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.

Abstract (VO)
Résumé

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

by

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

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 .

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

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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     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/

Bibliographie
Cité par

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

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

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

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

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

[8] S. Gustafson; K. Nakanishi; T.-P. Tsai 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] J. Krieger; W. Schlag; D. Tataru Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math., Volume 171 (2008) no. 3, pp. 543-615

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

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

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

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

[18] P. Raphaël 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] P. Raphaël; J. Szeftel 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] M. Struwe On the evolution of harmonic mappings of Riemannian surfaces, Comm. Math. Helv., Volume 60 (1985), pp. 558-581

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

Joachim Krieger; Shuang Miao On the stability of blowup solutions for the critical corotational wave-map problem, Duke Mathematical Journal, Volume 169 (2020) no. 3, pp. 435-532 | DOI:10.1215/00127094-2019-0053 | Zbl:1441.35071
Tej-eddine Ghoul; Slim Ibrahim; Van Tien Nguyen On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow, Analysis PDE, Volume 12 (2019) no. 1, pp. 113-187 | DOI:10.2140/apde.2019.12.113 | Zbl:1397.35129
Antoine Hocquet Finite-time singularity of the stochastic harmonic map flow, Annales de l'Institut Henri Poincaré. Probabilités et Statistiques, Volume 55 (2019) no. 2, pp. 1011-1041 | DOI:10.1214/18-aihp907 | Zbl:1427.60124
Slim Tayachi; Hatem Zaag Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term, Transactions of the American Mathematical Society, Volume 371 (2019) no. 8, pp. 5899-5972 | DOI:10.1090/tran/7631 | Zbl:1423.35186
Tej-Eddine Ghoul; Van Tien Nguyen; Hatem Zaag Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 35 (2018) no. 6, pp. 1577-1630 | DOI:10.1016/j.anihpc.2018.01.003 | Zbl:1394.35222
Nai-Heng Chang The Cauchy problem for a spin-liquid model in three space dimensions, Applicable Analysis, Volume 97 (2018) no. 10, pp. 1771-1796 | DOI:10.1080/00036811.2017.1341973 | Zbl:1394.35468
Antoine Hocquet Struwe-like solutions for the stochastic harmonic map flow, Journal of Evolution Equations, Volume 18 (2018) no. 3, pp. 1189-1228 | DOI:10.1007/s00028-018-0437-3 | Zbl:1434.60157
Van Tien Nguyen; Hatem Zaag Finite degrees of freedom for the refined blow-up profile of the semilinear heat equation, Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, Volume 50 (2017) no. 5, pp. 1241-1282 | DOI:10.24033/asens.2644 | Zbl:1395.35125
Fethi Mahmoudi; Nejla Nouaili; Hatem Zaag Construction of a stable periodic solution to a semilinear heat equation with a prescribed profile, Nonlinear Analysis. Theory, Methods Applications. Series A: Theory and Methods, Volume 131 (2016), pp. 300-324 | DOI:10.1016/j.na.2015.09.002 | Zbl:1334.35145
Sohrab Shahshahani Renormalization and blow-up for wave maps from ${S}^2{{\times}} {{\mathbb}} {R}$ to $S^{2}$ , Transactions of the American Mathematical Society, Volume 368 (2016) no. 8, pp. 5621-5654 | DOI:10.1090/tran/6524 | Zbl:1339.35063
Nejla Nouaili; Hatem Zaag Profile for a simultaneously blowing up solution to a complex valued semilinear heat equation, Communications in Partial Differential Equations, Volume 40 (2015) no. 7, pp. 1197-1217 | DOI:10.1080/03605302.2015.1018997 | Zbl:1335.35126
Jishan Fan; Tohru Ozawa A regularity criterion for the Schrödinger map, Current trends in analysis and its applications. Proceedings of the 9th ISAAC congress, Kraków, Poland, August 5–9, 2013, Cham: Birkhäuser/Springer, 2015, pp. 217-223 | DOI:10.1007/978-3-319-12577-0_26 | Zbl:1327.35290
Pierre Raphaël; Remi Schweyer Quantized slow blow-up dynamics for the corotational energy-critical harmonic heat flow, Analysis PDE, Volume 7 (2014) no. 8, p. 1713 | DOI:10.2140/apde.2014.7.1713
Paul Smith An unconstrained Lagrangian formulation and conservation laws for the Schrödinger map system, Journal of Mathematical Physics, Volume 55 (2014) no. 5, p. 051502 | DOI:10.1063/1.4874106 | Zbl:1348.58021
Pierre Raphaël; Rémi Schweyer On the stability of critical chemotactic aggregation, Mathematische Annalen, Volume 359 (2014) no. 1-2, pp. 267-377 | DOI:10.1007/s00208-013-1002-6 | Zbl:1320.35100
Paul Smith Conditional global regularity of Schrödinger maps: Subthreshold dispersed energy, Analysis PDE, Volume 6 (2013) no. 3, p. 601 | DOI:10.2140/apde.2013.6.601
Penghong Zhong; Shu Wang; Boling Guo Blowup rate of isotropic anti-ferromagnetic equation near the equivariant data, Communications in Nonlinear Science and Numerical Simulation, Volume 18 (2013) no. 8, p. 2222 | DOI:10.1016/j.cnsns.2012.12.013
Pierre Raphaël; Remi Schweyer Stable blowup dynamics for the 1-corotational energy critical harmonic heat flow, Communications on Pure and Applied Mathematics, Volume 66 (2013) no. 3, pp. 414-480 | DOI:10.1002/cpa.21435 | Zbl:1270.35136
I. Bejenaru; A. Ionescu; C. Kenig; D. Tataru Equivariant Schrödinger maps in two spatial dimensions, Duke Mathematical Journal, Volume 162 (2013) no. 11 | DOI:10.1215/00127094-2293611
Jan Bouwe van den Berg; J. F. Williams (In-)stability of singular equivariant solutions to the Landau-Lifshitz-Gilbert equation, European Journal of Applied Mathematics, Volume 24 (2013) no. 6, pp. 921-948 | DOI:10.1017/s0956792513000247 | Zbl:1293.35059
Baoxiang Wang Globally well and ill posedness for non-elliptic derivative Schrödinger equations with small rough data, Journal of Functional Analysis, Volume 265 (2013) no. 12, pp. 3009-3052 | DOI:10.1016/j.jfa.2013.08.009 | Zbl:1348.35248
Chong Song; Xiaowei Sun; Youde Wang Geometric solitons of Hamiltonian flows on manifolds, Journal of Mathematical Physics, Volume 54 (2013) no. 12, p. 121505 | DOI:10.1063/1.4848775 | Zbl:1337.37056
Yvan Martel; Frank Merle; Pierre Raphaël Blow up and near soliton dynamics for the $L^{2}$ critical gKdV equation, Séminaire Laurent Schwartz. EDP et Applications, Volume 2011-2012 (2013), p. ex | DOI:10.5802/slsedp.28 | Zbl:1319.35224
Jishan Fan; Tohru Ozawa Regularity criteria for hyperbolic Navier-Stokes and related system, ISRN Mathematical Analysis, Volume 2012 (2012), p. 7 (Id/No 796368) | DOI:10.5402/2012/796368 | Zbl:1254.35042
Rémi Schweyer Type II blow-up for the four dimensional energy critical semi linear heat equation, Journal of Functional Analysis, Volume 263 (2012) no. 12, p. 3922 | DOI:10.1016/j.jfa.2012.09.015

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