Some remarks on the nonlinear Schr\"odinger equation with fractional dissipation

We consider the Cauchy problem for the $L^2$-critical nonlinear Schr"odinger equation with a fractional dissipation. According to the order of the fractional dissipation, we prove the global existence or the existence of finite time blowup dynamics with the log-log blow-up speed for $|\nabla u(t)|_{L^2}$.