Geometric & Functional Analysis GAFA

, Volume 3, Issue 2, pp 107–156 | Cite as

Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations

Part I: Schrödinger equations
  • J. Bourgain


Evolution Equation Nonlinear Evolution Nonlinear Evolution Equation Lattice Subset Restriction Phenomenon 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BP]E. Bombieri, J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59, 337–357 (1989).Google Scholar
  2. [Bo1]J. Bourgain, On λ(p)-subsets of squares, Israel J. Math. 67:3 (1989), 291–311.Google Scholar
  3. [Bo2]J. Bourgain, Exponential sums and nonlinear Schrödinger equations, in this issue.Google Scholar
  4. [CW]T. Cazenave, F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation inH 3, Nonlinear Analysis, Theory Methods and Applications 14:10 (1990), 807–836.Google Scholar
  5. [GiV]J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation, H. Poincaré Analyse Non Linéaire 2 (1985), 309–327.Google Scholar
  6. [Gr]E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag (1985).Google Scholar
  7. [K1]T. Kato, StrongL p-solutions of the Navier-Stokes equations inR m with applications to weak solutions, Math. Z 187 (1984), 471–480.Google Scholar
  8. [K2]T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Physique Théorique 46 (1987), 113–129.Google Scholar
  9. [K3]T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Math. Suppl. Studies, Studies in Applied Math. 8 (1983), 93–128.Google Scholar
  10. [KePoVe1]C. Kenig, G. Ponce, L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. AMS 4 (1991), 323–347.Google Scholar
  11. [KePoVe2]C. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, preprint.Google Scholar
  12. [KruF]S. Kruzhkov, A. Faminskii, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Math. USSR Sbornik 48 (1984), 93–138.Google Scholar
  13. [L]P. Lax, Periodic solutions of the KDV equations, Comm. Pure and Applied Math. 28, 141–188 (1975).Google Scholar
  14. [LeRSp]J. Lebowitz, H. Rose, E. Speer, Statistical Mechanics of the Nonlinear Schrödinger Equation, J. Statistical Physics 50:3/4 (1988), 657–687.Google Scholar
  15. [MTr]H.P. McKean, E. Trubowitz, Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), 143–226.Google Scholar
  16. [MiGKr]R. Miura, M. Gardner, K. Kruskal, Korteweg-de Vries Equation and Generalizations II. Existence of Conservation Laws and Constant of Motion, J. Math. Physics 9:8 (1968), 1204–1209.Google Scholar
  17. [PosTr]J. Poschel, E. Trubowitz, Inverse Spectral Theory, Academic Press 1987.Google Scholar
  18. [Sj]A. Sjöberg, On the Korteweg-de Vries equation: existence and uniqueness, J. Math. Anal. Appl. 29 (1970), 569–579.Google Scholar
  19. [St]R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705–714.Google Scholar
  20. [T]P. Tomas, A restriction for the Fourier transform, Bull. AMS 81 (1975), 477–478.Google Scholar
  21. [Tr]E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), 325–341.Google Scholar
  22. [Ts]Y. Tsutsumi,L 2-solutions for nonlinear Schrödinger Equations and nonlinear groups, Funkcialaj Ekvacioj 30 (1987), 115–125.Google Scholar
  23. [Vi]I.M. Vinogradov, The Method of Trigonometric Sums in the Theory of Numbers, Intersciences, NY (1954).Google Scholar
  24. [ZS]V. Zakharov, A. Shabat, Exact theory of two-dimenional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34:1, 62–69 (1972).Google Scholar

Copyright information

© Birkhäuser Verlag 1993

Authors and Affiliations

  • J. Bourgain
    • 1
  1. 1.IHESBures sur YvetteFrance

Personalised recommendations