Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map

Frank Merle; Pierre Raphaël; Igor Rodnianski

doi:10.1016/j.crma.2011.01.026

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]

Présenté par : Jean-Michel Bony

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

Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 279-283.

Résumé
Abstract

Nous considérons lʼapplication de Schrödinger sur la 2-sphère énergie critique $\partial_{t} u = u \land Δ u$ 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 ${\dot{H}}^{1}$ de lʼapplication harmonique dʼénergie minimale $Q_{1}$ 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) - e^{Θ^{⁎} R} Q_{1} (\frac{x}{λ (t)}) \to u^{⁎} in {\dot{H}}^{1}

où

Θ^{⁎} \in R

u^{⁎} \in {\dot{H}}^{1}

, R est une rotation et la vitesse de concentration est donnée pour une certain

κ (u) > 0

par :

λ (t) = κ (u) \frac{T - t}{{| \log (T - t) |}^{2}} (1 + o (1)) quand t \to T .

We consider the energy critical Schrödinger map $\partial_{t} u = u \land Δ u$ 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 $Q_{1}$ in the scale invariant norm ${\dot{H}}^{1}$ 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) - e^{Θ^{⁎} R} Q_{1} (\frac{x}{λ (t)}) \to u^{⁎} in {\dot{H}}^{1} as t \to T

where

Θ^{⁎} \in R

u^{⁎} \in {\dot{H}}^{1}

, R is a rotation and the concentration rate is given for some

κ (u) > 0

λ (t) = κ (u) \frac{T - t}{{| \log (T - t) |}^{2}} (1 + o (1)) as t \to T .

Reçu le : 2010-12-17
Accepté le : 2011-01-26
Publié le : 2011-02-18

DOI : 10.1016/j.crma.2011.01.026

Affiliations des auteurs :

Frank Merle ¹ ; Pierre Raphaël ² ; Igor Rodnianski ³

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     author = {Frank Merle and Pierre Rapha\"el and Igor Rodnianski},
     title = {Blow up dynamics for smooth equivariant solutions to the energy critical {Schr\"odinger} map},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {279--283},
     publisher = {Elsevier},
     volume = {349},
     number = {5-6},
     year = {2011},
     doi = {10.1016/j.crma.2011.01.026},
     language = {en},
}

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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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