Global well-posedness and scattering for the defocusing Ḣ1∕2-critical nonlinear Schrödinger equation in ℝ2

Yu, Xueying

doi:10.2140/apde.2021.14.2225

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Global well-posedness and scattering for the defocusing $\dot{H}^{1/2}$-critical nonlinear Schrödinger equation in $\mathbb{R}^2$

Xueying Yu

Vol. 14 (2021), No. 7, 2225–2268

DOI: 10.2140/apde.2021.14.2225

Abstract

We consider the Cauchy initial value problem for the defocusing quintic nonlinear Schrödinger equation in two dimensions with general data in the critical space $Ḣ^{1 ∕ 2} (ℝ^{2})$ . We show that if a solution remains bounded in $Ḣ^{1 ∕ 2} (ℝ^{2})$ in its maximal interval of existence, then the interval is infinite and the solution scatters.

Keywords

NLS, scattering, Morawetz inequality, concentration-compactness

Mathematical Subject Classification 2010

Primary: 35Q55

Milestones

Received: 30 September 2019

Revised: 21 April 2020

Accepted: 31 July 2020

Published: 10 November 2021

Authors

Xueying Yu
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States	Department of Mathematics and Statistics University of Massachusetts Amherst, MA United States

Vol. 14, No. 7, 2021