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

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Abstract

We consider finite time blow-up solutions to the critical nonlinear Schrödinger equation iu t =-Δu-|u|4/N u with initial condition u 0∈H 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.

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  1. Département de Mathématiques, Université de Cergy-Pontoise, 2, avenue Adolphe Chauvin, BP 222, France

    Frank Merle & Pierre Raphael

  2. Institut Universitaire de France, France

    Frank Merle

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  1. Frank Merle
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  2. Pierre Raphael
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Correspondence to Frank Merle.

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Merle, F., Raphael, P. On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation. Invent. math. 156, 565–672 (2004). https://doi.org/10.1007/s00222-003-0346-z

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