Inventiones mathematicae

, Volume 156, Issue 3, pp 565–672 | Cite as

On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation

  • Frank MerleEmail author
  • Pierre Raphael


We consider finite time blow-up solutions to the critical nonlinear Schrödinger equation iu t =-Δu-|u|4/N u with initial condition u 0H 1. Existence of such solutions is known, but the complete blow-up dynamic is not understood so far. For a specific set of initial data, finite time blow-up with a universal sharp upper bound on the blow-up rate has been proved in [22], [23].

We establish in this paper the existence of a universal blow-up profile which attracts blow-up solutions in the vicinity of blow-up time. Such a property relies on classification results of a new type for solutions to critical NLS. In particular, a new characterization of soliton solutions is given, and a refined study of dispersive effects of (NLS) in L 2 will remove the possibility of self similar blow-up in energy space H 1.


Soliton Initial Data Classification Result Finite Time Soliton Solution 
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© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de Cergy-PontoiseFrance
  2. 2.Institut Universitaire de FranceFrance

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