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Transactions of the American Mathematical Society Series B

ISSN 2330-0000



On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $ \mathbb{R}^d$, $ d \geq 3$

Authors: Árpád Bényi, Tadahiro Oh and Oana Pocovnicu
Journal: Trans. Amer. Math. Soc. Ser. B 2 (2015), 1-50
MSC (2010): Primary 35Q55
Published electronically: May 26, 2015
MathSciNet review: 3350022
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Abstract: We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) $ : i \partial _t u + \Delta u = \pm \vert u\vert^{2}u$ on $ \mathbb{R}^d$, $ d \geq 3$, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity $ s_\textup {crit} = \frac {d-2}{2}$. More precisely, given a function on $ \mathbb{R}^d$, we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scaling-critical regularity along with small data global existence and scattering. (ii) We implement a probabilistic perturbation argument and prove `conditional' almost sure global well-posedness for $ d = 4$ in the defocusing case, assuming an a priori energy bound on the critical Sobolev norm of the nonlinear part of a solution; when $ d \ne 4$, we show that conditional almost sure global well-posedness in the defocusing case also holds under an additional assumption of global well-posedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity. (iii) Lastly, we prove global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

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Additional Information

Árpád Bényi
Affiliation: Department of Mathematics, Western Washington University, 516 High Street, Bellingham, Washington 98225

Tadahiro Oh
Affiliation: 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

Oana Pocovnicu
Affiliation: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540 – and – Department of Mathematics, Fine Hall, Princeton University, Washington Road, Princeton, New Jersey 08544

Keywords: Nonlinear Schr\"odinger equation, almost sure well-posedness, modulation space, Wiener decomposition
Received by editor(s): October 27, 2014
Received by editor(s) in revised form: April 20, 2015
Published electronically: May 26, 2015
Additional Notes: This work was partially supported by a grant from the Simons Foundation (No. 246024 to the first author). The second author was supported by the European Research Council (grant no. 637995 “ProbDynDispEq”). The third author was supported by the NSF grant under agreement No. DMS-1128155. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF
Article copyright: © Copyright 2015 by the author under Creative Commons Attribution 3.0 License (CC BY 3.0)