Center for diffusion of mathematic journals

 
 
 
 

Séminaire Laurent Schwartz — EDP et applications

Table of contents for this volume | Previous article | Next article
Patrick Gérard; Sandrine Grellier
On the growth of Sobolev norms for the cubic Szegő equation
Séminaire Laurent Schwartz — EDP et applications (2014-2015), Exp. No. 11, 20 p., doi: 10.5802/slsedp.70
Article PDF

Résumé - Abstract

We report on a recent result establishing that trajectories of the cubic Szegő equation in Sobolev spaces with high regularity are generically unbounded, and moreover that, on solutions generated by suitable bounded subsets of initial data, every polynomial bound in time fails for high Sobolev norms. The proof relies on an instability phenomenon for a new nonlinear Fourier transform describing explicitly the solutions to the initial value problem, which is inherited from the Lax pair structure enjoyed by the equation.

Bibliography

[1] Adamyan, V. M., Arov, D. Z., Krein, M. G., Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem. (Russian) Mat. Sb. (N.S.) 86(128) (1971), 34–75; English transl. Math USSR. Sb. 15 (1971), 31–73.  MR 298453 |  Zbl 0248.47019
[2] Bourgain, J., Problems in Hamiltonian PDE’s, Geom. Funct. Anal. (2000), Special Volume, Part I, 32–56.  MR 1826248 |  Zbl 1050.35016
[3] Bourgain, J., On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices, (1996), 277-304.  MR 1386079 |  Zbl 0934.35166
[4] Burq, N., Gérard, P., Tzvetkov, N.: Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds Amer. J. Math. 126, 569–605 (2004).  MR 2058384 |  Zbl 1067.58027
[5] Colliander J., Keel M., Staffilani G., Takaoka H., Tao, T., Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrodinger equation, Inventiones Math.181 (2010), 39–113.  MR 2651381 |  Zbl 1197.35265
[6] Dodson, B., Global well-posedness and scattering for the defocusing, $L^2$ -critical, nonlinear Schrödinger equation when $d= 2$, arXiv:1006.1375, preprint, 2011.  MR 2869023 |  Zbl 1236.35163
[7] Gérard, P., Grellier, S., The cubic Szegő equation , Ann. Scient. Éc. Norm. Sup. 43 (2010), 761–810. Numdam |  MR 2721876 |  Zbl 1228.35225
[8] Gérard, P., Grellier, S., Invariant Tori for the cubic Szegő equation, Invent. Math. 187 (2012), 707–754.  MR 2944951 |  Zbl 1252.35026
[9] Gérard, P., Grellier, S., Inverse spectral problems for compact Hankel operators, J. Inst. Math. Jussieu 13 (2014), 273–301.  MR 3177280
[10] Gérard, P., Grellier, S., An explicit formula for the cubic Szegő equation, to appear in Trans. A.M.S.
[11] Gérard, P., Grellier, S., Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDEs 5 (2012), 1139–1155.  MR 3022852 |  Zbl 1268.35013
[12] Gérard, P., Pushnitski, A., An inverse problem for self-adjoint positive Hankel operators, arXiv:1401.2042, to appear in IMRN.
[13] Ginibre, J., Velo, G. Scattering theory in the energy space for a class of nonlinear Schršdinger equations, J. Math. Pures Appl. (9) 64 (1985), no. 4, 363–401.  MR 839728 |  Zbl 0535.35069
[14] Guardia, M., Kaloshin, V., Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation, arXiv:1205.5188[math.AP], to appear in J. European Math. Soc.  MR 3312404
[15] Grébert, B., Kappeler, T., The defocusing NLS equation and Its Normal Form, EMS series of Lectures in Mathematics, European Mathematical Society, 2014.  MR 3203027
[16] Hani, Z., Long-time strong instability and unbounded orbits for some periodic nonlinear Schrödinger equations, to appear Acce in Archives for Rational Mechanics and Analysis. arXiv:1210.7509.
[17] Hani, Z., Pausader, B., Tzvetkov, N., Visciglia, N., Modified scattering for the cubic Schrödinger equations on product spaces and applications, arXiv:1311.2275, 2013.
[18] Hernández B. A., Frías-Armenta M. E., Verduzco F., On differential structures of polynomial spaces in control theory, Journal of Systems Science and Systems Engineering 21 (2012), 372–382.
[19] Kappeler, T., Pöschel, J., KdV & KAM, A Series of Modern Surveys in Mathematics, vol. 45, Springer-Verlag, 2003.  MR 1997070 |  Zbl 1187.35237
[20] Killip, R., Tao, T., Visan, M., The cubic nonlinear Schršdinger equation in two dimensions with radial data. J. Eur. Math. Soc. 11 (2009), 1203–1258.  MR 2557134 |  Zbl 0162.41103
[21] Lax, P. : Integrals of Nonlinear equations of Evolution and Solitary Waves, Comm. Pure and Applied Math. 21, 467–490 (1968).  MR 235310 |  Zbl 0882.76035
[22] Majda, A., Mc Laughlin, D., Tabak, E., A one dimensional model for dispersive wave turbulence, J. Nonlinear Sci. 7 (1997) 9–44.  MR 1431687 |  Zbl 1007.47001
[23] Nikolskii, N. K., Operators, functions, and systems: an easy reading. Vol. 1. Hardy, Hankel, and Toeplitz. Translated from the French by Andreas Hartmann. Mathematical Surveys and Monographs, 92. American Mathematical Society, Providence, RI, 2002.  MR 1864396 |  Zbl 1030.47002
[24] Peller, V.V., Hankel Operators and their applications Springer Monographs in Mathematics. Springer-Verlag, New York, 2003.  MR 1949210 |  Zbl 1235.35263
[25] Pocovnicu, O. Explicit formula for the solution of the Szegő equation on the real line and applications, Discrete Cont. Dyn. Syst. 31 (2011), 607–649.  MR 2825631 |  Zbl 1270.65060
[26] Pocovnicu, O. First and second order approximations of a nonlinear wave equation J. Dynam. Differential Equations, article no. 9286 (2013), 29 pp, DOI:10.1007/s10884-013-9286-5.  MR 3054639 |  Zbl 1160.35067
[27] Ryckman, E., Visan, M., Global well-posedness and scattering for the defocusing energy-critical nonlinear Schršdinger equation in $\mathbb{R}^{1+4}$. Amer. J. Math. 129 (2007), 1–60.  MR 2288737 |  Zbl 0874.35114
[28] Staffilani, G., On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J., 86 (1997), 109-142.  MR 1427847
[29] Thirouin, J., work in preparation.
[30] Xu, H., Large time blow up for a perturbation of the cubic Szegő equation, Anal. PDE, 7 (2014), No. 3, 717–731.  MR 3227431
[31] Zakharov, V. E., Shabat, A. B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Physics JETP 34 (1972), no. 1, 62–69.  MR 406174 |  Zbl 0986.76032
[32] Zakharov, V., Guyenne, P., Pushkarev, A., Dias, F., Wave turbulence in one-dimensional models, Physica D 152–153 (2001) 573–619.  MR 1837930
Copyright Cellule MathDoc 2017 | Credit | Site Map