Vol. 6, No. 8, 2013

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ISSN: 1948-206X (e-only)
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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
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 u0 H1() satisfies u0L2 < 2π + ε. This result shows us that there are no blow-up solutions whose masses slightly exceed 2π, 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 +, 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