Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data. (April 2023)

• Record Type: Journal Article
• Authors: Spitz, Martin
• Is Part Of: Nonlinear analysis. Volume 229(2023)
• Publication Date: 2023-04
• Subjects: Nonlinear Schrödinger equation -- Random initial data -- Almost sure wellposedness -- Almost sure scattering; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113204
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the cubic defocusing nonlinear Schrödinger equation on R 4 with supercritical initial data. For randomized initial data in H s ( R 4 ), we prove almost sure local wellposedness for 1 7 < s < 1 and almost sure scattering for 5 7 < s < 1 . The randomization is based on a unit-scale decomposition in frequency space, a decomposition in the angular variable, and – for the almost sure scattering result – an additional unit-scale decomposition in physical space. We employ new probabilistic estimates for the linear Schrödinger flow with randomized data, where we effectively combine the advantages of the different decompositions.