  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 $u(t)\in
\dot{H}^{s_c}\cap \dot{H}^1$ be a radially symmetric blow up solution which
blows up at $0<T<+\infty$, we prove that the scaling invariant $L^{p_c}$ norm
where $\dot{H}^{s_c}\rightharpoonup L^{p_c}$ also blows up with a lower bound
$|u(t)|_{L^{p_c}}\geq |\log(T-t)|^{C_{N,p}} $ as $t\to T$.
