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In this paper, we prove that there exists some small $\varepsilon_*>0$, suchthat the derivative nonlinear Schr\"{o}dinger equation (DNLS) is globalwell-posedness in the energy space, provided that the initial data $u_0\inH^1(\mathbb{R})$ satisfies $\|u_0\|_{L^2}<\sqrt{2\pi}+\varepsilon_*$. Thisresult shows us that there are no blow up solutions whose masses slightlyexceed $2\pi$, even if their energies are negative. This phenomenon is muchdifferent from the behavior of nonlinear Schr\"odinger equation with criticalnonlinearity. The technique is a variational argument together with themomentum conservation law. Further, for the DNLS on half-line $\mathbb{R}^+$,we show the blow-up for the solution with negative energy.

Global well-posedness for the nonlinear Schr\"{o}dinger equation with derivative in energy space