Unconditional well-posedness for subcritical NLS in $ {H}^{s}$

Keith M. Rogers

doi:10.1016/j.crma.2007.09.003

Partial Differential Equations

Unconditional well-posedness for subcritical NLS in

H^{s}

[Unicité inconditionnelle pour l'équation de Schrödinger non-linéaire sous-critique dans

H^{s}

]

Présenté par : Jean Bourgain

Keith M. Rogers ¹

¹ Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Comptes Rendus. Mathématique, Volume 345 (2007) no. 7, pp. 395-398.

Résumé
Abstract

On considère l'équation de Schrödinger linéaire sous-critique $i \partial_{t} u + Δ u = {| u |}^{α} u$ , sur $R^{n}$ , $n ⩾ 3$ , à donnée initiale $u_{0}$ dans $H^{s} (R^{n})$ . Si $s ⩾ 1$ , Kato a démontré que si il existe une solution maximale, elle est unique dans $C ([0, T_{\max}), H^{s})$ . Les seuls résultats d'unicité connus auparavant étaient dans des sous-espaces stricts de cet espace. L'existence d'une solution étant connue pour $s \in [0, 1]$ , l'équation de Schrödinger sous-critique est localement bien posée dans $H^{1}$ sans condition supplémentaire pour l'unicité. Dans cette Note, nous généralisons le résultat d'unicité de Kato, montrant que l'équation est bien posée avec unicité inconditionnelle dans tous les espaces $H^{s}$ , $s \in [\frac{n}{2 (n - 1)}, 1]$ .

Let $n ⩾ 3$ and consider the subcritical nonlinear Schrödinger equation, $i \partial_{t} u + Δ u = {| u |}^{α} u$ , with initial data $u_{0} \in H^{s} (R^{n})$ . When $s ⩾ 1$ , Kato proved that if a maximal solution exists, then it is unique in $C ([0, T_{\max}), H^{s})$ . Previously, uniqueness had only been proven in strictly smaller subspaces. The existence of a solution is assured when $s \in [0, 1]$ , so that the subcritical nonlinear Schrödinger equation is unconditionally locally well-posed in $H^{1}$ . We extend the uniqueness result so that the subcritical nonlinear Schrödinger equation is unconditionally locally well-posed in $H^{s}$ when $s \in [\frac{n}{2 (n - 1)}, 1]$ .

Reçu le : 2007-06-27
Accepté le : 2007-09-12
Publié le : 2007-10-01

DOI : 10.1016/j.crma.2007.09.003

Affiliations des auteurs :

Keith M. Rogers ¹

¹ Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

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     title = {Unconditional well-posedness for subcritical {NLS} in $ {H}^{s}$},
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     pages = {395--398},
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Keith M. Rogers. Unconditional well-posedness for subcritical NLS in $ {H}^{s}$. Comptes Rendus. Mathématique, Volume 345 (2007) no. 7, pp. 395-398. doi : 10.1016/j.crma.2007.09.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.09.003/

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