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Schrödinger maps. (English) Zbl 1028.35134
The authors consider the Cauchy problem for Schrödinger maps into a compact Riemann surface. In one space dimension, for finite energy data, it is shown that the problem has a global unique solution, and that for smooth initial data the solution is smooth. In two space dimensions, for radial or equivariant maps, small energy is shown to imply global existence and uniqueness and further, for smooth initial data, regularity.

35Q55 NLS equations (nonlinear Schrödinger equations)
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
58J90 Applications of PDEs on manifolds
Full Text: DOI
[1] An introduction to nonlinear Schrödinger equations. Textos de Métodos Matemáticos No. 22, Instituto de Matemática Universidade Federal do Rio de Janeiro, Rio de Janeiro, 1996.
[2] Daniel, J Math Phys 35 pp 6498– (1994)
[3] Ding, Sci China Ser A 41 pp 746– (1998)
[4] ; Hamiltonian methods in the theory of solitons. Springer Series in Soviet Mathematics. Springer, Berlin-New York, 1987.
[5] Hayashi, Nonlinear Anal 31 pp 671– (1998)
[6] Kenig, Ann Inst H Poincaré Anal Non Linéaire 10 pp 255– (1993)
[7] Collected papers of L. D. Landau. Gordon and Breach, New York, 1965.
[8] Papanicolaou, Nuclear Phys B 360 pp 425– (1991)
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