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Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and $\mathbb T$


Authors: J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao
Journal: J. Amer. Math. Soc. 16 (2003), 705-749
MSC (2000): Primary 35Q53, 42B35, 37K10
DOI: https://doi.org/10.1090/S0894-0347-03-00421-1
Published electronically: January 29, 2003
MathSciNet review: 1969209
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Abstract: The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all $L^2$-based Sobolev spaces $H^s$ where local well-posedness is presently known, apart from the $H^{\frac{1}{4}} ({\mathbb R})$ endpoint for mKdV and the $H^{-\frac{3}{4}}$endpoint for KdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura's transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation.


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  • 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. J. E. Colliander, G. Staffilani, and H. Takaoka.
    Global Wellposedness of KdV below $l^2$.
    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 Korteweg-de 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. Gakkotosho, 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 $3\times 3$ 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. M. Keel and T. Tao.
    Local and Global Well-Posedness of Wave Maps on ${{\mathbb R}}^{1+1}$ 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. Y. Martel and F. Merle.
    Blow up in finite time and dynamics of blow up solutions for the $L^2$ 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. T. Tao.
    Multilinear weighted convolution of $L^2$ 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

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Additional Information

J. Colliander
Affiliation: Department of Mathematics, University of Toronto, Toronto, ON Canada, M5S 3G3

M. Keel
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455

G. Staffilani
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305-2125
Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02138

H. Takaoka
Affiliation: Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
Address at time of publication: Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

T. Tao
Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1555

DOI: https://doi.org/10.1090/S0894-0347-03-00421-1
Keywords: Korteweg-de Vries equation, nonlinear dispersive equations, bilinear estimates, multilinear harmonic analysis
Received by editor(s): November 7, 2001
Received by editor(s) in revised form: November 19, 2002
Published electronically: January 29, 2003
Additional Notes: The first author is supported in part by N.S.F. Grant DMS 0100595.
The second author is supported in part by N.S.F. Grant DMS 9801558
The third author is supported in part by N.S.F. Grant DMS 9800879 and by a grant from the Sloan Foundation
The fourth author is supported in part by J.S.P.S. Grant No. 13740087
The last author is a Clay Prize Fellow and is supported in part by grants from the Packard and Sloan Foundations
Article copyright: © Copyright 2003 American Mathematical Society

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