Comptes Rendus
Partial Differential Equations
Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map
[Dynamique explosive de solutions régulières équivariantes de lʼapplication de Schrödinger énergie critique]
Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 279-283.

We consider the energy critical Schrödinger map tu=uΔu 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 H˙1 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

u(t,x)eΘRQ1(xλ(t))uinH˙1astT
where ΘR, uH˙1, R is a rotation and the concentration rate is given for some κ(u)>0 by
λ(t)=κ(u)Tt|log(Tt)|2(1+o(1))astT.

Nous considérons lʼapplication de Schrödinger sur la 2-sphère énergie critique tu=uΔu pour des données initiales à symétrie équivariante et de degré k=1. Nous exhibons un ensemble de données initiales régulières arbitrairement proches dans la topologie invariante dʼéchelle H˙1 de lʼapplication harmonique dʼénergie minimale Q1 qui engendrent des solutions explosives en temps fini. Nous donnons une description fine de la formation de singularité qui correspond à la concentration dʼune bulle universelle dʼénergie

u(t,x)eΘRQ1(xλ(t))uinH˙1
ΘR, uH˙1, R est une rotation et la vitesse de concentration est donnée pour une certain κ(u)>0 par :
λ(t)=κ(u)Tt|log(Tt)|2(1+o(1))quandtT.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.01.026

Frank Merle 1 ; Pierre Raphaël 2 ; Igor Rodnianski 3

1 Université de Cergy Pontoise et IHES, 2, avenue Adolphe-Chauvin, 95302 Cergy Pontoise, France
2 Institut de mathématiques, université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France
3 Mathematics Department, Princeton University, Fine Hall, Washington road, NJ 08544-1000, USA
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Frank Merle; Pierre Raphaël; Igor Rodnianski. Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map. Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 279-283. doi : 10.1016/j.crma.2011.01.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.01.026/

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