Proof of a Spectral Property related to the singularity formation for the [formula omitted] critical nonlinear Schrödinger equation

Abstract

We give a proof of a Spectral Property related to the description of the singularity formation for the L 2 critical nonlinear Schrödinger equation i u t + Δ u + u | u | 4 N = 0 in dimensions N = 2 , 3 , 4 . Assuming this property, the rigorous mathematical analysis developed in a recent series of papers by Merle and Raphaël provides a complete description of the collapse dynamics for a suitable class of initial data. In particular, this implies in dimension N = 2 the existence of a large class of solutions blowing up with the log–log speed | u ( t ) | H 1 ∼ log | log ( T − t ) T − t where T > 0 is the blow up time. This Spectral Property is equivalent to the coercivity of some Schrödinger type operators. An analytic proof is given in [F. Merle, P. Raphaël, Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Ann. of Math. 161 (1) (2005) 157–222] in dimension N = 1 and in this paper, we give a computer assisted proof in dimensions N = 2 , 3 , 4 . We propose in particular a rigorous mathematical frame to reduce the check of this type of coercivity property to accessible and robust numerical results.