×
Nahmod, Andrea; Stefanov, Atanas; Uhlenbeck, Karen

On Schrödinger maps. (English) Zbl 1028.58018

Commun. Pure Appl. Math. 56, No. 1, 114-151 (2003).
The Schrödinger map equation is an evolution problem associated with the equation for harmonic maps into a Kähler manifold. It resembles the harmonic map heat flow equation where the time derivative undergoes an additional rotation determined by the complex structure of the target. In this paper, Schrödinger maps from \(R^1\times R^2\) to the sphere \(S^2\) are considered; since the equation does not depend on the sign of curvatures, the results carry over for maps to the hyperbolic plane \(H^2\).
In these special cases, there is a modification of the Schrödinger map equation by a (non-explicit) choice of gauge. This leads to a certain nonlinear Schrödinger system for the derivatives of the original map, which is called the modified Schrödinger map system. In principle, there is a (fairly complicated) transformation which recovers the Schrödinger map from the solutions of the modified system.
For the modified system, local well-posedness of the Cauchy problem is proven under minimal regularity assumptions on the data, which are only assumed to be in \(H^\varepsilon\) for any \(\varepsilon>0\).
Reviewer: Andreas Gastel (Erlangen)

Cited in 1 Review
Cited in 41 Documents

MSC:

58E20 Harmonic maps, etc.
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)

Keywords:

evolution problem; harmonic maps; nonlinear Schrödinger equation; local well-posedness; Cauchy problem; Schrödinger maps
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom Funct Anal 3 (2) pp 107– (1993) · Zbl 0787.35097
[2] Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom Funct Anal 3 (3) pp 209– (1993) · Zbl 0787.35098
[3] Bourgain, American Mathematical Society Colloquium Publications 46, in: Global solutions of nonlinear Schrödinger equations (1999) · Zbl 0933.35178
[4] Cazenave, The Cauchy problem for the critical nonlinear Schrödinger equation in Hs, Nonlinear Anal 14 (10) pp 807– (1990) · Zbl 0706.35127 · doi:10.1016/0362-546X(90)90023-A
[5] Chang, Schrödinger maps, Comm Pure Appl Math 53 (5) pp 590– (2000) · Zbl 1028.35134 · doi:10.1002/(SICI)1097-0312(200005)53:5<590::AID-CPA2>3.0.CO;2-R
[6] Chihara, The initial value problem for cubic semilinear Schrödinger equations, Publ Res Inst Math Sci 32 (3) pp 445– (1996) · Zbl 0870.35095
[7] Chihara, Gain of regularity for semilinear Schrödinger equations, Math Ann 315 (4) pp 529– (1999) · Zbl 0942.35161 · doi:10.1007/s002080050328
[8] Christodoulou, On the regularity of spherically symmetric wave maps, Comm Pure Appl Math 46 (7) pp 1041– (1993) · Zbl 0744.58071
[9] Colliander, Bilinear estimates and applications to 2D NLS, Trans Amer Math Soc 353 (8) pp 3307– (2001) · Zbl 0970.35142 · doi:10.1090/S0002-9947-01-02760-X
[10] Costakis , G. Grillakis , M. Local and global existence for the Schrödinger map problem 2001
[11] Ding, Local Schrödinger flow into Kähler manifolds, Sci China Ser A 44 (11) pp 1446– (2001)
[12] Eells, Harmonic mappings of Riemannian manifolds, Amer J Math 86 pp 109– (1964) · Zbl 0122.40102
[13] Grillakis , M. G. A priori estimates and regularity of nonlinear waves Proceedings of the International Congress of Mathematicians 1 1187 1194 Birkhäuser Basel 1995
[14] Hasimoto, A soliton on a vortex filament, J Fluid Mech 51 (3) pp 477– (1972) · Zbl 0237.76010
[15] Hayashi, Global, small radially symmetric solutions to nonlinear Schrödinger equations and a gauge transformation, Differential Integral Equations 8 (5) pp 1061– (1995) · Zbl 0823.35157
[16] Ishimori , Y. Multivortex solutions of a two-dimensional nonlinear wave equation 1984 1 33 37 · Zbl 1074.35573
[17] Kenig, Small solutions to nonlinear Schrödinger equations, Ann Inst H Poincaré, Anal Non Linéaire 10 (3) pp 255– (1993) · Zbl 0786.35121 · doi:10.1016/S0294-1449(16)30213-X
[18] Kenig, Quadratic forms for the 1-D semilinear Schrödinger equation, Trans Amer Math Soc 348 (8) pp 3323– (1996) · Zbl 0862.35111 · doi:10.1090/S0002-9947-96-01645-5
[19] Kenig, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent Math 134 (3) pp 489– (1998) · Zbl 0928.35158 · doi:10.1007/s002220050272
[20] Kenig, On the initial value problem for the Ishimori system, Ann Henri Poincaré 1 (2) pp 341– (2000) · Zbl 1021.35105
[21] Klainerman, Space-time estimates for null forms and the local existence theorem, Comm Pure Appl Math 46 (9) pp 1221– (1993) · Zbl 0803.35095
[22] Klainerman, Smoothing estimates for null forms and applications, Duke Math J 81 (1) pp 99– (1995) · Zbl 0909.35094 · doi:10.1215/S0012-7094-95-08109-5
[23] Klainerman, On the optimal local regularity for gauge field theories, Differential Integral Equations 10 (6) pp 1019– (1997) · Zbl 0940.35011
[24] Klainerman, Remark on the optimal regularity for equations of wave maps type, Comm Partial Differential Equations 22 (5-6) pp 901– (1997) · Zbl 0884.35102
[25] Landau, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys A (Soviet Union) 153 (8) (1935) · Zbl 0012.28501
[26] Collected papers of L. D. Landau (1967)
[27] Nahmod , A. Stefanov , A. Uhlenbeck , K. On the well-posedness of the wave map problem in high dimensions 2001 · Zbl 1085.58022
[28] Papanicolaou, Dynamics of magnetic vortices, Nuclear Phys B 360 (2-3) pp 425– (1991) · doi:10.1016/0550-3213(91)90410-Y
[29] Shatah, On the Cauchy problem for equivariant wave maps, Comm Pure Appl Math 47 (5) pp 719– (1994) · Zbl 0811.58059
[30] Shatah, On the stability of stationary wave maps, Comm Math Phys 185 (1) pp 231– (1997) · Zbl 0877.58054 · doi:10.1007/s002200050089
[31] Shatah, Courant Lecture Notes in Mathematics 2, in: Geometric wave equations (1998) · Zbl 0993.35001
[32] Staffilani , G. Doctoral dissertation
[33] Struwe, Mathematical and Physical Sciences 332, in: Nematics (Orsay, 1990) pp 357– (1991)
[34] Struwe, IAS/Park City Mathematics Series 2, in: Nonlinear partial differential equations in differential geometry (Park City, UT, 1992) pp 257– (1996) · doi:10.1090/pcms/002/07
[35] Sulem, On the continuous limit for a system of classical spins, Comm Math Phys 107 (3) pp 431– (1986) · Zbl 0614.35087
[36] Tataru, Local and global results for wave maps. I, Comm Partial Differential Equations 23 (9) pp 1781– (1998) · Zbl 0914.35083
[37] Tataru, On global existence and scattering for the wave maps equation, Amer J Math 123 (1) pp 37– (2001) · Zbl 0979.35100
[38] Tao, Multilinear weighted convolution of L2-functions, and applications to nonlinear dispersive equations, Amer J Math 123 (5) pp 839– (2001)
[39] Tao, Global regularity of wave maps. I. Small critical Sobolev norm in high dimension, Internat Math Res Notices 2001 (6) pp 299– · Zbl 0983.35080
[40] Tao, Global regularity of wave maps. II. Small energy in two dimensions, Comm Math Phys 224 (2) pp 443– (2001) · Zbl 1020.35046
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.