On the growth of Sobolev norms of solutions of the fractional defocusing NLS equation on the circle
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 509-531.

This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schrödinger equation on the torus:

i∂tu=|D|αu+|u|2u,u(0,⋅)=u0,
where α is a real parameter. We show that, apart from the case α=1, which corresponds to a half-wave equation with no dispersive property at all, solutions of this equation grow at a polynomial rate at most. We also address the case of the cubic and quadratic half-wave equations.

DOI : 10.1016/j.anihpc.2016.02.002
Classification : 37K40, 35B45
Mots-clés : Hamiltonian systems, Fractional nonlinear Schrödinger equation, Nonlinear wave equation, Dispersive properties 
@article{AIHPC_2017__34_2_509_0,
     author = {Thirouin, Joseph},
     title = {On the growth of {Sobolev} norms of solutions of the fractional defocusing {NLS} equation on the circle},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {509--531},
     publisher = {Elsevier},
     volume = {34},
     number = {2},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.02.002},
     zbl = {1370.37133},
     mrnumber = {3610943},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.02.002/}
}
TY  - JOUR
AU  - Thirouin, Joseph
TI  - On the growth of Sobolev norms of solutions of the fractional defocusing NLS equation on the circle
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2017
SP  - 509
EP  - 531
VL  - 34
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2016.02.002/
DO  - 10.1016/j.anihpc.2016.02.002
LA  - en
ID  - AIHPC_2017__34_2_509_0
ER  - 
%0 Journal Article
%A Thirouin, Joseph
%T On the growth of Sobolev norms of solutions of the fractional defocusing NLS equation on the circle
%J Annales de l'I.H.P. Analyse non linéaire
%D 2017
%P 509-531
%V 34
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2016.02.002/
%R 10.1016/j.anihpc.2016.02.002
%G en
%F AIHPC_2017__34_2_509_0
Thirouin, Joseph. On the growth of Sobolev norms of solutions of the fractional defocusing NLS equation on the circle. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 509-531. doi : 10.1016/j.anihpc.2016.02.002. http://www.numdam.org/articles/10.1016/j.anihpc.2016.02.002/

[1] Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geom. Funct. Anal., Volume 3 (1993) no. 3, pp. 209–262 | DOI | MR | Zbl

[2] Bourgain, J. On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Int. Math. Res. Not. (1996) no. 6, pp. 277–304 | DOI | MR | Zbl

[3] Bourgain, J. Problems in Hamiltonian PDE's, Geom. Funct. Anal., Volume Special Volume (2000) no. Part I, pp. 32–56 GAFA 2000 (Tel Aviv, 1999) | MR

[4] Brezis, H.; Gallouët, T. Nonlinear Schrödinger evolution equations, Nonlinear Anal., Volume 4 (1980) no. 4, pp. 677–681 | DOI | MR | Zbl

[5] Burq, N.; Gérard, P.; Tzvetkov, N. Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math., Volume 159 (2005) no. 1, pp. 187–223 | DOI | MR | Zbl

[6] Burq, N.; Gérard, P.; Tzvetkov, N. Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. Sci. Éc. Norm. Super., Volume 38 (2005), pp. 255–301 | Numdam | MR | Zbl

[7] Demirbas, S.; Erdoğan, M.B.; Tzirakis, N. Existence and Uniqueness theory for the fractional Schrödinger equation on the torus, 2013 (preprint) | arXiv | MR

[8] Gérard, P.; Grellier, S. The cubic Szegő equation, Ann. Sci. Éc. Norm. Super. (4), Volume 43 (2010) no. 5, pp. 761–810 | Numdam | MR | Zbl

[9] Gérard, P.; Grellier, S. Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDE, Volume 5 (2012) no. 5, pp. 1139–1155 | DOI | MR | Zbl

[10] Gérard, P.; Grellier, S. On the growth of Sobolev norms for the cubic Szegő equation, Séminaire Laurent Schwartz — EDP et applications, 2014 (Exp. No. 11, 20 p) | MR

[11] Gérard, P.; Grellier, S. The cubic Szegő equation and Hankel operators, 2015 (preprint) | arXiv | MR | Zbl

[12] Ginibre, J. Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), Séminaire Bourbaki, Vol. 1994/95, Exp. No. 796, 4, Astérisque, vol. 237, 1996, pp. 163–187 | Numdam | MR | Zbl

[13] Gottwald, G.; Grimshaw, R.; Malomed, B.A. Stable two-dimensional parametric solitons in fluid systems, Phys. Lett. A, Volume 248 (1998) no. 2, pp. 208–218

[14] Grébert, B.; Kappeler, T. The Defocusing NLS Equation and Its Normal Form, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2014 | DOI | MR | Zbl

[15] Hani, Z.; Pausader, B.; Tzvetkov, N.; Visciglia, N. Modified scattering for the cubic Schrödinger equation on product spaces and applications, 2013 (preprint) | arXiv | MR

[16] Hayashi, N.; Ozawa, T.; Tanaka, K. On a system of nonlinear Schrödinger equations with quadratic interaction, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 30 (2013), pp. 661–690 | Numdam | MR | Zbl

[17] Kappeler, T.; Schaad, B.; Topalov, P. Scattering-like phenomena of the periodic defocusing NLS equation, 2015 (preprint) | arXiv | MR | Zbl

[18] Kenig, C.E.; Ponce, G.; Vega, L. Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Commun. Pure Appl. Math., Volume 46 (1993) no. 4, pp. 527–620 | DOI | MR | Zbl

[19] Kivshar, Yu.S. Bright and dark spatial solitons in non-Kerr media, Opt. Quantum Electron., Volume 30 (1998) no. 7, pp. 571–614

[20] Majda, A.J.; McLaughlin, D.W.; Tabak, E.G. A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci., Volume 7 (1997) no. 1, pp. 9–44 | DOI | MR | Zbl

[21] Ozawa, T.; Visciglia, N. An improvement on the Brezis–Gallouët technique for 2D NLS and 1D half-wave equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire (2015) | DOI | Numdam | MR | Zbl

[22] Staffilani, G. On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J., Volume 86 (1997) no. 1, pp. 109–142 | DOI | MR | Zbl

[23] Stein, E.M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993 (With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III) | MR | Zbl

[24] Tsutsumi, M. On smooth solutions to the initial-boundary value problem for the nonlinear Schrödinger equation in two space dimensions, Nonlinear Anal., Theory Methods Appl., Volume 13 (1989) no. 9, pp. 1051–1056 | DOI | MR | Zbl

[25] Xu, H. Large time blowup for a perturbation of the cubic Szegő equation, Anal. PDE, Volume 7 (2014) no. 3, pp. 717–731 | MR | Zbl

[26] Xu, H. Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation, 2015 (preprint) | arXiv | MR

[27] Zakharov, V.E.; Shabat, A.B. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Zh. Èksp. Teor. Fiz., Volume 61 (1971) no. 1, pp. 118–134 | MR

Cité par Sources :