Dynamique explosive de solutions régulières équivariantes de l'application de Schrödinger énergie critique

Frank Merle; Pierre Raphaël; Igor Rodnianski

doi:10.1016/j.crma.2011.01.026

Dynamique explosive de solutions régulières équivariantes de l'application de Schrödinger énergie critique

Translated title of the contribution: Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map

Frank Merle, Pierre Raphaël, Igor Rodnianski

Mathematics

Princeton University

Research output: Contribution to journal › Article › peer-review

23 Scopus citations

Abstract

We consider the energy critical Schrödinger map ∂_tu=u∧δu to the 2-sphere for equivariant initial data of homotopy number k=1. We show the existence of a set of smooth initial data arbitrarily close to the ground state harmonic map Q1 in the scale invariant norm Ḣ1 which generate finite time blow up solutions. We give in addition a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy. where θ^*∈R, u^*∈Ḣ¹, R is a rotation and the concentration rate is given for some κ(u)>0 by.

Translated title of the contribution	Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map
Original language	French
Pages (from-to)	279-283
Number of pages	5
Journal	Comptes Rendus Mathematique
Volume	349
Issue number	5-6
DOIs	https://doi.org/10.1016/j.crma.2011.01.026
State	Published - Mar 2011

ASJC Scopus subject areas

General Mathematics

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title = "Dynamique explosive de solutions r{\'e}guli{\`e}res {\'e}quivariantes de l'application de Schr{\"o}dinger {\'e}nergie critique",

abstract = "We consider the energy critical Schr{\"o}dinger map ∂tu=u∧δu to the 2-sphere for equivariant initial data of homotopy number k=1. We show the existence of a set of smooth initial data arbitrarily close to the ground state harmonic map Q1 in the scale invariant norm Ḣ1 which generate finite time blow up solutions. We give in addition a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy. where θ*∈R, u*∈Ḣ1, R is a rotation and the concentration rate is given for some κ(u)>0 by.",

author = "Frank Merle and Pierre Rapha{\"e}l and Igor Rodnianski",

note = "Funding Information: P.R. is supported by ANR Jeune Chercheur SWAP. I.R. is supported by NSF grant DMS-0702270.",

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doi = "10.1016/j.crma.2011.01.026",

language = "French",

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T1 - Dynamique explosive de solutions régulières équivariantes de l'application de Schrödinger énergie critique

AU - Merle, Frank

AU - Raphaël, Pierre

AU - Rodnianski, Igor

N1 - Funding Information: P.R. is supported by ANR Jeune Chercheur SWAP. I.R. is supported by NSF grant DMS-0702270.

PY - 2011/3

Y1 - 2011/3

N2 - We consider the energy critical Schrödinger map ∂tu=u∧δu to the 2-sphere for equivariant initial data of homotopy number k=1. We show the existence of a set of smooth initial data arbitrarily close to the ground state harmonic map Q1 in the scale invariant norm Ḣ1 which generate finite time blow up solutions. We give in addition a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy. where θ*∈R, u*∈Ḣ1, R is a rotation and the concentration rate is given for some κ(u)>0 by.

AB - We consider the energy critical Schrödinger map ∂tu=u∧δu to the 2-sphere for equivariant initial data of homotopy number k=1. We show the existence of a set of smooth initial data arbitrarily close to the ground state harmonic map Q1 in the scale invariant norm Ḣ1 which generate finite time blow up solutions. We give in addition a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy. where θ*∈R, u*∈Ḣ1, R is a rotation and the concentration rate is given for some κ(u)>0 by.

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JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

IS - 5-6

ER -

Dynamique explosive de solutions régulières équivariantes de l'application de Schrödinger énergie critique

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