  In this paper, we prove that there exists some small $\varepsilon_*>0$, such
that the derivative nonlinear Schr\"{o}dinger equation (DNLS) is global
well-posedness in the energy space, provided that the initial data $u_0\in
H^1(\mathbb{R})$ satisfies $\|u_0\|_{L^2}<\sqrt{2\pi}+\varepsilon_*$. 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 Schr\"odinger 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.
