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) : itu + 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

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

Included in

Mathematics Commons

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