On the blow up phenomenon of the critical nonlinear Schrödinger equation

In this paper we consider the blow up phenomenon of critical nonlinear Schrödinger equations in dimension 1D and 2D. We define the minimal mass as the L2 norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism. Asymptotic compactness properties and L2-concentration are proved. The proof relies on linear and nonlinear profile decompositions.