We consider the critical nonlinear Schr\"odinger equation $iu_t=-\Delta u-|u|^{\frac{4}{N}}u$ with initial condition $u(0,x)=u_0$.\ For $u_0\in H^1$, local existence in time of solutions on an interval $[0,T)$ is known, and there exist finite time blow up solutions, that is $u_0$ such that $\lim_{t\uparrow T<+\infty}|\nabla u(t)|_{L^2}=+\infty$. This is the smallest power in the nonlinearity for which blow up occurs, and is critical in this sense.\The question we address is to control the blow up rate from above for small in a certain sense blow up solutions with negative energy.
We prove the sharp upper bound expected from numerics as $$|\nabla u(t)|_{L^2}\leq C\left(\frac{\ln|\ln(T-t)|}{T-t}\right)^{\frac{1}{2}}$$ by exhibiting the exact geometrical structure of dispersion for the problem.
Copyright. All Rights Reserved.