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

Abstract

In this paper, we prove that there exists some small ${\epsilon }_{\ast }>0$, such that the derivative nonlinear Schrödinger equation (DNLS) is global well-posedness in the energy space, provided that the initial data $u_0\in H^1(\mathbb{R})$ satisfies $\Vert u_0{\Vert }_{L^2}<\sqrt{2\pi }+{\epsilon }_{\ast }$. 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 nonlinear Schrödinger equation with critical nonlinearity. The technique is a variational argument together with the momentum conservation law. Further, for the DNLS on half-line ${\mathbb{R}}^{+}$, we show the blow-up for the solution with negative energy.

To appear in Analysis & PDE. We add some references, and change some expressions in English

References in corpus (3)

• Low regularity local well-posedness of the Derivative Nonlinear Schrödinger Equation with periodic initial data
• Global well-posedness for Schrödinger equation with derivative in $H^{1/2}(\mathbb{R})$
• On Blow-up criterion for the Nonlinear Schrödinger Equation