The defocusing energy-critical nonlinear schrödinger equation in higher dimensions

Duke Mathematical Journal

Volumn 138, Issue 2, 2007, Pages 281-374

  Visan, Monica ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

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[No Author keywords available]

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EID: 34547304016     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-07-13825-0     Document Type: Article

References (37)

1. J. BOURGAIN, Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity, Internat. Math. Res. Notices 1998, no. 5, 253-283. MR 1616917
2. _, Global Solutions of Nonlinear Schrödinger Equations, Amer. Math. Soc. Colloq. Publ. 46, Amer. Math. Soc., Providence, 1999. MR 1691575
3. _, Global well-posedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), 145-171. MR 1626257
4. _, A remark on normal forms and the "I-Method" for periodic NLS, J. Anal. Math. 94 (2004), 125-157. MR 2124457
5. T. CAZENAVE, Semilinear Schrödinger Equations, Courant Lect. Notes Math. 10, Amer. Math. Soc., Providence, 2003. MR 2002047
6. T. CAZENAVE and F. B. WEISSLER, "Some remarks on the nonlinear Schrödinger equation in the critical case" in Nonlinear Semigroups, Partial Differential Equations and Attractors (Washington, D.C., 1987), Lecture Notes in Math. 1394, Springer, Berlin, 1989, 18-29. MR 1021011
7. s, Nonlinear Anal. 14 (1990), 807-836. MR 1055532
8. M. CHRIST [F. M. CHRIST] and A. KISELEV, Maximal functions associated to filiations, J. Funct. Anal. 179 (2001), 409-425. MR 1809116
9. F. M. CHRIST and M. I. WEINSTEIN, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal. 100 (1991), 87-109. MR 1124294
10. R. R. COIFMAN and Y. MEYER, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315-331. MR 0380244
11. J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, and T. TAO, Global well-posedness for Schrödinger equations with derivative, Siam J. Math. Anal. 33 (2001), 649-669. MR 1871414
12. 3 en dessous l'espace d'énergie" in Journées "Équations aux dérivées partielles" (Forges-les-Eaux, France, 2002), Univ. Nantes, Nantes, 2002, exp. 10. MR 1968206
13. 3, Comm. Pure Appl. Math. 57 (2004), 987-1014. MR 2053757
14. 3, to appear in Ann. of Math. (2), preprint, arXiv:math/0402129v7 [math.AP]
15. D. FOSCHI, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ. 2 (2005), 1-24. MR 2134950
16. n), Commun. Contemp. Math. 5 (2003), 349-367. MR 1992354
17. J. GINIBRE and G. VELO, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 64 (1985), 363 -401. MR 0839728
18. R. T. GLASSEY, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), 1794-1797. MR 0460850
19. M. G. GRILLAKIS, On nonlinear Schrödinger equations, Comm. Partial Differential Equations 25 (2000), 1827-1844. MR 1778782
20. A. HASSELL, T. TAO, and J. WUNSCH, A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds, Comm. Partial Differential Equations 30 (2005), 157-205. MR 2131050
21. T. KATO, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), 113-129. MR 0877998
22. s -solutions and unconditional well-posedness, J. Anal. Math. 67 (1995), 281-306. MR 1383498
23. T. KATO and G. PONCE, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891-907. MR 0951744
24. M. KEEL and T. TAO, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955-980. MR 1646048
25. J. E. LIN and W. A. STRAUSS, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal. 30 (1978), 245-263. MR 0515228
26. Y. MEYER and R. R. COIFMAN, Ondelettes et operateurs, III: Operateurs multilineaires, Actualites Math., Hermann, Paris, 1991. MR 1160989
27. C. S. MORAWETZ, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A 306 (1968), 291 -296. MR 0234136
28. K. NAKANISHI, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal. 169 (1999), 201 -225. MR 1726753
29. 1+4, Amer. J. Math. 129 (2007), 1-60.
30. H. F. SMITH and C. D. SOGGE, Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations 25 (2000), 2171-2183. MR 1789924
31. E. M. STEIN, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, 1970. MR 0290095
32. _, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Monogr. Harmon. Anal. 3, Princeton Univ. Press, Princeton, 1993. MR 1232192
33. R. S. STRICHARTZ, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031-1060. MR 0215084
34. T. TAO, On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation, Dyn. Partial Differ. Equ. 1 (2004), 1 -48. MR 2091393
35. _, Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data, New York J. Math. 11 (2005), 57-80. MR 2154347
36. T. TAO and M. VISAN, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations 2005, no. 118. MR 2174550
37. M. VISAN. The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Ph.D. dissertation, University of California, Los Angeles, Los Angeles, Calif., 2006.