Growing Sobolev norms for the cubic defocusing Schrödinger equation
Zaher Hani1; Benoit Pausader2; Nikolay Tzvetkov3; Nicola Visciglia4
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), Talk no. 16, 11 p.

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.

DOI: 10.5802/slsedp.60
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 
@article{SLSEDP_2013-2014____A16_0,
     author = {Zaher Hani and Benoit Pausader and Nikolay Tzvetkov and Nicola Visciglia},
     title = {Growing {Sobolev} norms for the cubic defocusing {Schr\"odinger} equation},
     journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
     note = {talk:16},
     publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2013-2014},
     doi = {10.5802/slsedp.60},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.60/}
}
TY  - JOUR
AU  - Zaher Hani
AU  - Benoit Pausader
AU  - Nikolay Tzvetkov
AU  - Nicola Visciglia
TI  - Growing Sobolev norms for the cubic defocusing Schrödinger equation
JO  - Séminaire Laurent Schwartz — EDP et applications
N1  - talk:16
PY  - 2013-2014
PB  - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.60/
DO  - 10.5802/slsedp.60
LA  - en
ID  - SLSEDP_2013-2014____A16_0
ER  - 
%0 Journal Article
%A Zaher Hani
%A Benoit Pausader
%A Nikolay Tzvetkov
%A Nicola Visciglia
%T Growing Sobolev norms for the cubic defocusing Schrödinger equation
%J Séminaire Laurent Schwartz — EDP et applications
%Z talk:16
%D 2013-2014
%I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
%U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.60/
%R 10.5802/slsedp.60
%G en
%F SLSEDP_2013-2014____A16_0
Zaher Hani; Benoit Pausader; Nikolay Tzvetkov; Nicola Visciglia. Growing Sobolev norms for the cubic defocusing Schrödinger equation. Séminaire Laurent Schwartz — EDP et applications (2013-2014), Talk no. 16, 11 p. doi : 10.5802/slsedp.60. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.60/

[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. | MR | Zbl

[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. | MR | Zbl

[3] J. Bourgain, Problems in Hamiltonian PDE’s, Geom. Funct. Anal., 2000. (Special volume, Part I), 32–56. | MR | Zbl

[4] J. Bourgain, Refinements of Strichartz inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Res. Not., (1998), 253-283. | MR | Zbl

[5] J. Bourgain, Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces. Israel J. Math., 193 (2013), no. 1, 441–458. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[8] Z. Hani, Long-time strong instability and unbounded orbits for some periodic nonlinear Schödinger equations, Arch. Rat. Mech. Anal., to appear (). | DOI | MR

[9] Z. Hani, B. Pausader. N. Tzvetkov. N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, Preprint 2013.

[10] S. Herr, D. Tataru, and N. Tzvetkov, Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications, J. Ang. Math., to appear, . | DOI

[11] M. Guardia and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation, J. Eur. Math. Soc, to appear.

[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). | MR | Zbl

[13] T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Comm. Math. Phys., 139 (1991), pp. 479–493. | MR | Zbl

Cited by Sources: