An Lp-theory for almost sure local well-posedness of the nonlinear Schrödinger equations

Pocovnicu, Oana; Wang, Yuzhao

doi:10.1016/j.crma.2018.04.009

Partial differential equations

An L^p-theory for almost sure local well-posedness of the nonlinear Schrödinger equations
[Une théorie L^p pour le problème de Cauchy de l'équation de Schrödinger non linéaire à données initiales aléatoires]

Présenté par : Haïm Brézis

Pocovnicu, Oana ¹ ; Wang, Yuzhao ^2,³

¹ Department of Mathematics, Heriot-Watt University and The Maxwell Institute for the Mathematical Sciences, Edinburgh, EH14 4AS, United Kingdom
² School of Mathematics, University of Birmingham, Watson Building, Edgbaston, Birmingham B15 2TT, United Kingdom
³ 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

Comptes Rendus. Mathématique, Tome 356 (2018) no. 6, pp. 637-643.

Résumé
Abstract

Dans cet article, nous considérons l'équation de Schrödinger non linéaire (NLS) sur $R^{d}$ à données initiales aléatoires et surcritiques. En travaillant dans des espaces de $L^{p} (R^{d})$ , $p > 2$ , nous améliorons les résultats précédents de la littérature, en ce sens que nous prouvons que l'équation NLS est localement bien posée presque sûrement pour des données initiales à régularité plus basse. L'ingrédient principal de la preuve est l'estimation dispersive.

We consider the nonlinear Schrödinger equations (NLS) on $R^{d}$ with random and rough initial data. By working in the framework of $L^{p} (R^{d})$ spaces, $p > 2$ , we prove almost sure local well-posedness for rougher initial data than those considered in the existing literature. The main ingredient of the proof is the dispersive estimate.

Reçu le : 2018-01-21
Accepté le : 2018-04-12
Publié le : 2018-04-25

DOI : 10.1016/j.crma.2018.04.009

Affiliations des auteurs :

Pocovnicu, Oana ¹ ; Wang, Yuzhao ^{2, 3}

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     author = {Pocovnicu, Oana and Wang, Yuzhao},
     title = {An {\protect\emph{L}\protect\textsuperscript{\protect\emph{p}}-theory} for almost sure local well-posedness of the nonlinear {Schr\"odinger} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {637--643},
     publisher = {Elsevier},
     volume = {356},
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     doi = {10.1016/j.crma.2018.04.009},
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Pocovnicu, Oana; Wang, Yuzhao. An L^p-theory for almost sure local well-posedness of the nonlinear Schrödinger equations. Comptes Rendus. Mathématique, Tome 356 (2018) no. 6, pp. 637-643. doi : 10.1016/j.crma.2018.04.009. http://www.numdam.org/articles/10.1016/j.crma.2018.04.009/

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