On the role of quadratic oscillations in nonlinear schrödinger equations II. the l2-critical case

Transactions of the American Mathematical Society

Volumn 359, Issue 1, 2007, Pages 33-62

  Carles, Rémi ^a   Keraani, Sahbi ^b  

^a UNIVERSITÉ BORDEAUX 1   (France)

^b UNIVERSITÉ DE RENNES 1   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34547208398     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-06-03955-9     Document Type: Article

References (34)

1. H. Bahouri, P. Gerard High frequency approximation of solutions to critical nonlinear wave equations Amer. J. Math., 121, 1999; 1, 131-175. MR2000i: 35123.
2. J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no.2, 147-187. MR1097257 (92g:42010)
3. Some new estimates on oscillatory integrals, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton Math. Ser., vol.42, Princeton Univ. Press, Princeton, NJ, 1995, pp. 83-112. MR96c:42028
4. Refinements of Strichartz' inequality and applications to 2D-NLS with critical non-linearity, Internat. Math. Res. Notices (1998), no.5, 253-283. MR1616917 (99f:35184)
5. J. Bourgain and W. Wang, Construction of blowup solutions for the nonlinear Schrodinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no.1-2, 197-215. MR1655515 (99m:35219)
6. R. Carles, Geometric optics with caustic crossing for some nonlinear Schrodinger equations, Indiana Univ. Math. J. 49 (2000), no.2, 475-551. MR1793681 (2001k:35265)
7. R. Carles, C. Fermanian, and I. Gallagher, On the role of quadratic oscillations in nonlinear Schrodinger equations, J. Funct. Anal. 203 (2003), no.2, 453-493. MR2003356 (2004k:35344)
8. T. Cazenave, Semilinear Schrodinger equations, Courant Lecture Notes in Mathematics, vol.10, New York University, Courant Institute of Mathematical Sciences, New York, 2003. MR2002047 (2004j:35266)
9. T. Cazenave and F. Weissler, Some remarks on the nonlinear Schrodinger equation in the critical case, Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987), Lecture Notes in Math., vol.1394, Springer, Berlin, 1989, pp. 18-29. MR1021011 (91a:35149)
10. s, Nonlin ear Anal. 14 (1990), no.10, 807-836. MR1055532 (91j:35252)
11. C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no.2, 129-206. MR707957 (85f:35001)
12. I. Gallagher, Profile decomposition for solutions of the Navier-Stokes equations, Bull. Soc. Math. France 129 (2001), no.2, 285-316. MR2002h:35235
13. P. Gerard, Description du defaut de compacite de l'injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998), 213-233 (electronic). MR99h:46051
14. J. Ginibre and G. Velo, Sur une equation de Schrodinger non lineaire avec interaction non locale, Nonlinear partial differential equations and their applications. College de France Seminar, Vol.II (Paris, 1979/1980), Res. Notes in Math., vol.60, Pitman, Boston, Mass., 1982, pp. 155-199, 391-392. MR652511 (84m:81040)
15. The global Cauchy problem for the nonlinear Schrodinger equation revisited, Ann. Inst. H. Poincare Anal. Non Lineaire 2 (1985), no.4, 309-327. MR801582 (87b:35150)
16. C. Kenig, G. Ponce, and L. Vega, On the concentration of blow up solutions for the generalized KdV equation critical in L, Nonlinear wave equations (Providence, RI, 1998), Con-temp. Math., vol.263, Amer. Math. Soc, Providence, RI, 2000, pp. 131-156. MR1777639 (2001g:35224)
17. S. Keraani, Etudes de quelques regimes asymptotiques de l'́equation de Schrodinger, Ph.D. thesis, Universite Paris-Sud, Orsay, 2000.
18. On the defect of compactness for the Strichartz estimates of the Schrodinger equa tions, J. Differential Equations 175 (2001), no.2, 353-392. MR1855973 (2002j:35281)
19. n, Arch. Rational Mech. Anal. 105 (1989), no.3, 243-266. MR90d:35015
20. F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrodinger equations with critical power, Duke Math. J. 69 (1993), no.2, 427-454. MR94b:35262
21. 2 critical nonlinear Schrodinger equation, Invent. Math. 156 (2004), no.3, 565-672. MR2061329
22. 2 solutions of the critical nonlinear Schrodinger equation in 2D, Internat. Math. Res. Notices (1998), no.8, 399-425. MR99d:35156
23. G. Metivier and S. Schochet, Trilinear resonant interactions of semilinear hyperbolic waves, Duke Math. J. 95 (1998), no.2, 241-304. MR1652009 (99m:35144)
24. A. Moyua, A. Vargas, and L. Vega, Restriction theorems and maximal operators related to oscillatory integrals inM., Duke Math. J. 96 (1999), no.3, 547-574. MR2000b:42017
25. U. Niederer, The maximal kinematical invariance groups of Schrodinger equations with arbitrary potentials, Helv. Phys. Acta 47 (1974), 167-172. MR0366263 (51:2511)
26. G. Perelman, On the formation of singularities in solutions of the critical nonlinear Schrodinger equation, Ann. Henri Poincare 2 (2001), no.4, 605-673. MR1852922 (2002m:35205) (Pubitemid 33570417)
27. J. Rauch, Partial differential equations, Graduate Texts in Mathematics, vol.128, Springer-Verlag, New York, 1991. MR1223093 (94e:35002)
28. W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no.2, 149-162. MR0454365 (56:12616)
29. R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no.3, 705-714. MR0512086 (58:23577)
30. T. Tao, Recent progress on the restriction conjecture, arXiv:math.CA/0311181, 2003, Lecture notes, Park City, Utah.
31. A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), no.6, 1359-1384. MR2033842 (2004m:47111)
32. M. I. Weinstein, Nonlinear Schrodinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/1983), no.4, 567-576. MR84d:35140
33. On the structure and formation of singularities in solutions to nonlinear disper sive evolution equations, Comm. in Partial Diff. Eq. 11;1986(5)545-565, MR829596 (87i:35026)
34. K. Yajima, Existence of solutions for Schrodinger evolution equations, Comm. Math. Phys. 110 (1987), no.3, 415-426. MR891945 (88e:35048)