Growing Sobolev norms for the cubic defocusing Schrödinger equation

Zaher Hani[1]; Benoit Pausader[2]; Nikolay Tzvetkov[3]; Nicola Visciglia[4]

  • [1] Courant Institute of Mathematical Sciences 251 Mercer Street New York NY 10012
  • [2] Université Paris-Nord
  • [3] Université Cergy-Pontoise
  • [4] Universita di Pisa

Séminaire Laurent Schwartz — EDP et applications (2013-2014)

  • Volume: 3, page 1-11
  • ISSN: 2266-0607

Abstract

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This text aims to describe results of the authors on the long time behavior of NLS on product spaces with a particular emphasis on the existence of solutions with growing higher Sobolev norms.

How to cite

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Hani, Zaher, et al. "Growing Sobolev norms for the cubic defocusing Schrödinger equation." Séminaire Laurent Schwartz — EDP et applications 3 (2013-2014): 1-11. <http://eudml.org/doc/275798>.

@article{Hani2013-2014,
abstract = {This text aims to describe results of the authors on the long time behavior of NLS on product spaces with a particular emphasis on the existence of solutions with growing higher Sobolev norms.},
affiliation = {Courant Institute of Mathematical Sciences 251 Mercer Street New York NY 10012; Université Paris-Nord; Université Cergy-Pontoise; Universita di Pisa},
author = {Hani, Zaher, Pausader, Benoit, Tzvetkov, Nikolay, Visciglia, Nicola},
journal = {Séminaire Laurent Schwartz — EDP et applications},
keywords = {toroidal space; resonant system; strong solutions},
language = {eng},
pages = {1-11},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Growing Sobolev norms for the cubic defocusing Schrödinger equation},
url = {http://eudml.org/doc/275798},
volume = {3},
year = {2013-2014},
}

TY - JOUR
AU - Hani, Zaher
AU - Pausader, Benoit
AU - Tzvetkov, Nikolay
AU - Visciglia, Nicola
TI - Growing Sobolev norms for the cubic defocusing Schrödinger equation
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2013-2014
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 3
SP - 1
EP - 11
AB - This text aims to describe results of the authors on the long time behavior of NLS on product spaces with a particular emphasis on the existence of solutions with growing higher Sobolev norms.
LA - eng
KW - toroidal space; resonant system; strong solutions
UR - http://eudml.org/doc/275798
ER -

References

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  1. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal.3, (1993), 107–156. Zbl0787.35097MR1209299
  2. J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices 1996, no. 6, 277–304. Zbl0934.35166MR1386079
  3. J. Bourgain, Problems in Hamiltonian PDE’s, Geom. Funct. Anal., 2000. (Special volume, Part I), 32–56. Zbl1050.35016MR1826248
  4. J. Bourgain, Refinements of Strichartz inequality and applications to 2 D-NLS with critical nonlinearity, Int. Math. Res. Not., (1998), 253-283. Zbl0917.35126MR1616917
  5. J. Bourgain, Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces. Israel J. Math., 193 (2013), no. 1, 441–458. Zbl1271.42039MR3038558
  6. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649–669. Zbl1002.35113MR1871414
  7. 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. Zbl1197.35265MR2651381
  8. Z. Hani, Long-time strong instability and unbounded orbits for some periodic nonlinear Schödinger equations, Arch. Rat. Mech. Anal., to appear (http://dx.doi.org/10.1007/s00205-013-0689-6). Zbl1293.35298MR3158811
  9. Z. Hani, B. Pausader. N. Tzvetkov. N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, Preprint 2013. Zbl1326.35348
  10. S. Herr, D. Tataru, and N. Tzvetkov, Strichartz estimates for partially periodic solutions to Schrödinger equations in 4 d and applications, J. Ang. Math., to appear, http://dx.doi.org/10.1515/crelle-2012-0013. Zbl1293.35299
  11. M. Guardia and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation, J. Eur. Math. Soc, to appear. Zbl1311.35284
  12. J. Kato and F. Pusateri, A new proof of long range scattering for critical nonlinear Schrödinger equations, J. Diff. Int. Equ., Vol. 24, no. 9–10 (2011). Zbl1249.35307MR2850346
  13. T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys., 139 (1991), pp. 479–493. Zbl0742.35043MR1121130

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