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

Merle, Frank; Raphaël, Pierre; Rodnianski, Igor

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

Merle, Frank ¹ ; Raphaël, Pierre ² ; Rodnianski, Igor 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, Tome 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 :

Merle, Frank ¹ ; Raphaël, Pierre ² ; Rodnianski, Igor ³

@article{CRMATH_2011__349_5-6_279_0,
     author = {Merle, Frank and Rapha\"el, Pierre and Rodnianski, Igor},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2011.01.026/}
}

TY  - JOUR
AU  - Merle, Frank
AU  - Raphaël, Pierre
AU  - Rodnianski, Igor
TI  - Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 279
EP  - 283
VL  - 349
IS  - 5-6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2011.01.026/
DO  - 10.1016/j.crma.2011.01.026
LA  - en
ID  - CRMATH_2011__349_5-6_279_0
ER  -

%0 Journal Article
%A Merle, Frank
%A Raphaël, Pierre
%A Rodnianski, Igor
%T Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map
%J Comptes Rendus. Mathématique
%D 2011
%P 279-283
%V 349
%N 5-6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2011.01.026/
%R 10.1016/j.crma.2011.01.026
%G en
%F CRMATH_2011__349_5-6_279_0

Merle, Frank; Raphaël, Pierre; Rodnianski, Igor. Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map. Comptes Rendus. Mathématique, Tome 349 (2011) no. 5-6, pp. 279-283. doi : 10.1016/j.crma.2011.01.026. http://www.numdam.org/articles/10.1016/j.crma.2011.01.026/

Bibliographie
Cité par

[1] Bejenaru, I.; Tataru, D. Near soliton evolution for equivariant Schrödinger maps in two spatial dimensions | arXiv

[2] I. Bejenaru, A. Ionescu, C. Kenig, D. Tataru, Global Schrödinger maps, Annals of Math., in press.

[3] Van den Bergh, J.; Hulshof, J.; King, J. Formal asymptotics of bubbling in the harmonic map heat flow, SIAM J. Appl. Math., Volume 63 (2003), pp. 1682-1717

[4] Chang, N.-H.; Shatah, J.; Uhlenbeck, K. Schrödinger maps, Comm. Pure Appl. Math., Volume 53 (2000) no. 5

[5] Grotowski, J.; Shatah, J. Geometric evolution equations in critical dimensions, Calc. Var. Partial Differential Equations, Volume 30 (2007) no. 4, pp. 499-512

[6] Gustafson, S.; Kang, K.; Tsai, T.-P. Schrödinger flow near harmonic maps, Comm. Pure Appl. Math., Volume 60 (2007) no. 4, pp. 463-499

[7] Gustafson, S.; Kang, K.; Tsai, T.-P. Asymptotic stability of harmonic maps under the Schrödinger flow, Duke Math. J., Volume 145 (2008) no. 3, pp. 537-583

[8] Gustafson, S.; Nakanishi, K.; Tsai, T.-P. Asymptotic stability, concentration and oscillations in harmonic map heat flow, Landau Lifschitz and Schrödinger maps on $R^{2}$ , Comm. Math. Phys., Volume 300 (2010) no. 1, pp. 205-242

[9] Krieger, J.; Schlag, W.; Tataru, D. Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math., Volume 171 (2008) no. 3, pp. 543-615

[10] Merle, F.; Raphaël, P. Sharp upper bound on the blow up rate for critical nonlinear Schrödinger equation, Geom. Funct. Anal., Volume 13 (2003), pp. 591-642

[11] Merle, F.; Raphaël, P. On universality of blow up profile for $L^{2}$ critical nonlinear Schrödinger equation, Invent. Math., Volume 156 (2004), pp. 565-672

[12] Merle, F.; Raphaël, P. Profiles and quantization of the blow up mass for critical non linear Schrödinger equation, Comm. Math. Phys., Volume 253 (2004) no. 3, pp. 675-704

[13] Merle, F.; Raphaël, P. Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Annals of Math., Volume 161 (2005) no. 1, pp. 157-222

[14] Merle, F.; Raphaël, P. Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation, J. Amer. Math. Soc., Volume 19 (2006) no. 1, pp. 37-90

[15] F. Merle, P. Raphaël, I. Rodnianski, Blow up for the energy critical corotational Schrödinger map, preprint 2011.

[16] Qing, J. On singularities of the heat flow for harmonic maps from surfaces into spheres, Comm. Anal. Geom., Volume 3 (1995), pp. 297-315

[17] Qing, J.; Tian, G. Bubbling of the heat flows for harmonic maps from surface, Comm. Pure Appl. Math., Volume 50 (1997), pp. 295-310

[18] Raphaël, P. Stability of the log-log bound for blow up solutions to the critical nonlinear Schrödinger equation, Math. Ann., Volume 331 (2005), pp. 577-609

[19] P. Raphaël, I. Rodnianski, Stable blow up dynamics for the critical corotational Wave map and equivariant Yang–Mills problems, Publi. I.H.E.S., in press.

[20] Raphaël, P.; Szeftel, J. Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc., Volume 24 (2011), pp. 471-546

[21] Struwe, M. On the evolution of harmonic mappings of Riemannian surfaces, Comm. Math. Helv., Volume 60 (1985), pp. 558-581

[22] Topping, P.M. Winding behaviour of finite-time singularities of the harmonic map heat flow, Math. Z., Volume 247 (2004)

Cité par Sources :