Remark on the periodic mass critical nonlinear Schrödinger equation

Kishimoto, Nobu

doi:10.1090/S0002-9939-2014-12024-7

Remark on the periodic mass critical nonlinear Schrödinger equation
HTML articles powered by AMS MathViewer

by Nobu Kishimoto

Proc. Amer. Math. Soc. 142 (2014), 2649-2660

DOI: https://doi.org/10.1090/S0002-9939-2014-12024-7

Published electronically: May 2, 2014

PDF | Request permission

Abstract:

We consider the mass critical NLS on $\mathbb {T}$ and $\mathbb {T} ^2$. In the $\mathbb {R}^d$ case the Strichartz estimates enable us to show well-posedness of the IVP in $L^2$ (at least for small data) via the Picard iteration method. However, counterexamples to the $L^6$ Strichartz on $\mathbb {T}$ and the $L^4$ Strichartz on $\mathbb {T}^2$ were given by Bourgain (1993) and Takaoka-Tzvetkov (2001), respectively, which means that the Strichartz spaces are not suitable for iteration in these problems. In this note, we show a slightly stronger result, namely, that the IVP on $\mathbb {T}$ and $\mathbb {T}^2$ cannot have a smooth data-to-solution map in $L^2$ even for small initial data. The same results are also obtained for most of the two dimensional irrational tori.

References

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156. MR 1209299, DOI 10.1007/BF01896020
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 (1993), no. 3, 209–262. MR 1215780, DOI 10.1007/BF01895688
J. Bourgain, Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. (N.S.) 3 (1997), no. 2, 115–159. MR 1466164, DOI 10.1007/s000290050008
J. Bourgain, On Strichartz’s inequalities and the nonlinear Schrödinger equation on irrational tori, Mathematical aspects of nonlinear dispersive equations, Ann. of Math. Stud., vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, pp. 1–20. MR 2331676
F. Catoire and W.-M. Wang, Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori, Commun. Pure Appl. Anal. 9 (2010), no. 2, 483–491. MR 2600446, DOI 10.3934/cpaa.2010.9.483
M. Christ, J. Colliander, and T. Tao, Instability of the periodic nonlinear Schrödinger equation, preprint (2003). arXiv:0311227
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$, J. Amer. Math. Soc. 16 (2003), no. 3, 705–749. MR 1969209, DOI 10.1090/S0894-0347-03-00421-1
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal. 211 (2004), no. 1, 173–218. MR 2054622, DOI 10.1016/S0022-1236(03)00218-0
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math. 181 (2010), no. 1, 39–113. MR 2651381, DOI 10.1007/s00222-010-0242-2
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 6th ed., Oxford University Press, Oxford, 2008. Revised by D. R. Heath-Brown and J. H. Silverman; With a foreword by Andrew Wiles. MR 2445243
Sebastian Herr, Daniel Tataru, and Nikolay Tzvetkov, Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in $H^1(\Bbb T^3)$, Duke Math. J. 159 (2011), no. 2, 329–349. MR 2824485, DOI 10.1215/00127094-1415889
Alexandru D. Ionescu and Benoit Pausader, The energy-critical defocusing NLS on $\Bbb T^3$, Duke Math. J. 161 (2012), no. 8, 1581–1612. MR 2931275, DOI 10.1215/00127094-1593335
H. Takaoka and N. Tzvetkov, On 2D nonlinear Schrödinger equations with data on ${\Bbb R}\times \Bbb T$, J. Funct. Anal. 182 (2001), no. 2, 427–442. MR 1828800, DOI 10.1006/jfan.2000.3732