Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem

Frank Merle; Pierre Raphaël; Igor Rodnianski

doi:10.1007/s00222-012-0427-y

Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem

Frank Merle, Pierre Raphaël, Igor Rodnianski

Mathematics

Princeton University

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

58 Scopus citations

Abstract

We consider the energy critical Schrödinger map problem with the 2-sphere target for equivariant initial data of homotopy index k=1. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blowup solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy.

Original language	American English
Pages (from-to)	249-365
Number of pages	117
Journal	Inventiones Mathematicae
Volume	193
Issue number	2
DOIs	https://doi.org/10.1007/s00222-012-0427-y
State	Published - Aug 2013

ASJC Scopus subject areas

General Mathematics

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title = "Blowup dynamics for smooth data equivariant solutions to the critical Schr{\"o}dinger map problem",

abstract = "We consider the energy critical Schr{\"o}dinger map problem with the 2-sphere target for equivariant initial data of homotopy index k=1. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blowup solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy.",

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

note = "Funding Information: The authors are indebted to the anonymous referees for their careful reading of the manuscript which considerably helped to improve the presentation. P.R. is supported by the Junior ERC-ANR program SWAP. I.R. is partially supported by the NSF grant DMS-1001500. This work was completed while P.R. was visiting the Mathematics Department at ETH, Zurich, which he would like to thank for its kind hospitality.",

year = "2013",

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doi = "10.1007/s00222-012-0427-y",

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T1 - Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem

AU - Merle, Frank

AU - Raphaël, Pierre

AU - Rodnianski, Igor

N1 - Funding Information: The authors are indebted to the anonymous referees for their careful reading of the manuscript which considerably helped to improve the presentation. P.R. is supported by the Junior ERC-ANR program SWAP. I.R. is partially supported by the NSF grant DMS-1001500. This work was completed while P.R. was visiting the Mathematics Department at ETH, Zurich, which he would like to thank for its kind hospitality.

PY - 2013/8

Y1 - 2013/8

N2 - We consider the energy critical Schrödinger map problem with the 2-sphere target for equivariant initial data of homotopy index k=1. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blowup solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy.

AB - We consider the energy critical Schrödinger map problem with the 2-sphere target for equivariant initial data of homotopy index k=1. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blowup solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy.

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EP - 365

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

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ER -

Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem

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