Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation

We consider the critical nonlinear Schrödinger equation i u t = - Δ u - | u | 4 N u with initial condition u ( 0 , x ) = u 0 in dimension N . For u 0 ∈ H 1 , local existence in time of solutions on an interval [ 0 , T ) is known, and there exists finite time blow up solutions, that is u 0 such that lim t → T < + ∞ | u x ( t ) | L 2 = + ∞ . 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 understand the blow up dynamic. Even though there exists an explicit example of blow up solution and a class of initial data known to lead to blow up, no general understanding of the blow up dynamic is known. At first, we propose in this paper a general setting to study and understand small in a certain sense blow up solutions. Blow up in finite time follows for the whole class of initial data in H 1 with strictly negative energy, and one is able to prove a control from above of the blow up rate below the one of the known explicit explosive solution, which has strictly positive energy.