Norm inflation for the derivative nonlinear Schrödinger equation

Yuzhao Wang; Younes Zine

doi:10.5802/crmath.566

Article de recherche - Équations aux dérivées partielles

Norm inflation for the derivative nonlinear Schrödinger equation
[Inflation de la norme pour l’équation de Schrödinger non linéaire dérivée]

Yuzhao Wang ¹ ; Younes Zine ^2,³

¹ School of Mathematics, University of Birmingham, Watson Building, Edgbaston, Birmingham, B15 2TT, United Kingdom
² School 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
³ École Polytechnique Fédérale de Lausanne, 1015 Lausanne Switzerland

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1857-1871.

Résumé
Abstract

Dans cette note, nous étudions le caractère mal posé de l’équation de Schrödinger avec perte de dérivée dans la non-linéarité (DNLS) en une dimension d’espace. Plus précisément, en utilisant un développement ternaire-quinaire de la formule de Duhamel, nous prouvons l’inflation de la norme des solutions dans les espaces de Sobolev en dessous de la régularité critique pour l’équation DNLS gaugée. Ce résultat est optimal puisque l’équation DNLS est connue pour être globalement en régularité positive [16]. La principale nouveauté de notre approche est de contrôler la perte de dérivée de la non-linéarité cubique par la non-linéarité quintique avec des données initiales soigneusement choisies.

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 $L^{2} (ℝ)$ [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.

Reçu le : 2022-07-21
Révisé le : 2023-03-02
Accepté le : 2023-09-05
Publié le : 2024-12-03

DOI : 10.5802/crmath.566

Classification : 35Q55, 35R25

Affiliations des auteurs :

Yuzhao Wang ¹ ; Younes Zine ^{2, 3}

¹ School of Mathematics, University of Birmingham, Watson Building, Edgbaston, Birmingham, B15 2TT, United Kingdom
² School 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
³ École Polytechnique Fédérale de Lausanne, 1015 Lausanne Switzerland

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2024__362_G13_1857_0,
     author = {Yuzhao Wang and Younes Zine},
     title = {Norm inflation for the derivative nonlinear {Schr\"odinger} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1857--1871},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.566},
     language = {en},
}

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Yuzhao Wang; Younes Zine. Norm inflation for the derivative nonlinear Schrödinger equation. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1857-1871. doi : 10.5802/crmath.566. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.566/

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