Skip to main content
SpringerLink
Log in

Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS

  • Chapter
Excursions in Harmonic Analysis, Volume 4

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

We introduce a randomization of a function on \( \mathbb{R}^{d} \) that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schrödinger equation. As an example, we also show that the energy-critical cubic nonlinear Schrödinger equation on \( \mathbb{R}^{4} \) is almost surely locally well posed with respect to randomized initial data below the energy space.

This work is partially supported by a grant from the Simons Foundation (No. 246024 to Árpád Bényi).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For NLW, one needs to specify \( (u,\partial _{t}u)\vert _{t=0} \) as an initial condition. For simplicity of presentation, we only displayed u |  t = 0in (3).

References

  1. T. Alazard, R. Carles, Loss of regularity for supercritical nonlinear Schrödinger equations. Math. Ann. 343(2), 397–420 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Ayache, N. Tzvetkov, L pproperties for Gaussian random series. Trans. Am. Math. Soc. 360(8), 4425–4439 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Á. Bényi, T. Oh, O. Pocovnicu, On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on \( \mathbb{R}^{d} \) , d ≥ 3. Trans. Am. Math. Soc. Ser. B 2, 1–50 (2015)

    Google Scholar 

  4. Á. Bényi, K. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces. Bull. Lond. Math. Soc. 41(3), 549–558 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal. 3, 107–156 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation. Commun. Math. Phys. 176(2), 421–445 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Bourgain, Invariant measures for the Gross-Piatevskii equation. J. Math. Pures Appl. (9) 76(8), 649–702 (1997)

    Google Scholar 

  8. J. Bourgain, Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity. Int. Math. Res. Not. 1998(5), 253–283 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Bourgain, A. Bulut, Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3D case. J. Eur. Math. Soc. 16(6), 1289–1325 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Bourgain, A. Bulut, Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3D ball. J. Funct. Anal. 266(4), 2319–2340 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. N. Burq, P. Gérard, N. Tzvetkov, Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations. Ann. Sci. École Norm. Sup. (4) 38(2), 255–301 (2005)

    Google Scholar 

  12. N. Burq, L. Thomann, N. Tzvetkov, Long time dynamics for the one dimensional nonlinear Schrödinger equation. Ann. Inst. Fourier (Grenoble) 63(6), 2137–2198 (2013)

    Google Scholar 

  13. N. Burq, L. Thomann, N. Tzvetkov, Global infinite energy solutions for the cubic wave equation. Bull. Soc. Math. France. 143(2) 301–313 (2015)

    MathSciNet  Google Scholar 

  14. N. Burq, N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory. Invent. Math. 173(3), 449–475 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. N. Burq, N. Tzvetkov, Probabilistic well-posedness for the cubic wave equation. J. Eur. Math. Soc. 16(1), 1–30 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Carles, Geometric optics and instability for semi-classical Schrödinger equations. Arch. Ration. Mech. Anal. 183(3), 525–553 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. T. Cazenave, F. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, in Nonlinear Semigroups, Partial Differential Equations and Attractors (Washington, DC, 1987). Lecture Notes in Mathematics, vol. 1394 (Springer, Berlin, 1989), pp. 18–29

    Google Scholar 

  18. M. Christ, J. Colliander, T. Tao, Asymptotics, frequency modulation, and low-regularity illposedness of canonical defocusing equations. Am. J. Math. 125(6), 1235–1293 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Colliander, T. Oh, Almost sure well-posedness of the cubic nonlinear Schrödinger equation below \( L^{2}(\mathbb{T}) \) . Duke Math. J. 161(3), 367–414 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Deng, Two-dimensional nonlinear Schrödinger equation with random radial data. Anal. PDE 5(5), 913–960 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. A.-S. de Suzzoni, Invariant measure for the cubic wave equation on the unit ball of \( \mathbb{R}^{3} \) . Dyn. Partial Differ. Equ. 8(2), 127–147 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. A.-S. de Suzzoni, Consequences of the choice of a particular basis of L 2(S 3) for the cubic wave equation on the sphere and the Euclidian space. Commun. Pure Appl. Anal. 13(3), 991–1015 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Feichtinger, Modulation spaces of locally compact Abelian groups, Technical report, University of Vienna (1983), in Proc. Internat. Conf. on Wavelets and Applications (Chennai, 2002), ed. by R. Radha, M. Krishna, S. Thangavelu (New Delhi Allied Publishers, New Delhi, 2003), pp. 1–56.

    Google Scholar 

  24. H. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I. J. Funct. Anal. 86, 307–340 (1989)

    Article  MATH  Google Scholar 

  25. H. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, II. Monatsh. Math. 108, 129–148 (1989)

    Article  MATH  Google Scholar 

  26. J. Ginibre, G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Commun. Math. Phys. 144(1), 163–188 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. K. Gröchenig, Foundations of Time-Frequency Analysis(Birkhäuser, Boston, 2001), xvi+359 pp

    Google Scholar 

  28. M. Hadac, S. Herr, H. Koch, Well-posedness and scattering for the KP-II equation in a critical space. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(3), 917–941 (2009); Erratum to “Well-posedness and scattering for the KP-II equation in a critical space”. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(3), 971–972 (2010)

    Google Scholar 

  29. S. Herr, D. Tataru, N. Tzvetkov, Global well-posedness of the energy critical Nonlinear Schrödinger equation with small initial data in \( H^{1}(\mathbb{T}^{3}) \) . Duke Math. J. 159, 329–349 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. J.P. Kahane, Some Random Series of Functions. Cambridge Studies in Advanced Mathematics, vol. 5, 2nd edn. (Cambridge University Press, Cambridge, 1985), xiv+305 pp

    Google Scholar 

  31. M. Keel, T. Tao, Endpoint Strichartz estimates. Am. J. Math. 120(5), 955–980 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  32. C. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math. 166(3), 645–675 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. C. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math. J. 71(1), 1–21 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Kobayashi, M. Sugimoto, The inclusion relation between Sobolev and modulation spaces. J. Funct. Anal. 260(11), 3189–3208 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. H. Koch, D. Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces. Int. Math. Res. Not. IMRN 2007(16), Art. ID rnm053, 36 pp (2007)

    Google Scholar 

  36. J. Lührmann, D. Mendelson, Random data Cauchy theory for nonlinear wave equations of power-type on \( \mathbb{R}^{3} \) . Commun. Partial Differ. Equs. 39(12), 2262–2283 (2014)

    Article  MATH  Google Scholar 

  37. A. Nahmod, G. Staffilani, Almost sure well-posedness for the periodic 3D quintic NLS below the energy space. J. Eur. Math. Soc. (2012, to appear)

    Google Scholar 

  38. A. Nahmod, N. Pavlović, G. Staffilani, Almost sure existence of global weak solutions for supercritical Navier-Stokes equations. SIAM J. Math. Anal. 45(6), 3431–3452 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. K. Okoudjou, Embeddings of some classical Banach spaces into modulation spaces. Proc. Am. Math. Soc. 132, 1639–1647 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. T. Ozawa, Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations. Differ. Integr. Equ. 11(2), 201–222 (1998)

    MathSciNet  MATH  Google Scholar 

  41. R.E.A.C. Paley, A. Zygmund, On some series of functions (1), (2), (3), Proc. Camb. Philos. Soc. 26, 337–357, 458–474 (1930); 28, 190–205 (1932)

    Google Scholar 

  42. O. Pocovnicu, Almost sure global well-posedness for the energy-critical defocusing cubic nonlinear wave equation on \( \mathbb{R}^{d},d = 4 \) and 5. J. Eur. Math. Soc. (2014, to appear)

    Google Scholar 

  43. A. Poiret, D. Robert, L. Thomann, Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator. Anal. PDE 7(4), 997–1026 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. G. Richards, Invariance of the Gibbs measure for the periodic quartic gKdV. Ann. Inst. H. Poincaré Anal. Non Linćaire (2012, to appear)

    Google Scholar 

  45. E. Ryckman, M. Vişan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in \( \mathbb{R}^{1+4} \) . Am. J. Math. 129(1), 1–60 (2007)

    Article  MATH  Google Scholar 

  46. R.S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44(3), 705–714 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  47. M. Sugimoto, N. Tomita, The dilation property of modulation spaces and their inclusion relation with Besov spaces. J. Funct. Anal. 248(1), 79–106 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  48. T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis. CBMS Regional Conference Series in Mathematics, vol. 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC (American Mathematical Society, Providence, 2006), xvi+373 pp

    Google Scholar 

  49. L. Thomann, Random data Cauchy problem for supercritical Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(6), 2385–2402 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  50. J. Toft, Convolution and embeddings for weighted modulation spaces, in Advances in Pseudo-Differential Operators. Oper. Theory Adv. Appl., vol. 155 (Birkhäuser, Basel, 2004), pp. 165–186

    Google Scholar 

  51. M. Vişan, Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions. Int. Math. Res. Not. IMRN 2012(5), 1037–1067 (2012)

    MATH  Google Scholar 

  52. B.X. Wang, L. Han, C. Huang, Global well-posedness and scatering for the derivative nonlinear Schrödinger equation with small rough data. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 2253–2281 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  53. N. Wiener, Tauberian theorems. Ann. Math. (2) 33(1), 1–100 (1932)

    Google Scholar 

  54. K. Yajima, Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110(3), 415–426 (1987)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Western Washington University, 516 High Street, Bellingham, WA, 98225, USA

    Árpád Bényi

  2. School of Mathematics, The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK

    Tadahiro Oh

  3. School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ, 08540, USA

    Oana Pocovnicu

  4. Department of Mathematics, Princeton University, Fine Hall, Washington Rd., Princeton, NJ, 08544-1000, USA

    Oana Pocovnicu

  5. Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Oana Pocovnicu

Authors
  1. Árpád Bényi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tadahiro Oh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Oana Pocovnicu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Árpád Bényi .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    Radu Balan

  2. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    Matthew Begué

  3. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    John J. Benedetto

  4. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    Wojciech Czaja

  5. Department of Mathematics, Norbert Wiener Center University of Maryland, College Park, Maryland, USA

    Kasso A. Okoudjou

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bényi, Á., Oh, T., Pocovnicu, O. (2015). Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS. In: Balan, R., Begué, M., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 4. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-20188-7_1

Download citation

Publish with us

Policies and ethics

Search

Navigation