Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space

Wu, Yi

doi:10.2140/apde.2013.6.1989

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Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space

Yifei Wu

Vol. 6 (2013), No. 8, 1989–2002

DOI: 10.2140/apde.2013.6.1989

Abstract

In this paper, we prove that there exists some small $ε_{*} > 0$ such that the derivative nonlinear Schrödinger equation (DNLS) is globally well-posed in the energy space, provided that the initial data $u_{0} \in H^{1} (ℝ)$ satisfies $∥ u_{0} ∥_{L^{2}} < \sqrt{2 π} + ε_{*}$ . This result shows us that there are no blow-up solutions whose masses slightly exceed $2 π$ , even if their energies are negative. This phenomenon is much different from the behavior of the nonlinear Schrödinger equation with critical nonlinearity. The technique used is a variational argument together with the momentum conservation law. Further, for the DNLS on the half-line $ℝ^{+}$ , we show the blow-up for the solution with negative energy.

Keywords

nonlinear Schrödinger equation with derivative, global well-posedness, blow-up, half-line

Mathematical Subject Classification 2010

Primary: 35Q55

Secondary: 35A01, 35B44

Milestones

Received: 9 March 2013

Revised: 11 September 2013

Accepted: 4 October 2013

Published: 20 April 2014

Authors

Yifei Wu
School of Mathematical Science Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education Beijing, 100875 China

Vol. 6, No. 8, 2013