Geodesic
MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS
Forum of Mathematics, Pi, Tome 3 (2015)

Voir la notice de l'article provenant de la source Cambridge University Press

We consider the cubic nonlinear Schrödinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^{d}$. We prove modified scattering and construct modified wave operators for small initial and final data respectively ($1\leqslant d\leqslant 4$). The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when $d\geqslant 2$. As a consequence, we obtain global strong solutions (for $d\geqslant 2$) with infinitely growing high Sobolev norms $H^{s}$. 
@article{10_1017_fmp_2015_5,
     author = {ZAHER HANI and BENOIT PAUSADER and NIKOLAY TZVETKOV and NICOLA VISCIGLIA},
     title = {MODIFIED {SCATTERING} {FOR} {THE} {CUBIC} {SCHR\"ODINGER} {EQUATION} {ON} {PRODUCT} {SPACES} {AND} {APPLICATIONS}},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {3},
     year = {2015},
     doi = {10.1017/fmp.2015.5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2015.5/}
}
TY  - JOUR
AU  - ZAHER HANI
AU  - BENOIT PAUSADER
AU  - NIKOLAY TZVETKOV
AU  - NICOLA VISCIGLIA
TI  - MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS
JO  - Forum of Mathematics, Pi
PY  - 2015
VL  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2015.5/
DO  - 10.1017/fmp.2015.5
LA  - en
ID  - 10_1017_fmp_2015_5
ER  - 
%0 Journal Article
%A ZAHER HANI
%A BENOIT PAUSADER
%A NIKOLAY TZVETKOV
%A NICOLA VISCIGLIA
%T MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS
%J Forum of Mathematics, Pi
%D 2015
%V 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2015.5/
%R 10.1017/fmp.2015.5
%G en
%F 10_1017_fmp_2015_5
ZAHER HANI; BENOIT PAUSADER; NIKOLAY TZVETKOV; NICOLA VISCIGLIA. MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS. Forum of Mathematics, Pi, Tome 3 (2015). doi: 10.1017/fmp.2015.5

[1] Alazard, T. and Delort, J. M., ‘Global solutions and asymptotic behavior for two dimensional gravity water waves’, Preprint, 2013, arXiv:1305.4090.Google Scholar | DOI

[2] Antonelli, P., Carles, R. and Silva, J. D., ‘Scattering for nonlinear Schrödinger equation under partial harmonic confinement’, Preprint, 2013, arXiv:1310.1352.Google Scholar | DOI

[3] Banica, V., Carles, R. and Duyckaerts, T., ‘On scattering for NLS: from Euclidean to hyperbolic space’, Discrete Contin. Dyn. Syst. 24(4) (2009), 1113–1127.Google Scholar | DOI

[4] Bourgain, J., ‘Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations’, Geom. Funct. Anal. 3 (1993), 107–156.Google Scholar | DOI

[5] Bourgain, J., ‘Exponential sums and nonlinear Schrödinger equations’, Geom. Funct. Anal. 3 (1993), 157–178.Google Scholar | DOI

[6] Bourgain, J., ‘Periodic nonlinear Schrödinger equation and invariant measures’, Comm. Math. Phys. 166 (1994), 1–26.Google Scholar | DOI

[7] Bourgain, J., ‘Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations’, Geom. Funct. Anal. 5(2) (1995), 105–140.Google Scholar | DOI

[8] Bourgain, J., ‘On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE’, Int. Math. Res. Not. IMRN 6 (1996), 277–304.Google Scholar | DOI

[9] Bourgain, J., ‘Invariant measures for the 2d-defocusing nonlinear Schrödinger equation’, Comm. Math. Phys. 176 (1996), 421–445.Google Scholar | DOI

[10] Bourgain, J., ‘On growth in time of Sobolev norms of smooth solutions of nonlinear Schrödinger equations in ℝD’, J. Anal. Math. 72 (1997), 299–310.Google Scholar | DOI

[11] Bourgain, J., ‘Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations’, Ann. of Math. (2) 148(2) (1998), 363–439.Google Scholar | DOI

[12] Bourgain, J., ‘Refinements of Strichartz inequality and applications to 2D-NLS with critical nonlinearity’, Int. Math. Res. Not. IMRN (1998), 253–283.Google Scholar | DOI

[13] Bourgain, J., ‘Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case’, J. Amer. Math. Soc. 12 (1999), 145–171.Google Scholar | DOI

[14] Bourgain, J., ‘Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential’, Comm. Math. Phys. 204(1) (1999), 207–247.Google Scholar | DOI

[15] Bourgain, J., ‘On growth of Sobolev norms in linear Schrödinger equations with smooth, time-dependent potential’, J. Anal. Math. 77 (1999), 315–348.Google Scholar | DOI

[16] Bourgain, J., ‘Problems in Hamiltonian PDE’s’, Geom. Funct. Anal. 2000 (2000), 32–56 (special volume, Part I).Google Scholar

[17] Bourgain, J., ‘Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations’, Ergod. Th. & Dynam. Syst. 24(5) (2004), 1331–1357.Google Scholar | DOI

[18] Bourgain, J., ‘Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces’, Israel J. Math. 193(1) (2013), 441–458.Google Scholar | DOI

[19] Bourgain, J. and Bulut, A., ‘Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3D case’, J. Eur. Math. Soc. 16 (2014), 1289–1325.Google Scholar | DOI

[20] Brézis, H. and Gallouët, T., ‘Nonlinear Schrödinger evolution equations’, Nonlinear Anal. Theory Methods Appl. 4 (1980), 677–681.Google Scholar | DOI

[21] Burq, N., Gérard, P. and Tzvetkov, N., ‘An instability property of the nonlinear Schrödinger equation on Sd’, Math. Res. Lett. 9 (2002), 323–335.Google Scholar | DOI

[22] Burq, N., Gérard, P. and Tzvetkov, N., ‘Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds’, Amer. J. Math. 126 (2004), 569–605.Google Scholar | DOI

[23] Burq, N., Gérard, P. and Tzvetkov, N., ‘Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces’, Invent. Math. 159 (2005), 187–223.Google Scholar | DOI

[24] Burq, N., Gérard, P. and Tzvetkov, N., ‘Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations’, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), 255–301.Google Scholar | DOI

[25] Carles, R., ‘Geometric optics and long range scattering for one-dimensional nonlinear Schrödinger equations’, Commun. Math. Phys. 220(1) (2001), 41–67.Google Scholar | DOI

[26] Carles, R. and Faou, E., ‘Energy cascade for NLS on the torus’, Discrete Contin. Dyn. Syst. 32(6) (2012), 2063–2077.Google Scholar | DOI

[27] Cazenave, T., Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10 (New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003).Google Scholar | DOI

[28] Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T., ‘Global well-posedness for Schrödinger equations with derivative’, SIAM J. Math. Anal. 33 (2001), 649–669.Google Scholar | DOI

[29] Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T., ‘Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ℝ3’, Ann. of Math. (2) 167 (2008), 767–865.Google Scholar | DOI

[30] Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T., ‘Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation’, Invent. Math. 181(1) (2010), 39–113.Google Scholar | DOI

[31] Colliander, J., Kwon, S. and Oh, T., ‘A remark on normal forms and the ‘upside-down’ I-method for periodic NLS: growth of higher Sobolev norms’, J. Anal. Math 118(1) (2012), 55–82.Google Scholar | DOI

[32] Deift, P. and Zhou, X., ‘Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space, Dedicated to the memory of Jürgen K. Moser’, Comm. Pure Appl. Math. 56(8) (2003), 1029–1077.Google Scholar | DOI

[33] Delort, J.-M., ‘Existence globale et comportement asymptotique pour l’équation de Klein–Gordon quasi linéaire à données petites en dimension 1’, Ann. Sci. Éc. Norm. Supér. (4) 34(1) (2001), 1–61.Google Scholar | DOI

[34] Dodson, B., ‘Global well-posedness and scattering for the defocusing, -critical, nonlinear Schrd̈inger equation when ’, Preprint, 2010, arXiv:1010.0040.Google Scholar

[35] Dodson, B., ‘Global well-posedness and scattering for the defocusing, -critical, nonlinear Schrödinger equation when ’, Preprint, 2010, arXiv:1006.1375.Google Scholar

[36] Dodson, B., ‘Global well-posedness and scattering for the defocusing, energy-critical, nonlinear Schrödinger equation in the exterior of a convex obstacle when ’, Preprint, 2011, arXiv:1112.0710.Google Scholar

[37] Eliasson, L. H. and Kuksin, S., ‘KAM for the nonlinear Schrödinger equation’, Ann. of Math. (2) 172(1) (2010), 371–435.Google Scholar | DOI

[38] Faou, E., Germain, P. and Hani, Z., ‘The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation’, J. Amer. Math. Soc. (JAMS), to appear, Preprint, 2013, arXiv:1308.6267.Google Scholar

[39] Gérard, P. and Grellier, S., ‘The Szegö cubic equation’, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 761–809.Google Scholar | DOI

[40] Gérard, P. and Grellier, S., ‘An explicit formula for the cubic Szegö equation’, Trans. Amer. Math. Soc. 367 (2015), 2979–2995.Google Scholar | DOI

[41] Gérard, P. and Grellier, S., ‘Effective integrable dynamics for some nonlinear wave equation’, Anal. PDE 5 (2012), 1139–1155.Google Scholar | DOI

[42] Germain, P., Masmoudi, N. and Shatah, J., ‘Global solutions for 3D quadratic Schrödinger equations’, Int. Math. Res. Not. IMRN (2009), 414–432.Google Scholar

[43] Germain, P., Masmoudi, N. and Shatah, J., ‘Global solutions for the gravity water waves equation in dimension 3’, Ann. of Math. (2) 175 (2012), 691–754.Google Scholar | DOI

[44] Grebert, B., Paturel, E. and Thomann, L., ‘Beating effects in cubic Schrödinger systems and growth of Sobolev norms’, Nonlinearity 26 (2013), 1361–1376.Google Scholar | DOI

[45] Guardia, M. and Kaloshin, V., ‘Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation’, J. Eur. Math. Soc. 17(1) (2015), 71–149.Google Scholar | DOI

[46] Guardia, M. and Kaloshin, V., ‘Erratum to “Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation”’, personal communication.Google Scholar

[47] Guo, S., ‘On the 1D cubic NLS in an almost critical space’, Master Thesis, University of Bonn (2012).Google Scholar

[48] Guo, Y., Ionescu, A. and Pausader, B., ‘The Euler–Maxwell 2 fluid in ’, Preprint, 2013, arXiv:1303.1060.Google Scholar

[49] Hani, Z., ‘Global well-posedness of the cubic nonlinear Schrödinger equation on compact manifolds without boundary’, Comm. Partial Differential Equations 37(7) (2012), 1186–1236.Google Scholar | DOI

[50] Hani, Z., ‘Long-time strong instability and unbounded orbits for some periodic nonlinear Schödinger equations’, Arch. Ration. Mech. Anal. 211(3) (2014), 929–964.Google Scholar | DOI

[51] Hani, Z. and Pausader, B., ‘On scattering for the quintic defocusing nonlinear Schrödinger equation on ℝ ×T2’, Comm. Pure Appl. Math. 67(9) (2014), 1466–1542.Google Scholar | DOI

[52] Hani, Z. and Thomann, L., ‘Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping’, Commun. Pure Appl. Math. (CPAM) Preprint, 2014, arXiv:1408.6213, published online: 15 July 2015, doi:.Google Scholar | DOI

[53] Hayashi, N., Li, C. and Naumkin, P., ‘Modified wave operator for a system of nonlinear Schrödinger equations in 2d’, Comm. Partial Differential Equations 37(6) (2012), 947–968.Google Scholar | DOI

[54] Hayashi, N. and Naumkin, P., ‘Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations’, Amer. J. Math. 120(2) (1998), 369–389.Google Scholar | DOI

[55] Hayashi, N., Naumkin, P., Shimomura, A. and Tonegawa, S., ‘Modified wave operators for nonlinear Schrödinger equations in one and two dimensions’, Electron. J. Differential Equations 62 (2004), 16 pp.Google Scholar

[56] Herr, S., ‘The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds’, Amer. J. Math. 135(5) (2013), 1271–1290.Google Scholar | DOI

[57] Herr, S., Tataru, D. and Tzvetkov, N., ‘Global well-posedness of the energy critical nonlinear Schrödinger equation with small initial data in H 1(T3)’, Duke Math. J. 159 (2011), 329–349.Google Scholar | DOI

[58] Herr, S., Tataru, D. and Tzvetkov, N., ‘Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d and applications’, J. Ang. Math. 2014(690) (2012), 65–78.Google Scholar

[59] Ionescu, A. D. and Pausader, B., ‘Global wellposedness of the energy-critical defocusing NLS on ℝ ×T3’, Commun. Math. Phys. 312(3) (2012), 781–831.Google Scholar | DOI

[60] Ionescu, A. D. and Pausader, B., ‘The energy-critical defocusing NLS on T3’, Duke Math. J. 161(8) (2012), 1581–1612.Google Scholar | DOI

[61] Ionescu, A. D., Pausader, B. and Staffilani, G., ‘On the global well-posedness of energy-critical Schrödinger equations in curved spaces’, Anal. PDE 5(4) (2012), 705–746.Google Scholar | DOI

[62] Ionescu, A. and Pausader, B., ‘Global solutions of quasilinear systems of Klein–Gordon equations in 3D’, J. Eur. Math. Soc. 16(11) (2014), 2355–2431.Google Scholar | DOI

[63] Ionescu, A. D. and Pusateri, F., ‘Nonlinear fractional Schrödinger equations in one dimensions’, J. Funct. Anal. 266(1) (2014), 139–176.Google Scholar | DOI

[64] Ionescu, A. D. and Pusateri, F., ‘Global solutions for the gravity water waves system in 2d’, Invent. Math. 199(3) (2015), 653–804.Google Scholar | DOI

[65] Ionescu, A. D. and Staffilani, G., ‘Semilinear Schrödinger flows on hyperbolic spaces: scattering in H 1’, Math. Ann. 345 (2009), 133–158.Google Scholar | DOI

[66] Kato, J. and Pusateri, F., ‘A new proof of long range scattering for critical nonlinear Schrödinger equations’, J. Diff. Int. Equ. 24(9–10) (2011).Google Scholar

[67] Kenig, C. E. and Merle, F., ‘Global well-posedness, scattering and blow-up for the energy-critical, focusing, nonlinear Schrödinger equation in the radial case’, Invent. Math. 166 (2006), 645–675.Google Scholar | DOI

[68] Killip, R., Tao, T. and Visan, M., ‘The cubic nonlinear Schrödinger equation in two dimensions with radial data’, J. Eur. Math. Soc. 11 (2009), 1203–1258.Google Scholar | DOI

[69] Killip, R. and Visan, M., ‘Global well-posedness and scattering for the defocusing quintic NLS in three dimensions’, Anal. PDE 5 (2012), 855–885.Google Scholar | DOI

[70] Killip, R., Visan, M. and Zhang, X., ‘Quintic NLS in the exterior of a strictly convex obstacle’, Preprint, 2012, arXiv:1208.4904.Google Scholar

[71] Kuksin, S., ‘Oscillations in space-periodic nonlinear Schrödinger equations’, Geom. Funct. Anal. 7(2) (1997), 338–363.Google Scholar | DOI

[72] Kuksin, S. and Pöschel, J., ‘Invariant Cantor manifolds of quasi periodic oscillations for a nonlinear Schrödinger equation’, Ann. of Math. (2) 143 (1996), 149–179.Google Scholar | DOI

[73] Majda, A., Mclaughlin, D. and Tabak, E., ‘A one-dimensional model for dispersive wave turbulence’, J. Nonlinear Sci. 7(1) (1997), 9–44.Google Scholar | DOI

[74] Ozawa, T., ‘Long range scattering for nonlinear Schrödinger equations in one space dimension’, Commun. Math. Phys. 139 (1991), 479–493.Google Scholar | DOI

[75] Pausader, B., Tzvetkov, N. and Wang, X., ‘Global regularity for the energy-critical NLS on S3’, Ann. Inst. H. Poincaré Anal. Non Linéaire 31(2) (2014), 315–338.Google Scholar

[76] Pocovnicu, O., ‘Explicit formula for the solution of the Szegö equation on the real line and applications’, Discrete Contin. Dyn. Syst. 31 (2011), 607–649.Google Scholar | DOI

[77] Pocovnicu, O., ‘First and second order approximations for a nonlinear wave equation’, J. Dynam. Differential Equations 25(2) (2013), 305–333. 29.Google Scholar | DOI

[78] Procesi, M. and Procesi, C., ‘A KAM algorithm for the resonant nonlinear Schrödinger equation’, Preprint arXiv:1211.4242.Google Scholar

[79] Ryckman, E. and Visan, M., ‘Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in ℝ1+4’, Amer. J. Math. 129 (2007), 1–60.Google Scholar | DOI

[80] Sohinger, V., ‘Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on S 1’, Differential Integral Equations 24(7–8) (2011), 653–718.Google Scholar | DOI

[81] Staffilani, G., ‘On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations’, Duke Math. J. 86(1) (1997), 109–142.Google Scholar | DOI

[82] Takaoka, H. and Tzvetkov, N., ‘On 2D Nonlinear Schrödinger equations with data on ℝ×T’, J. Funct. Anal. 182 (2001), 427–442.Google Scholar | DOI

[83] Terracini, S., Tzvetkov, N. and Visciglia, N., ‘The NLS ground states on product spaces’, Anal. PDE 7(1) (2014), 73–96.Google Scholar | DOI

[84] Tzvetkov, N., ‘Invariant measures for the defocusing NLS’, Ann. Inst. Fourier 58 (2008), 2543–2604.Google Scholar | DOI

[85] Tzvetkov, N. and Visciglia, N., ‘Small data scattering for the nonlinear Schrödinger equation on product spaces’, Comm. Partial Differential Equations 37(1) (2012), 125–135.Google Scholar | DOI

[86] Tzvetkov, N. and Visciglia, N., ‘Well-posedness and scattering for NLS on in the energy space’, Preprint, 2014, arXiv:1409.3938.Google Scholar

[87] Visan, M., ‘Global well-posedness and scattering for the defocusing cubic NLS in four dimensions’, Int. Math. Res. Not. IMRN 2011 (2011), doi:.Google Scholar | DOI

[88] Xu, H., ‘Large time blow up for a perturbation of the cubic Szegö equation’, Preprint, 2013, arXiv:1307.5284.Google Scholar

[89] Zakharov, V. E. and Shabat, A. B., ‘Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media’, Sov. Phys. JEPT 34 (1972), 62–69.Google Scholar

[90] Zakharov, V. E., L’Vov, V. and Falkovich, G., Kolmogorov Spectra of Turbulence 1. Wave Turbulence, Springer Series in Nonlinear Dynamics, (Springer, Berlin, Heidelberg, 1992).Google Scholar | DOI

Cité par Sources :