Sharp global well-posedness for KDV and modified KDV on ℝ and double-struck T sign

Journal of the American Mathematical Society

Volumn 16, Issue 3, 2003, Pages 705-749

  Colliander, J ^a   Keel, M ^a   Staffilani, G ^a   Takaoka, H ^a   Tao, T ^a  

^a NONE

Author keywords

Bilinear estimates; Korteweg de Vries equation; Multilinear harmonic analysis; Nonlinear dispersive equations

Indexed keywords

EID: 0038806261     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0894-0347-03-00421-1     Document Type: Article

References (50)

1. M. Beals. Self-spreading and strength of singularities for solutions to semilinear wave equations. Ann. of Math. (2), 118(1):187-214, 1983. MR 85c:35057
2. B. Birnir, C. Kenig, G. Ponce, N. Svanstedt, L. Vega. On the ill-posedness of the Initial Value Problem for the generalizedKorteweg-de Vries and nonlinear Schrödinger equations. J. London Math. Soc. (2). 53(3):551-559, 1996. MR 97d:35233
3. B. Birnir, G. Ponce, N. Svanstedt. The local ill-posedness of the modified KdV equation. Ann. Inst. H. Poincaré Anal. Non-Linéaire. 13(4):529-535, 1996. MR 97e:35152
4. J. L. Bona and R. Smith. The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London Ser. A, 278(1287):555-601, 1975. MR 52:6219
5. J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I, II. Geom. Funct. Anal., 3:107-156, 209-262, 1993. MR 95d:35160a, MR 95d:35160b
6. J. Bourgain. Approximation of solutions of the cubic nonlinear Schrödinger equations by finite-dimensional equations and nonsqueezing properties. Internat. Math. Res. Notices, 2:79-88, 1994. MR 95f:35237
7. J. Bourgain. Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations Geom. Funct. Anal., 5:105-140, 1995. MR 96f:35151
8. J. Bourgain. On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE. Internat. Math. Res. Notices, 6:277-304, 1996. MR 97k:35016
9. J. Bourgain. Periodic Korteweg de Vries equation with measures as initial data. Selecta Math. (N.S.), 3(2):115-159, 1997. MR 2000i:35173
10. J. Bourgain. Refinements of Strichartz' inequality and applications to 2D-NLS with critical nonlinearity. International Mathematical Research Notices, 5:253-283, 1998. MR 99f:35184
11. L. Carleson and P. Sjölin. Oscillatory integrals and a multiplier problem for the disc. Studia Math., 44:287-299. (errata insert), 1972. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, III. MR 50:14052
12. A. Cohen. Existence and regularity for solutions of the Korteweg-de Vries equation. Arch. Rational Mech. Anal., 71(2):143-175, 1979. MR 80g:35109
13. M. Christ, J. Colliander, and T. Tao Asymptotics, frequency modulation and low regularity ill-posedness for canonical defocusing equations. To appear Amer. J. Math., 2002.
14. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. A refined global wellposedness result for Schrödinger equations with derivative. To appear SIAM J. Math. Anal., 2002.
15. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. Global well-posedness for KdV in Sobolev spaces of negative index. Electron. J. Diff. Eqns., 2001(26):1-7, 2001. MR 2001m:35269
16. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. Global wellposedness for Schrödinger equations with derivative. SIAM Journal of Mathematical Analysis, 2001. SIAM J. Math. Anal. 33(3):649-669, 2001. MR 2002j:35278
17. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. Multilinear estimates for periodic KdV equations and applications. To appear J. Funct. Anal., 2002.
18. 2. Mathematical Research Letters, 6(5,6):755-778, 1999. MR 2000m:35159
19. C. Fefferman. A note on spherical summation multipliers. Israel J. Math., 15:44-52, 1973. MR 47:9160
20. G. Fonseca, F. Linares, and G. Ponce. Global well-posedness for the modified Kortewegde Vries equation. Comm. Partial Differential Equations, 24(3-4):683-705, 1999. MR 2000a:35210
21. G. Fonseca, F. Linares, and G. Ponce. Global existence for the critical generalized KdV equation. Preprint, 2002.
22. J. Ginibre. An introduction to nonlinear Schrödinger equations. In Nonlinear waves (Sapporo, 1995), pages 85-133. Gakkōtosho, Tokyo, 1997. MR 99a:35235
23. J. Ginibre, Y. Tsutsumi, and G. Velo. Existence and uniqueness of solutions for the generalized Korteweg de Vries equation. Math. Z., 203(1):9-36, 1990. MR 90m:35168
24. J. Ginibre. Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain). Astérisque, 237:Exp. No. 796, 4, 163-187, 1996. Séminaire Bourbaki, Vol. 1994/95. MR 98e:35154
25. A. Grünrock. A bilinear Airy-estimate with application to gKdV-3. Preprint, 2001.
26. H. Hofer and E. Zehnder. Symplectic invariants and Hamiltonian dynamics. Birkhäuser Verlag, Basel, 1994. MR 96g:58001
27. J.-L. Joly, G. Métivier, and J. Rauch. A nonlinear instability for 3 × 3 systems of conservation laws. Comm. Math. Phys., 162(1):47-59, 1994. MR 95f:35145
28. T. Kato. The Cauchy problem for the Korteweg-de Vries equation. Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. I (Paris, 1978/1979), pages 293-307. Pitman, Boston, Mass., 1981. MR 82m:35129
29. 1+1 for Rough Data. International Mathematical Research Notices, 21:1117-1156, 1998. MR 99k:58180
30. M. Keel and T. Tao. Global well-posedness for large data for the Maxwell-Klein-Gordon equation below the energy norm. Preprint, 2000.
31. C. Kenig, G. Ponce, and L. Vega. Well-Posedness and Scattering Results for the Generalized Korteweg-de Vries Equation via the Contraction Principle. Communications on Pure and Applied Mathematics, XLVI:527-620, 1993. MR 94h:35229
32. C. E. Kenig, G. Ponce, and L. Vega. A bilinear estimate with applications to the KdV equation. J. Amer. Math. Soc., 9:573-603, 1996. MR 96k:35159
33. C. E. Kenig, G. Ponce, and L. Vega. On the ill-posedness of some canonical dispersive equations. Duke Math. J., 106(3):617-633, 2001. MR 2002c:35265
34. C. E. Kenig, G. Ponce, and L. Vega. Global well-posedness for semi-linear wave equations. Comm. Partial Differential Equations, 25(9-10):1741-1752, 2000. MR 2001h:35128
35. S. B. Kuksin. On squeezing and flow of energy for nonlinear wave equations. Geom. Funct. Anal., 5:668-711, 1995. MR 96d:35091
36. S. B. Kuksin. Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDEs. Comm. Math. Phys. 167(3):531-552, 1995. MR 96e:58060
37. S. Klainerman and M. Machedon. Smoothing estimates for null forms and applications. Internat. Math. Res. Notices, 9, 1994. MR 95i:58174
38. 2 critical generalized KdV equation. J. Amer. Math. Soc. 15(3):617-664, 2002.
39. F. Merle. Personal communication. 2002.
40. Y. Meyer and R. R. Coifman. Ondelettes et opérateurs. III. Hermann, Paris, 1991. Opérateurs multilinéaires, [Multilinear operators]. MR 93m:42004
41. Y. Meyer and R. Coifman. Wavelets. Calderón-Zygmund and multilinear operators, translated from the 1990 and 1991 French originals by David Salinger, Cambridge University Press, Cambridge, 1997. MR 98e:42001
42. R. M. Miura. Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Mathematical Phys., 9:1202-1204, 1968. MR 40:6042a
43. R. M. Miura. The Korteweg-de Vries Equation: A Survey of Results. SIAM Review, 18(3):412-459, 1976. MR 53:8689
44. R. M. Miura. Errata: "The Korteweg-deVries equation: a survey of results" (SIAM Rev. 18 (1976), no. 3, 412-459). SIAM Rev., 19(4):vi, 1977. MR 57:6908
45. K. Nakanishi, H. Takaoka, and Y. Tsutsumi, Counterexamples to bilinear estimates related to the KdV equation and the nonlinear Schrödinger equation. Methods of Appl. Anal. 8(4):569-578, 2001.
46. J. Rauch and M. Reed. Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension. Duke Math. J., 49:397-475, 1982. MR 83m:35098
47. R. R. Rosales. I. Exact solution of some nonlinear evolution equations, II. The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendent. PhD thesis, California Institute of Technology, 1977.
48. H. Takaoka. Global Well-posedness for the Kadomtsev-Petviashvili II Equation. Discrete Cont. Dynam. Systems. 6(2):483-499, 1999. MR 2000m:35163
49. 2 functions and applications to nonlinear dispersive equations. Amer. J. Math., 123(5):839-908, 2001. MR 2002k:35283
50. N. Tzvetkov. Global low regularity solutions for Kadomtsev-Petviashvili equation. Differential Integral Equations., 13(10-12):1289-1320, 2001. MR 2001g:35227