×
Bejenaru, Ioan; Ionescu, Alexandru; Kenig, Carlos; Tataru, Daniel

Equivariant Schrödinger maps in two spatial dimensions: the \(\mathbb{H}^2\) target. (English) Zbl 1370.35235

Kyoto J. Math. 56, No. 2, 283-323 (2016).
Summary: We consider equivariant solutions for the Schrödinger map problem from \(\mathbb{R}^{2+1}\) to \(\mathbb{H}^{2}\) with finite energy and show that they are global in time and scatter.

Cited in 9 Documents

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
35B65 Smoothness and regularity of solutions to PDEs

Keywords:

Schrödinger maps; equivariant; large data; global well-posedness; scattering
Cite Review PDF
Full Text: DOI arXiv Euclid

References:

[1] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations , Amer. J. Math. 121 (1999), 131-175. · Zbl 0919.35089 · doi:10.1353/ajm.1999.0001
[2] P. Bégout and A. Vargas, Mass concentration phenomena for the \(L^{2}\)-critical nonlinear Schrödinger equation , Trans. Amer. Math. Soc. 359 , no. 11 (2007), 5257-5282. · Zbl 1171.35109 · doi:10.1090/S0002-9947-07-04250-X
[3] I. Bejenaru, Global results for Schrödinger maps in dimensions \(n\geq3\) , Comm. Partial Differential Equations 33 (2008), 451-477. · Zbl 1148.35083 · doi:10.1080/03605300801895225
[4] I. Bejenaru, On Schrödinger maps , Amer. J. Math. 4 (2008), 1033-1065. · Zbl 1159.35065 · doi:10.1353/ajm.0.0014
[5] I. Bejenaru, A. D. Ionescu, and C. E. Kenig, Global existence and uniqueness of Schrödinger maps in dimensions \(d\geq4\) , Adv. Math. 215 (2007), 263-291. · Zbl 1152.35049 · doi:10.1016/j.aim.2007.04.009
[6] I. Bejenaru, A. D. Ionescu, C. E. Kenig, and D. Tataru, Global Schrödinger maps in dimensions \(d\geq2\): Small data in the critical Sobolev spaces , Ann. of Math. (2) 173 (2011), 1443-1506. · Zbl 1233.35112 · doi:10.4007/annals.2011.173.3.5
[7] I. Bejenaru, A. D. Ionescu, C. E. Kenig, and D. Tataru, Equivariant Schrödinger Maps in two spatial dimensions , Duke Math. J. 162 (2013), 1967-2025. · Zbl 1326.35087 · doi:10.1215/00127094-2293611
[8] I. Bejenaru and D. Tataru, Near Soliton Evolution for Equivariant Schrödinger Maps in Two Spatial Dimensions , Mem. Amer. Math. Soc. 228 , Amer. Math. Soc., Providence, 2014. · Zbl 1303.58009 · doi:10.1090/memo/1069
[9] N.-H. Chang, J. Shatah, and K. Uhlenbeck, Schrödinger maps , Comm. Pure Appl. Math. 53 (2000), 590-602. · Zbl 1028.35134 · doi:10.1002/(SICI)1097-0312(200005)53:5<590::AID-CPA2>3.0.CO;2-R
[10] M. G. Grillakis and V. Stefanopoulos, Lagrangian formulation, energy estimates, and the Schrödinger map problem , Comm. Partial Differential Equations 27 (2002), 1845-1877. · Zbl 1021.35103 · doi:10.1081/PDE-120016130
[11] S. Gustafson, K. Kang, and T.-P. Tsai, Schrödinger flow near harmonic maps , Comm. Pure Appl. Math. 60 (2007), 463-499. · Zbl 1144.53085 · doi:10.1002/cpa.20143
[12] S. Gustafson, K. Kang, and T. Tsai, Asymptotic stability of harmonic maps under the Schrödinger flow , Duke Math. J. 145 (2008), 537-583. · Zbl 1170.35091 · doi:10.1215/00127094-2008-058
[13] S. Gustafson and E. Koo, Global well-posedness for 2D radial Schrödinger maps into the sphere , preprint, [math.AP]. arXiv:1105.5659v1
[14] A. D. Ionescu and C. E. Kenig, Low-regularity Schrödinger maps , Differential Integral Equations 19 (2006), 1271-1300. · Zbl 1212.35449
[15] A. D. Ionescu and C. E. Kenig, Low-regularity Schrödinger maps, II: Global well-posedness in dimensions \(d\geq3\) , Comm. Math. Phys. 271 (2007), 523-559. · Zbl 1137.35068 · doi:10.1007/s00220-006-0180-4
[16] C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case , Invent. Math. 166 (2006), 645-675. · Zbl 1115.35125 · doi:10.1007/s00222-006-0011-4
[17] S. Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations , J. Differential Equations 175 (2001), 353-392. · Zbl 1038.35119 · doi:10.1006/jdeq.2000.3951
[18] H. McGahagan, An approximation scheme for Schrödinger maps , Comm. Partial Differential Equations 32 (2007), 375-400. · Zbl 1122.35138 · doi:10.1080/03605300600856758
[19] F. Merle, P. Raphaël, and I. Rodnianski, Blowup dynamics for smooth equivariant solutions to the critical Schrödinger map problem , Invent. Math. 193 (2013), 249-365. · Zbl 1326.35052 · doi:10.1007/s00222-012-0427-y
[20] F. Merle and L. Vega, Compactness at blow-up time for \(L^{2}\) solutions of the critical nonlinear Schrödinger equation in 2D , Int. Math. Res. Not. IMRN 1998 , no. 8, 399-425. · Zbl 0913.35126 · doi:10.1155/S1073792898000270
[21] P. Smith, Conditional global regularity of Schrödinger maps: Subthreshold dispersed energy , Anal. PDE 6 (2013), 601-686. · Zbl 1287.35085 · doi:10.2140/apde.2013.6.601
[22] P.-L. Sulem, C. Sulem, and C. Bardos, On the continuous limit for a system of classical spins , Comm. Math. Phys. 107 (1986), 431-454. · Zbl 0614.35087 · doi:10.1007/BF01220998
[23] T. Tao, M. Visan, and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS , Forum Math. 20 (2008), 881-919. · Zbl 1154.35085 · doi:10.1515/FORUM.2008.042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.