On Schrödinger Maps

Communications on Pure and Applied Mathematics

Volumn 56, Issue 1, 2003, Pages 114-151

  Nahmod, Andrea ^a   Stefanov, Atanas ^a   Uhlenbeck, Karen ^b  

^a UNIVERSITY OF MASSACHUSETTS   (United States)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2442616113     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.10054     Document Type: Article

References (40)

1. Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3 (1993), no. 2, 107-156.
2. Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation. Geom. Funct. Anal. 3 (1993), no. 3, 209-262.
3. Bourgain, J. Global solutions of nonlinear Schrödinger equations. American Mathematical Society Colloquium Publications, 46. American Mathematical Society, Providence, R.I., 1999.
4. S. Nonlinear Anal. 14 (1990), no. 10, 807-836.
5. Chang, N.-H.; Shatah, J.; Uhlenbeck, K. Schrödinger maps. Comm. Pure Appl. Math. 53 (2000), no. 5, 590-602.
6. Chihara, H. The initial value problem for cubic semilinear Schrödinger equations. Publ. Res. Inst. Math. Sci. 32 (1996), no. 3, 445-471.
7. Chihara, H. Gain of regularity for semilinear Schrödinger equations. Math. Ann. 315 (1999), no. 4, 529-567.
8. Christodoulou, D.; Tahvildar-Zadeh, A. S. On the regularity of spherically symmetric wave maps. Comm. Pure Appl. Math. 46 (1993), no. 7, 1041-1091.
9. Colliander, J. E.; Delort, J.-M.; Kenig, C. E.; Staffilani, G. Bilinear estimates and applications to 2D NLS. Trans. Amer. Math. Soc. 353 (2001), no. 8, 3307-3325.
10. Costakis, G.; Grillakis, M. Local and global existence for the Schrödinger map problem. Preprint, 2001.
11. Ding, W.; Wang, Y. Local Schrödinger flow into Kähler manifolds. Sci. China Ser. A 44 (2001), no. 11, 1446-1464.
12. Eells, J., Jr.; Sampson, J. H. Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964), 109-160.
13. Grillakis, M. G. A priori estimates and regularity of nonlinear waves. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 1187-1194. Birkhäuser, Basel, 1995.
14. Hasimoto, H. A soliton on a vortex filament. J. Fluid Mech. 51 (1972), no. 3, 477-485.
15. Hayashi, N.; Ozawa, T. Global, small radially symmetric solutions to nonlinear Schrödinger equations and a gauge transformation. Differential Integral Equations 8 (1995), no. 5, 1061-1072.
16. Ishimori, Y. Multivortex solutions of a two-dimensional nonlinear wave equation. Progr. Theoret. Phys. 72 (1984), no. 1, 33-37.
17. Kenig, C. E.; Ponce, G.; Vega, L. Small solutions to nonlinear Schrödinger equations. Ann. Inst. H. Poincaré, Anal. Non Linéaire 10 (1993), no. 3, 255-288.
18. Kenig, C. E.; Ponce, G.; Vega, L. Quadratic forms for the 1-D semilinear Schrödinger equation. Trans. Amer. Math. Soc. 348 (1996), no. 8, 3323-3353.
19. Kenig, C. E.; Ponce, G.; Vega, L. Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations. Invent. Math. 134 (1998), no. 3, 489-545.
20. Kenig, C. E.; Ponce, G.; Vega, L. On the initial value problem for the Ishimori system. Ann. Henri Poincaré 1 (2000), no. 2, 341-384.
21. Klainerman, S.; Machedon, M. Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46 (1993), no. 9, 1221-1268.
22. Klainerman, S.; Machedon, M. Smoothing estimates for null forms and applications. Duke Math J. 81 (1995), no. 1, 99-133.
23. Klainerman, S.; Machedon, M. On the optimal local regularity for gauge field theories. Differential Integral Equations 10 (1997), no. 6, 1019-1030.
24. Klainerman, S.; Selberg, S. Remark on the optimal regularity for equations of wave maps type. Comm. Partial Differential Equations 22 (1997), no. 5-6, 901-918.
25. Landau, L. D.; Lifshitz, E. M. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. A (Soviet Union) 153 (1935), no. 8. Reproduced in Collected papers of L. D. Landau. Gordon and Breach, New York-London-Paris, 1967.
26. Landau, L. D.; Lifshitz, E. M. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. A (Soviet Union) 153 (1935), no. 8. Reproduced in Collected papers of L. D. Landau. Gordon and Breach, New York-London-Paris, 1967.
27. Nahmod, A.: Stefanov, A.; Uhlenbeck, K. On the well-posedness of the wave map problem in high dimensions. Preprint, 2001.
28. Papanicolaou, N.; Tomaras, T. N. Dynamics of magnetic vortices. Nuclear Phys. B 360 (1991), no. 2-3, 425-462.
29. Shatah, J.; Tahvildar-Zadeh, A. S. On the Cauchy problem for equivariant wave maps. Comm. Pure Appl. Math. 47 (1994), no. 5, 719-754.
30. Shatah, J.; Tahvildar-Zadeh, A. S. On the stability of stationary wave maps. Comm. Math. Phys. 185 (1997), no. 1, 231-256.
31. Shatah, J.; Struwe, M. Geometric wave equations. Courant Lecture Notes in Mathematics, 2. New York University, Courant Institute of Mathematical Sciences, New York, 1998.
32. Staffilani, G. Doctoral dissertation. University of Chicago.
33. Struwe, M. Regularity results for harmonic maps of Minkowski space. Nematics (Orsay, 1990), 357-369. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 332. Kluwer, Dordrecht, 1991.
34. Struwe, M. Geometric evolution problems. Nonlinear partial differential equations in differential geometry (Park City, UT, 1992), 257-339. IAS/Park City Mathematics Series, 2. American Mathematical Society, Providence, R.I., 1996.
35. Sulem, P.-L.; Sulem, C.; Bardos, C. On the continuous limit for a system of classical spins. Comm. Math. Phys. 107 (1986), no. 3, 431-454.
36. Tataru, D. Local and global results for wave maps. I. Comm. Partial Differential Equations 23 (1998), no. 9-10, 1781-1793.
37. Tataru, D. On global existence and scattering for the wave maps equation. Amer. J. Math. 123 (2001), no. 1, 37-77.
38. 2-functions, and applications to nonlinear dispersive equations. Amer. J. Math. 123 (2001), no. 5, 839-908.
39. Tao, T. Global regularity of wave maps. I. Small critical Sobolev norm in high dimension. Internat. Math. Res. Notices 2001, no. 6, 299-328.
40. Tao, T. Global regularity of wave maps. II. Small energy in two dimensions. Comm. Math. Phys. 224 (2001), no. 2, 443-544.