Partial differential equations
An Lp-theory for almost sure local well-posedness of the nonlinear Schrödinger equations
Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 637-643.

We consider the nonlinear Schrödinger equations (NLS) on Rd with random and rough initial data. By working in the framework of Lp(Rd) 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.

Dans cet article, nous considérons l'équation de Schrödinger non linéaire (NLS) sur Rd à données initiales aléatoires et surcritiques. En travaillant dans des espaces de Lp(Rd), 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.04.009
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
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     title = {An {\protect\emph{L}\protect\textsuperscript{\protect\emph{p}}-theory} for almost sure local well-posedness of the nonlinear {Schr\"odinger} equations},
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Pocovnicu, Oana; Wang, Yuzhao. An Lp-theory for almost sure local well-posedness of the nonlinear Schrödinger equations. Comptes Rendus. Mathématique, Volume 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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