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

In this paper, we prove that there exists some small $\varepsilon_*>0$, 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 $\|u_0\|_{L^2}<\sqrt{2π}+\varepsilon_*$. 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 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.