Random data theory for the cubic fourth-order nonlinear Schrödinger equation

Van Duong Dinh

doi:10.3934/cpaa.2020284

Article Contents

2021, Volume 20, Issue 2: 651-680. Doi: 10.3934/cpaa.2020284

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Random data theory for the cubic fourth-order nonlinear Schrödinger equation

Van Duong Dinh

Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France, Department of Mathematics, HCMC University of Education, 280 An Duong Vuong, Ho Chi Minh, Vietnam

Received: March 31, 2020

Revised: August 31, 2020

Early access: December 2020

Published: February 2021

This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01)

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Abstract

We consider the cubic nonlinear fourth-order Schrödinger equation

$ i \partial_t u - \Delta^2 u + \mu \Delta u = \pm |u|^2 u, \quad \mu \geq 0 $

on $ \mathbb R^N, N\geq 5 $ with random initial data. We prove almost sure local well-posedness below the scaling critical regularity. We also prove probabilistic small data global well-posedness and scattering. Finally, we prove the global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

Keywords:

Fourth-order nonlinear Schrödinger equation,
almost sure well-posedness,
wiener randomization,
probabilistic Strichartz estimates.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35A01.

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References

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