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Poincaré-Dulac Normal Form Reduction for Unconditional Well-Posedness of the Periodic Cubic NLS

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Abstract

We implement an infinite iteration scheme of Poincaré-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrödinger equation (NLS) in \({C_tL^2(\mathbb{T})}\), without using any auxiliary function space. This allows us to construct weak solutions of NLS in \({C_tL^2(\mathbb{T})}\) with initial data in \({L^2(\mathbb{T})}\) as limits of classical solutions. As a consequence of our construction, we also prove unconditional well-posedness of NLS in \({H^s(\mathbb{T})}\) for \({s \geq \frac{1}{6}}\).

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References

  1. Arnold, V.I.: Geometrical methods in the theory of ordinary differential equations. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 250. New York: Springer-Verlag, 1988

  2. Babin A., Ilyin A., Titi E.: On the regularization mechanism for the periodic Korteweg-de Vries equation. Comm. Pure Appl. Math. 64(5), 591–648 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bourgain J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I. Schrödinger equations. Geom. Funct. Anal. 3(2), 107–156 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bourgain J.: Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations. Erg. Th. Dyn. Systs. 24(5), 1331–1357 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bourgain J.: A remark on normal forms and the “I-method” for periodic NLS. J. Anal. Math. 94, 125–157 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burq N., Gérard P., Tzvetkov N.: An instability property of the nonlinear Schrödinger equation on \({\mathbb{S}^d}\). Math. Res. Lett. 9(2–3), 323–335 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Christ, M.: Nonuniqueness of weak solutions of the nonlinear Schrödinger equation. http://arxiv.org/abs/math/0503366v1 [math.AP], 2005

  8. Christ, M.: Power series solution of a nonlinear Schrödinger equation. In: Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., 163, Princeton, NJ: Princeton Univ. Press, 2007, pp. 131–155

  9. Christ M., Colliander J., Tao T.: Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125(6), 1235–1293 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Christ, M., Colliander, J., Tao, T.: Instability of the Periodic Nonlinear Schrödinger Equation. http://arxiv.org/abs/math/0311227v1 [math.AP], 2003

  11. Colliander J., Kwon S., Oh T.: A remark on normal forms and the “upside-down” I-method for periodic NLS: growth of higher Sobolev norms. J. Anal. Math. 118, 55–82 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Furioli, G., Planchon, F., Terraneo, E.: Unconditional well-posedness for semilinear Schrödinger and wave equations in H s. In: Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001), Contemp. Math., 320, Providence, RI: Amer. Math. Soc., 2003, pp. 147–156

  14. Germain P., Masmoudi N., Shatah J.: Global solutions for 3D quadratic Schrödinger equations. Int. Math. Res. Not. 2009(3), 414–432 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Ginibre, J.: An introduction to nonlinear Schrödinger equations. In: Nonlinear waves (Sapporo, 1995), GAKUTO Internat. Ser. Math. Sci. Appl., 10, Tokyo: Gakkōtosho, 1997, pp. 85–133

  16. Grünrock A., Herr S.: Low regularity local well-posedness of the derivative nonlinear Schrödinger equation with periodic initial data. SIAM J. Math. Anal. 39(6), 1890–1920 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Fifth edition. New York: The Clarendon Press/ Oxford University Press, 1979

  18. Molinet L.: On ill-posedness for the one-dimensional periodic cubic Schrödinger equation. Math. Res. Lett. 16(1), 111–120 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Kato T.: On nonlinear Schrödinger equations. II. H s-solutions and unconditional well-posedness. J. Anal. Math. 67, 281–306 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kwon S., Oh T.: On unconditional well-posedness of modified KdV. Internat. Math. Res. Not. 2012(15), 3509–3534 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Shatah J.: Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. Pure Appl. Math. 38(5), 685–696 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  22. Takaoka H., Tsutsumi Y.: Well-posedness of the Cauchy problem for the modified KdV equation with periodic boundary condition. Int. Math. Res. Not. 2004(56), 3009–3040 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhou Y.: Uniqueness of weak solution of the KdV equation. Int. Math. Res. Not. 1997(6), 271–283 (1997)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Science, Peking University, Beijing, 100871, China

    Zihua Guo

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

    Zihua Guo

  3. Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, Republic of Korea

    Soonsik Kwon

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

    Tadahiro Oh

Authors
  1. Zihua Guo
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  2. Soonsik Kwon
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  3. Tadahiro Oh
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Correspondence to Tadahiro Oh.

Additional information

Communicated by P. Constantin

Z.G. is 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.

S.K. is supported in part by NRF (Korea) grant 2010-0024017 and TJ Park science fellowship.

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Guo, Z., Kwon, S. & Oh, T. Poincaré-Dulac Normal Form Reduction for Unconditional Well-Posedness of the Periodic Cubic NLS. Commun. Math. Phys. 322, 19–48 (2013). https://doi.org/10.1007/s00220-013-1755-5

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  • DOI: https://doi.org/10.1007/s00220-013-1755-5

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