On a sharp lower bound on the blow-up rate for the $L^2$ critical nonlinear Schrödinger equation

We consider the $L^2$ critical nonlinear Schrödinger equation $iu_t=-\Delta u-\vert u\vert^{\frac{4}{N}}u$ with initial condition in the energy space $u(0,x)=u_0\in H^1$ and study the dynamics of finite time blow-up solutions. In an earlier sequence of papers, the authors established for a certain class of initial data on the basis of dispersive properties in $L^2_{loc}$ a sharp and stable upper bound on the blow-up rate: $\vert\nabla u(t)\vert _{L^2}\leq C\left(\frac{\log\vert\log(T-t)\vert}{T-t}\right)^{\frac{1}{2}}$.

In an earlier paper, the authors then addressed the question of a lower bound on the blow-up rate and proved for this class of initial data the nonexistence of self-similar solutions, that is, $\lim_{t\to T}\sqrt{T-t}\vert\nabla u(t)\vert _{L^2}=+\infty.$

In this paper, we prove the sharp lower bound

by exhibiting the dispersive structure in the scaling invariant space $L^2$ for this log-log regime. In addition, we will extend to the pure energy space $H^1$ a dynamical characterization of the solitons among the zero energy solutions.