derivative nonlinear Schrödinger equation
In this note, we study the ill-posedness problem for the derivative nonlinear Schrödinger equation (DNLS) in the one-dimensional setting. More precisely, by using a ternary-quinary tree expansion of the Duhamel formula we prove norm inflation in Sobolev spaces below the (scaling) critical regularity...
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| Language: | English |
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Académie des sciences
2024-12-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.566/ |
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| author | Wang, Yuzhao Zine, Younes |
| author_facet | Wang, Yuzhao Zine, Younes |
| author_sort | Wang, Yuzhao |
| collection | DOAJ |
| description | In this note, we study the ill-posedness problem for the derivative nonlinear Schrödinger equation (DNLS) in the one-dimensional setting. More precisely, by using a ternary-quinary tree expansion of the Duhamel formula we prove norm inflation in Sobolev spaces below the (scaling) critical regularity for the gauged DNLS. This ill-posedness result is sharp since DNLS is known to be globally well-posed in $L^2(\mathbb{R})$ [16]. The main novelty of our approach is to control the derivative loss from the cubic nonlinearity by the quintic nonlinearity with carefully chosen initial data. |
| format | Article |
| id | doaj-art-06dee7d1bb6146adbcc0d2f630c91991 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-06dee7d1bb6146adbcc0d2f630c919912025-02-07T11:27:00ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-12-01362G131857187110.5802/crmath.56610.5802/crmath.566Norm inflation for the derivative nonlinear Schrödinger equationWang, Yuzhao0Zine, Younes1School of Mathematics, University of Birmingham, Watson Building, Edgbaston, Birmingham, B15 2TT, United KingdomSchool of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom; École Polytechnique Fédérale de Lausanne, 1015 Lausanne SwitzerlandIn this note, we study the ill-posedness problem for the derivative nonlinear Schrödinger equation (DNLS) in the one-dimensional setting. More precisely, by using a ternary-quinary tree expansion of the Duhamel formula we prove norm inflation in Sobolev spaces below the (scaling) critical regularity for the gauged DNLS. This ill-posedness result is sharp since DNLS is known to be globally well-posed in $L^2(\mathbb{R})$ [16]. The main novelty of our approach is to control the derivative loss from the cubic nonlinearity by the quintic nonlinearity with carefully chosen initial data.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.566/ |
| spellingShingle | Wang, Yuzhao Zine, Younes Norm inflation for the derivative nonlinear Schrödinger equation Comptes Rendus. Mathématique |
| title | Norm inflation for the derivative nonlinear Schrödinger equation |
| title_full | Norm inflation for the derivative nonlinear Schrödinger equation |
| title_fullStr | Norm inflation for the derivative nonlinear Schrödinger equation |
| title_full_unstemmed | Norm inflation for the derivative nonlinear Schrödinger equation |
| title_short | Norm inflation for the derivative nonlinear Schrödinger equation |
| title_sort | norm inflation for the derivative nonlinear schrodinger equation |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.566/ |
| work_keys_str_mv | AT wangyuzhao norminflationforthederivativenonlinearschrodingerequation AT zineyounes norminflationforthederivativenonlinearschrodingerequation |