Global Regularity of Wave Maps, I: Small Critical Sobolev Norm in High Dimension

International Mathematics Research Notices

Volumn 2001, Issue 6, 2001, Pages

  Tao, Terence ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 18844483049     PISSN: 10737928     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (31)

1. S.-Y. A. Chang, L. Wang, and P. C. Yang, Regularity of harmonic maps, Comm. Pure Appl. Math. 52 (1999), 1099-1111.
2. D. Christodoulou and A. S. Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math. 46 (1993), 1041-1091.
3. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness below the energy norm for 2D NLS, preprint.
4. A. Freire, S. Müller, and M. Struwe, Weak compactness of wave maps and harmonic maps, Ann. Inst. H. Poincaré Anal. Non Lineaire 15 (1998), 725-754.
5. J. Ginibre and G. Velo, The Cauchy problem for the O(N), CP(N - 1),and Gc(N,P) models, Ann. Physics 142 (1982), 393-415.
6. M. Grillakis, Classical solutions for the equivariant wave map in 1 + 2 dimensions, to appear in Indiana Univ. Math. J.
7. C. H. Gu, On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space, Comm. Pure Appl. Math. 33 (1980), 727-737.
8. F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 591-596.
9. M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955-980.
10. 1-1 for rough data, Internat. Math. Res. Notices 1998, 1117-1156.
11. _, Global existence for the Maxwell-Klein-Gordon equation below the energy norm, in preparation.
12. 3+1" in Nonlinear Partial Differential Equations in Geometry and Physics (Knoxville, Tenn., 1995), Progr. Nonlinear Differential Equations Appl. 29, Birkhäuser, Basel, 1997, 29-69.
13. S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Duke Math. J. 81 (1995), 99-133.
14. _, On the optimal local regularity for gauge field theories, Differential Integral Equations 10 (1997), 1019-1030.
15. S. Klainerman and S. Selberg, Remark on the optimal regularity for equations of wave maps type, Comm. Partial Differential Equations 22 (1997), 901-918.
16. _, Bilinear estimates and applications to nonlinear wave equations, preprint, http://arXiv. org/abs/math.AP/0101118
17. O. A. Ladyzhenskaya and V. I. Shubov, Unique solvability of the Cauchy problem for the equations of the two dimensional chiral fields, taking values in complete Riemann manifolds, J. Soviet Math. 25 (1984), 855-864.
18. K. Nakanishi, Local wellposedness and illposedness in the critical Besov spaces for semilinear wave equations with quadratic forms, Funkcial Ekvac. 42 (1999), 261-279.
19. S. Seiberg, Multilinear space-time estimates and applications to local existence theory for non- linear wave equations, Ph.D. dissertation, Princeton Univ., Princeton.
20. _, Wave maps and bilinear spacetime estimates, preprint.
21. J. Shatah, Weak solutions and development of singularities of the SU(2) σ-model, Comm. Pure Appl. Math. 41 (1988), 459-469.
22. _, "The Cauchy problem for harmonic maps on Minkowski space" in Proceedings of the International Congress of Mathematicians, 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 1126- 1132.
23. J. Shatah and M. Struwe, Geometric Wave Equations, Courant Lect. Notes Math. 2, New York Univ., Courant Inst. Math. Sci., New York, 1998.
24. J. Shatah and A. Tavildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math. 47 (1994), 719-753.
25. T. C. Sideris, Global existence of harmonic maps in Minkowski space, Comm. Pure Appl. Math. 42 (1989), 1-13.
26. M. Struwe, "Wave Maps" in Nonlinear Partial Differential Equations in Geometry and Physics (Knoxville, 1995), Prog. Nonlinear Differential Equations Appl. 29, Birkhäuser, Basel, 1997, 113-153.
27. T. Tao, Ill-posedness for one-dimensional wave maps at the critical regularity, Amer. J. Math. 122 (2000), 451-463.
28. _, Global regularity of wave maps II. Small energy in two dimensions, submitted to Amer. J. Math.
29. D. Tataru, Local and global results for wave maps, I, Comm. Partial Differential Equations 23 (1998), 1781-1793.
30. 2 in 5 + 1 dimensions, Math. Res. Lett. 6 (1999), 469-485.
31. _, On global existence and scattering for the wave maps equation, preprint, http://arXiv. org/abs/math.AP/0011173