Modulation Spaces and Nonlinear Evolution Equations

  • Michael RuzhanskyEmail author
  • Mitsuru Sugimoto
  • Baoxiang Wang
Part of the Progress in Mathematics book series (PM, volume 301)


We survey some recent progress on modulation spaces and the wellposedness results for a class of nonlinear evolution equations by using the frequency-uniform localization techniques.


Frequency-uniform localization modulation spaces nonlinear Schrödinger equation. 


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  1. 1.
    M.J. Ablowitz, R. Haberman, Nonlinear evolution equations in two and three dimensions, Phys. Rev. Lett., 35 (1975), 1185–1188.MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Béenyi, K.A. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces, Bull. London Math. Soc., 41 (2009), 549–558.Google Scholar
  3. 3.
    A. Béenyi, K. Gröchenig, K.A. Okoudjou and L.G. Rogers, Unimodular Fourier multiplier for modulation spaces, J. Funct. Anal., 246 (2007), 366–384.Google Scholar
  4. 4.
    J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, 1976.Google Scholar
  5. 5.
    T. Cazenave and F.B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H s, Nonlinear Anal. TMA, 14 (1990), 807–836.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, Vol. 10, 2003.Google Scholar
  7. 7.
    J.C. Chen, D.S. Fan and L. Sun, Asymptotic estimates For unimodular Fourier multipliers on modulation spaces, preprint.Google Scholar
  8. 8.
    J. Colliander, Nonlinear Schrödinger equations, Lecture Notes, 2004.Google Scholar
  9. 9.
    P.A. Clarkson and J.A. Tuszyriski, Exact solutions of the multidimensional derivative nonlinear Schrödinger equation for many-body systems near criticality, J. Phys. A: Math. Gen. 23 (1990), 4269–4288.zbMATHCrossRefGoogle Scholar
  10. 10.
    E. Cordero, F. Nicola, Some new Strichartz estimates for the Schrödinger equation. J. Differential Equations, 245 (2008), 1945–1974.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    E. Cordero, F. Nicola, Remarks on Fourier multipliers and applications to the wave equation, J. Math. Anal. Appl., 353 (2009), 583–591.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    E. Cordero, F. Nicola, Sharpness of some properties of Wiener amalgam and modulation spaces. Bull. Aust. Math. Soc., 80 (2009), 105–116.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    C. Deng, J.H. Zhao, S.B. Cui, Well-posedness of a dissipative nonlinear electrohydrodynamic system in modulation spaces, Nonlinear Anal., TMA, 73 (2010), 2088–2100.MathSciNetzbMATHGoogle Scholar
  14. 14.
    J.M. Dixon and J.A. Tuszynski, Coherent structures in strongly interacting manybody systems: II, Classical solutions and quantum fluctuations, J. Phys. A: Math. Gen., 22 (1989), 4895–4920.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    H.G. Feichtinger, Modulation spaces on locally compact Abelian group, Technical Report, University of Vienna, 1983. Published in: “Proc. Internat. Conf. on Wavelet and Applications”, 99–140. New Delhi Allied Publishers, India, 2003.Google Scholar
  16. 16.
    L. Grafakos, Classical and modern Fourier analysis, Pearson/Prentice Hall, 2004.Google Scholar
  17. 17.
    P. Gröbner, Banachräume Glatter Funktionen und Zerlegungsmethoden, Doctoral thesis, University of Vienna, 1992.Google Scholar
  18. 18.
    K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, MA, 2001.Google Scholar
  19. 19.
    N. Hayashi and T. Ozawa, Finite energy solutions of nonlinear Schrödinger equations of derivative type, SIAM J. Math. Anal., 25 (1994), 1488–1503.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    A. Ionescu and C.E. Kenig, Low-regularity Schrödinger maps, II: Global wellposedness in dimensions d ≥ 3, Commun. Math. Phys., 271 (2007), 523–559.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    T. Iwabuchi, Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices. J. Differential Equations, 248 (2010), 1972–2002.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    C.E. Kenig, G. Ponce, C. Rolvent, L. Vega, The general quasilinear ultrahyperbolic Schrodinger equation, Adv. Math., 206 (2006), 402–433.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    C.E. Kenig, G. Ponce, L. Vega, Small solutions to nonlinear Schrödinger equation, Ann. Inst. Henri Poincarée, Sect C, 10 (1993), 255–288.Google Scholar
  24. 24.
    C.E. Kenig, G. Ponce and L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. Math., 134 (1998), 489–545.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    S. Klainerman, Long-time behavior of solutions to nonlinear evolution equations, Arch. Rational Mech. Anal., 78 (1982), 73–98.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., 46 (1993), no. 9, 1221–1268.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Duke Math. J., 81 (1995), 99–133.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    S. Klainerman, G. Ponce, Global small amplitude solutions to nonlinear evolution equations, Commun. Pure Appl. Math., 36 (1983), 133–141.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    M. Kobayashi, Modulation spaces M p,q for 0 < p,q ≤ ∞, J. of Funct. Spaces and Appl., 4 (2006), no.3, 329–341.Google Scholar
  30. 30.
    M. Kobayashi, Dual of modulation spaces, J. of Funct. Spaces and Appl., 5 (2007), 1–8.zbMATHCrossRefGoogle Scholar
  31. 31.
    M. Kobayashi, Y. Sawano, Molecular decomposition of the modulation spaces, Osaka J. Math., 47 (2010), 1029–1053.MathSciNetzbMATHGoogle Scholar
  32. 32.
    M. Kobayashi, M. Sugimoto, The inclusion relation between Sobolev and modulation spaces, J. Funct. Anal. 260 (2011), 3189–3208.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    F. Linares and G. Ponce, On the Davey–Stewartson systems, Ann. Inst. H. Poincarée, Anal. Non Linéeaire, 10 (1993), 523–548.Google Scholar
  34. 34.
    A. Miyachi, F. Nicola, S. Riveti, A. Taracco and N. Tomita, Estimates for unimodular Fourier multipliers on modulation spaces, Proc. Amer. Math. Soc., 137 (2009), 3869–3883.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    T. Ozawa, On the nonlinear Schrödinger equations of derivative type, Indiana Univ. Math. J., 45 (1996), 137–163.MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    T. Ozawa and J. Zhai, Global existence of small classical solutions to nonlinear Schrödinger equations, Ann. I. H. Poincarée, Anal. Non Linéeaire, 25 (2008), 303–311.Google Scholar
  37. 37.
    M. Ruzhansky, On the sharpness of Seeger-Sogge-Stein orders, Hokkaido Math. J., 28 (1999), 357–362.MathSciNetzbMATHGoogle Scholar
  38. 38.
    M.V. Ruzhansky, Singularities of affine fibrations in the regularity theory of Fourier integral operators, Russian Math. Surveys, 55 (2000), 93–161.MathSciNetCrossRefGoogle Scholar
  39. 39.
    M. Ruzhansky, J. Smith, Dispersive and Strichartz estimates for hyperbolic equations with constant coefficients, MSJMemoirs, 22, Mathematical Society of Japan, Tokyo, 2010.Google Scholar
  40. 40.
    M. Ruzhansky, M. Sugimoto, J. Toft and N. Tomita Changes of variables in modulation and Wiener amalgam spaces, Math. Nachr., 284 (2011), 2078–2092.MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    M. Ruzhansky and M. Sugimoto, Smoothing properties of evolution equations via canonical transforms and comparison principle, Proc. London Math. Soc. (2012), doi:  10.1112/plms/pds006.
  42. 42.
    M. Ruzhansky and M. Sugimoto, Structural resolvent estimates and derivative nonlinear Schrödinger equations, arXiv:1101.5026v1, to appear in Comm. Math. Phys.Google Scholar
  43. 43.
    J. Shatah, Global existence of small classical solutions to nonlinear evolution equations, J. Differential Equations, 46 (1982), 409–423.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    J. Shatah, Normal forms and quadratic nonlinear Klein–Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685–696.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    M. Sugimoto, M. Ruzhansky, A smoothing property of Schrödinger equations and a global existence result for derivative nonlinear equations, Advances in analysis, 315–320, World Sci. Publ., Hackensack, NJ, 2005.Google Scholar
  46. 46.
    M. Sugimoto amd N. Tomita, The dilation property of modulation spaces and their inclusion relation with Besov spaces, J. Funct. Anal., 248 (2007), 79–106.Google Scholar
  47. 47.
    M. Sugimoto; N. Tomita; B. X. Wang, Remarks on nonlinear operations on modulation spaces, Integral Transforms and Special Functions, 22 (2011), 351–358MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    J. Toft, Continuity properties for modulation spaces, with applications to pseudodifferential calculus, I, J. Funct. Anal., 207 (2004), 399–429.MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, 1983.Google Scholar
  50. 50.
    H. Triebel, Modulation spaces on the Euclidean n-spaces, Z. Anal. Anwendungen, 2 (1983), 443–457.MathSciNetzbMATHGoogle Scholar
  51. 51.
    J.A. Tuszynski and J.M. Dixon, Coherent structures in strongly interacting manybody systems: I, Derivation of dynamics, J. Phys. A: Math. Gen., 22 (1989), 4877–4894.MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    B.X. Wang, Sharp global well-posedness for non-elliptic derivative Schrödinger equations with small rough data, arXiv:1012.0370.Google Scholar
  53. 53.
    B.X. Wang, L.J. Han, C.Y. Huang, Global well-Posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data, Ann. I. H. Poincarée, Anal. Non Linéeaire, 26 (2009), 2253–2281.Google Scholar
  54. 54.
    B.X. Wang and C.Y. Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations, J. Differential Equations, 239 (2007), 213–250.MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    B.X. Wang and H. Hudzik, The global Cauchy problem for the NLS and NLKG with small rough data, J. Differential Equations, 231 (2007), 36–73.MathSciNetCrossRefGoogle Scholar
  56. 56.
    B.X.Wang, L.F. Zhao, B.L. Guo, Isometric decomposition operators, function spaces E λ p,q and their applications to nonlinear evolution equations, J. Funct. Anal., 233 (2006), 1–39.MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    N. Wiener, Tauberian theorems, Ann. of Math., 33 (1932), 1–100.MathSciNetCrossRefGoogle Scholar
  58. 58.
    V.E. Zakharov, E.A. Kuznetson, Multi-scale expansions in the theory of systems integrable by inverse scattering method, Physica D, 18 (1986), 455–463.MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    V.E. Zakharov, E.I. Schulman, Degenerated dispersion laws, motion invariant and kinetic equations, Physica D, 1 (1980), 185–250.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  • Michael Ruzhansky
    • 1
    Email author
  • Mitsuru Sugimoto
    • 2
  • Baoxiang Wang
    • 3
  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Graduate School of MathematicsNagoya UniversityNagoyaJapan
  3. 3.LMAM School of Mathematical SciencesPeking UniversityBeijingPR of China

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