#### Vol. 6, No. 8, 2013

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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 ${\epsilon }_{\ast }>0$ such that the derivative nonlinear Schrödinger equation (DNLS) is globally well-posed in the energy space, provided that the initial data ${u}_{0}\in {H}^{1}\left(ℝ\right)$ satisfies $\parallel {u}_{0}{\parallel }_{{L}^{2}}<\sqrt{2\pi }+{\epsilon }_{\ast }$. This result shows us that there are no blow-up solutions whose masses slightly exceed $2\pi$, 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