Skip to main content

Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations

Abstract

We prove some new Strichartz estimates for a class of dispersive equations with radial initial data. In particular, we obtain the full radial Strichartz estimates up to some endpoints for the Schrödinger equation. Using these estimates, we obtain some new results related to nonlinear problems, including small data scattering and large data LWP for the nonlinear Schrödinger and wave equations with radial critical initial data and the well-posedness theory for the fractional order Schrödinger equation in the radial case.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    B. Birnir, C. Kenig, G. Ponce, N. Svanstedt, and L. Vega, On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations, J. London Math. Soc. (2) 53 (1996), 551–559.

    Article  MATH  MathSciNet  Google Scholar 

  2. [2]

    T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in Hs, Nonlinear Anal. 14 (1990), 807–836.

    Article  MATH  MathSciNet  Google Scholar 

  3. [3]

    M. Christ, J. Colliander, and T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, arxiv:math/0311048[math.AP].

  4. [4]

    M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), 406–425.

    Google Scholar 

  5. [5]

    M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Kortewegde Vries equation, J. Funct. Anal. 100 (1991), 87–109.

    Article  MATH  MathSciNet  Google Scholar 

  6. [6]

    Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J. 62 (2013), 991–1020.

    Article  MATH  MathSciNet  Google Scholar 

  7. [7]

    D. Fang and C. Wang, Some remarks on Strichartz estimates for homogeneous wave equation, Nonlinear Anal. 65 (2006), 697–706.

    Article  MATH  MathSciNet  Google Scholar 

  8. [8]

    D. Fang and C. Wang, Weighted Strichartz estimates with angular regularity and their applications, Forum Math. 23 (2011), 181–205.

    Article  MATH  MathSciNet  Google Scholar 

  9. [9]

    J. Ginibre, A. Soffer, and G. Velo, The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal. 110 (1992), 96–130.

    Article  MATH  MathSciNet  Google Scholar 

  10. [10]

    Z. Guo, L. Peng, and B. Wang, Decay estimates for a class of wave equations, J. Funct. Anal. 254 (2008), 1642–1660.

    Article  MATH  MathSciNet  Google Scholar 

  11. [11]

    K. Hidono, Nonlinear Schrödinger equations with radially symmetric data of critical regularity, Funkcial. Ekvac. 51 (2008), 135–147.

    Article  MathSciNet  Google Scholar 

  12. [12]

    K. Hidano, Small solutions to semi-linear wave equations with radial data of critical regularity, Rev. Mat. Iberoam. 25 (2009), 693–708.

    Article  MATH  MathSciNet  Google Scholar 

  13. [13]

    K. Hidano and Y. Kurokawa, Weighted HLS inequalities for radial functions and Strichartz estimates for wave and Schrödinger equations, Illinois J. Math. 52 (2008), 365–388.

    MathSciNet  Google Scholar 

  14. [14]

    J.-C. Jiang, C. Wang, and X. Yu, Generalized and weighted Strichartz estimates, Commun. Pure Appl. Anal. 11 (2012), 1723–1752.

    Article  MATH  MathSciNet  Google Scholar 

  15. [15]

    Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl. 387 (2012), 857–861.

    Article  MATH  MathSciNet  Google Scholar 

  16. [16]

    M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 360–413.

    Article  MathSciNet  Google Scholar 

  17. [17]

    C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing non-linear wave equation in the radial case, Acta Math. 201 (2008), 147–212.

    Article  MATH  MathSciNet  Google Scholar 

  18. [18]

    C. E. Kenig, G. Ponce, and L. Vega, Well-posedness of the initial value problem for the Kortewegde Vries equation, J. Amer. Math. Soc. 4 (1991), 323–347.

    Article  MATH  MathSciNet  Google Scholar 

  19. [19]

    R. Killip and M. Visan, Nonlinear Schrödinger equations at critical regularity, Evolution Equations, Amer. Math. Soc., Providence, RI, 2013, pp. 325–437.

    Google Scholar 

  20. [20]

    S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), 1221–1268.

    Article  MATH  MathSciNet  Google Scholar 

  21. [21]

    H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal. 130 (1995), 357–426.

    Article  MATH  MathSciNet  Google Scholar 

  22. [22]

    S. Shao, A note on the cone restriction conjecture in the cylindrically symmetric case, Proc. Amer. Math. Soc. 137 (2009), 135–143.

    Article  MATH  MathSciNet  Google Scholar 

  23. [23]

    S. Shao, Sharp linear and bilinear restriction estimates for paraboloids in the cylindrically symmetric case, Rev. Mat. Iberoam. 25 (2009), 1127–1168.

    Article  MATH  MathSciNet  Google Scholar 

  24. [24]

    H. F. Smith and C. D. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations 25 (2000), 2171–2183.

    Article  MATH  MathSciNet  Google Scholar 

  25. [25]

    C. Sogge, Lectures on Nonlinear Wave Equations, International Press, Boston, MA, 1995.

    MATH  Google Scholar 

  26. [26]

    E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, 1971.

    MATH  Google Scholar 

  27. [27]

    E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.

    MATH  Google Scholar 

  28. [28]

    E. M. Stein and G. Weiss, Fractional integrals on n-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503–514.

    MATH  MathSciNet  Google Scholar 

  29. [29]

    J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, with an appendix by Igor Rodnianski, Int. Math. Res. Not. 2005, 187–231.

  30. [30]

    R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equation, Duke Math. J. 44 (1977), 705–714.

    Article  MATH  MathSciNet  Google Scholar 

  31. [31]

    T. Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations 25 (2000), 1471–1485.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Zihua Guo.

Additional information

This material is based upon work supported by the National Science Foundation under agreement No. DMS-0635607 and The S. S. Chern Fund. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or The S. S. Chern Fund.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Guo, Z., Wang, Y. Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations. JAMA 124, 1–38 (2014). https://doi.org/10.1007/s11854-014-0025-6

Download citation

Keywords

  • Dispersive Equation
  • Nonlinear Wave Equation
  • Critical Regularity
  • Strichartz Estimate
  • Radial Case