Blow up of the critical norm for some radial L2 super critical nonlinear Schrödinger equations

We consider the nonlinear Schr\"odinger equation $iu_t=-\Delta u-|u|^{p-1}u$ in dimension $N\geq 3$ in the $L^2$ super critical range $1+\frac{4}{N}< p<\frac{N+2}{N-2}$. The corresponding scaling invariant space is $\dot{H}^{s_c}$ with $0< s_c<1$ and this covers the physically relevant case $N=p=3$. The existence of finite time blow up solutions is known. Let $p_c=\frac{N}{2}(p-1)$ so that $\dot{H}^{s_c}\subset L^{p_c}$. Let $u(t)\in \dot{H}^{s_c}\cap \dot{H}^1$ be a radially symmetric blow up solution which blows up at $0<T<+\infty$, we prove that the scaling invariant $L^{p_c}$ norm also blows up with a lower bound: $$ |u(t)|_{L^{p_c}}\geq |\log(T-t)|^{C_{N,p}} {\rm as} \ \ t\to T.