Document Type
Article
Publication Date
5-26-2015
Keywords
Nonlinear Schrödinger equation, Almost sure well-posedness, Modulation space, Wiener decomposition
Abstract
We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) : i∂tu + Δu = ±|u|2u on R d, d ≥ 3, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity scrit = d-2/2. More precisely, given a function on 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 ≠ 4, we show that conditional almost sure global wellposedness 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.
Publication Title
Transactions of the American Mathematical Society Series B
Volume
2
First Page
1
Last Page
50
Required Publisher's Statement
© Copyright 2015, American Mathematical Society
Recommended Citation
Bényi, Árpád; Oh, Tadahiro; and Pocovnicu, Oana, "On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3" (2015). Mathematics Faculty Publications. 40.
https://cedar.wwu.edu/math_facpubs/40
Subjects - Topical (LCSH)
Cauchy problem; Gross-Pitaevskii equations; Decomposition (Mathematics); Probabilistic number theory
Genre/Form
articles
Type
Text
Rights
Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.
Language
English
Format
application/pdf
