Sharp upper bound on the blow-up rate for the critical nonlinear Schrodinger equation

We consider the critical nonlinear Schrodinger equation iu(t) = -Deltau - \u\(4/N)u with initial condition u(0, x) = u(0). For u(0) is an element of 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(tup arrowT<+&INFIN;),, \&DEL;u(t)\(L2) = +0&INFIN;. 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. In a previous paper [MeR], we established some blow-up properties of (NLS) in the energy space which implied a control \&DEL;u(t)\(L2) &LE; C \ln(T-t)\N/4/&RADIC;<(T-t)over bar> and removed the rate of the known explicit blow-up solutions which is C/T - t. In this paper, we prove the sharp upper bound expected from numerics as \delu(t)\(L2) less than or equal to C (ln\ln(T - t)\/(T - t)(1/2) by exhibiting the exact geometrical structure of dispersion for the problem.