On nonlinear Schrödinger equations with random initial data

Mitia Duerinckx,Laboratoire De Mathématiques D'orsay, Cnrs, Université Paris-Saclay, 91400 Orsay, France ,Département De Mathématique, Université Libre De Bruxelles, 1050 Brussels, Belgium

doi:10.3934/mine.2022030

Abstract

<abstract><p>This note is concerned with the global well-posedness of nonlinear Schrödinger equations in the continuum with spatially homogeneous random initial data.</p></abstract>

Highlights

Motivated by weak turbulence theory, e.g. [10], we consider nonlinear Schrodinger equations with spatially homogeneous statistical ensembles of initial data
This contrasts with the case of the nonlinear wave equation, as well as of the discrete nonlinear Schrodinger equation, for which there is an finite propagation speed and global well-posedness follows, see [3, Propositions 1–3]
The ball centered at x and of radius r in Rd v H1 (Rd) is denoted by Br(x), and we write for abbreviation B(x) := B1(x) and Br := Br(0)

Summary

Introduction

Motivated by weak turbulence theory, e.g. [10], we consider nonlinear Schrodinger equations with spatially homogeneous statistical ensembles of initial data. [10], we consider nonlinear Schrodinger equations with spatially homogeneous statistical ensembles of initial data. This divergence is a key aspect at the very core of weak turbulence: Strichartz’ estimates are not applicable in this infinite-energy setting, which is in sharp contrast with the finite-energy phenomenology and scattering results [7]. The main difficulty is related to the lack of a uniform bound on the propagation speed: mass that is initially spread out might move together and blow up This contrasts with the case of the nonlinear wave equation, as well as of the discrete nonlinear Schrodinger equation, for which there is an (approximate) finite propagation speed and global well-posedness follows, see [3, Propositions 1–3]. As explained in Examples 2.3 below, periodic and quasi-periodic initial data can be viewed as particular instances of the spatially homogeneous random setting.

DUERINCKX

Definition and examples

Functional setting