Long time dynamics for the one dimensional non linear Schrödinger equation

[Dynamiques en temps long pour l’équation de Schrödinger non linéaire]

Burq, Nicolas ; Thomann, Laurent ; Tzvetkov, Nikolay

Annales de l'Institut Fourier, Tome 63 (2013), p. 2137-2198 / Harvested from Numdam

Résumé

Nous présentons d’abord dans cet article la construction de mesures de Gibbs pour l’équation de Schrödinger non linéaire associée à un potentiel harmonique. Nous démontrons ensuite que le problème de Cauchy correspondant est globalement bien posé pour des données initiales très peu régulières (sur le support de cette mesure). Finalement, nous démontrons aussi que ces mesures de Gibbs sont invariantes par le flot ainsi défini. Nous obtenons comme conséquence de cette approche que l’équation de Schrödinger non linéaire $L^2$-critique et surcritique sur $\mathbb{R}$ (sans potentiel harmonique) est globalement bien posée et diffuse pour ces données initiales.

Abstract

In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the $L^2$ critical and super-critical NLS (without harmonic potential).

Détail

Publié le : 2013-01-01

MR 3237443 | Zbl 06325429 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2825
Classification:  35BXX,  37K05,  37L50,  35Q55
Mots clés: Equation de Schrödinger non linéaire, données aléatoires, mesure de Gibbs, mesures invariants, solutions globales

Bibliographie

[1] Ayache, Antoine; Tzvetkov, Nikolay $L^p$ properties for Gaussian random series, Trans. Amer. Math. Soc., Tome 360 (2008) no. 8, pp. 4425-4439 | Article | MR 2395179 | Zbl 1145.60019

[2] Betounes, David Differential equations: theory and applications, Springer, New York (2010), pp. xiv+626 | Article | MR 2571569 | Zbl 1192.34001

[3] Bouclet, Jean-Marc Distributions spectrales pour des opérateurs perturbés (2000) (PhD Thesis, Nantes University)

[4] Bourgain, J. Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., Tome 166 (1994) no. 1, pp. 1-26 http://projecteuclid.org/getRecord?id=euclid.cmp/1104271501 | Article | MR 1309539 | Zbl 0822.35126

[5] Bourgain, Jean Invariant measures for the $2$D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., Tome 176 (1996) no. 2, pp. 421-445 http://projecteuclid.org/getRecord?id=euclid.cmp/1104286005 | Article | MR 1374420 | Zbl 0852.35131

[6] Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. Sci. École Norm. Sup. (4), Tome 38 (2005) no. 2, pp. 255-301 | Article | Numdam | MR 2144988 | Zbl 1116.35109

[7] Burq, Nicolas; Tzvetkov, Nikolay Invariant measure for a three dimensional nonlinear wave equation, Int. Math. Res. Not. IMRN (2007) no. 22, pp. Art. ID rnm108, 26 | Article | MR 2376217 | Zbl 1134.35076

[8] Burq, Nicolas; Tzvetkov, Nikolay Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., Tome 173 (2008) no. 3, pp. 449-475 | Article | MR 2425133 | Zbl 1156.35062

[9] Burq, Nicolas; Tzvetkov, Nikolay Random data Cauchy theory for supercritical wave equations. II. A global existence result, Invent. Math., Tome 173 (2008) no. 3, pp. 477-496 | Article | MR 2425134 | Zbl 1187.35233

[10] Carles, Rémi Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Models Methods Appl. Sci., Tome 12 (2002) no. 10, pp. 1513-1523 | Article | MR 1933935 | Zbl 1029.35208

[11] Carles, Rémi Rotating points for the conformal NLS scattering operator, Dyn. Partial Differ. Equ., Tome 6 (2009) no. 1, pp. 35-51 | Article | MR 2517827 | Zbl 1191.35270

[12] Carles, Rémi Nonlinear Schrödinger equation with time dependent potential. 9 (2011), no. 4, 937–964., Commun. Math. Sci., Tome 9 (2011) no. 4, pp. 937-964 | Article | MR 2901811 | Zbl 1285.35105

[13] Cazenave, Thierry Semilinear Schrödinger equations, New York University Courant Institute of Mathematical Sciences, New York, Courant Lecture Notes in Mathematics, Tome 10 (2003), pp. xiv+323 | MR 2002047 | Zbl 1055.35003

[14] Christ, Michael; Colliander, Terry Jamesand Tao Ill-posedness for nonlinear Schrödinger and wave equations (2011) (Annales IHP, to appear)

[15] Colliander, Jim; Oh, Tadahiro Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^{2}(\mathbb{T})$, Duke Math. Journal, Tome 161 (2012) no. 3, pp. 367-414 | Article | MR 2881226 | Zbl 1260.35199

[16] Dodson, Benjamin Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d = 1$ (Preprint, http://fr.arxiv.org/abs/1010.0040)

[17] Dziubański, Jacek; Głowacki, Paweł Sobolev spaces related to Schrödinger operators with polynomial potentials, Math. Z., Tome 262 (2009) no. 4, pp. 881-894 | Article | MR 2511755 | Zbl 1177.47055

[18] Fukuizumi, Reika Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential, Discrete Contin. Dynam. Systems, Tome 7 (2001) no. 3, pp. 525-544 | Article | MR 1815766 | Zbl 0992.35094

[19] Ginibre, J.; Velo, G. The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, Tome 2 (1985) no. 4, pp. 309-327 | Numdam | MR 801582 | Zbl 0586.35042

[20] Hörmander, Lars The analysis of linear partial differential operators. III, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 274 (1985), pp. viii+525 (Pseudodifferential operators) | MR 781536 | Zbl 0601.35001

[21] Killip, Rowan; Tao, Terence; Visan, Monica The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS), Tome 11 (2009) no. 6, pp. 1203-1258 | Article | MR 2557134 | Zbl 1187.35237

[22] Koch, Herbert; Tataru, Daniel $L^p$ eigenfunction bounds for the Hermite operator, Duke Math. J., Tome 128 (2005) no. 2, pp. 369-392 | Article | MR 2140267 | Zbl 1075.35020

[23] Lebowitz, Joel L.; Rose, Harvey A.; Speer, Eugene R. Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys., Tome 50 (1988) no. 3-4, pp. 657-687 | Article | MR 939505 | Zbl 1084.82506

[24] Nakanishi, Kenji Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions $1$ and $2$, J. Funct. Anal., Tome 169 (1999) no. 1, pp. 201-225 | Article | MR 1726753 | Zbl 0942.35159

[25] Niederer, Ulrich The maximal kinematical invariance groups of the harmonic oscillator, Helv. Phys. Acta, Tome 46 (1973), pp. 191-200

[26] Oh, Tadahiro Invariant Gibbs measures and a.s. global well posedness for coupled KdV systems, Differential Integral Equations, Tome 22 (2009) no. 7-8, pp. 637-668 | MR 2532115 | Zbl 1240.35477

[27] Oh, Tadahiro Invariance of the Gibbs measure for the Schrödinger-Benjamin-Ono system, SIAM J. Math. Anal., Tome 41 (2009/10) no. 6, pp. 2207-2225 | Article | MR 2579711 | Zbl 1205.35268

[28] Oh, Yong-Geun Cauchy problem and Ehrenfest’s law of nonlinear Schrödinger equations with potentials, J. Differential Equations, Tome 81 (1989) no. 2, pp. 255-274 | Article | MR 1016082 | Zbl 0703.35158

[29] Parmeggiani, Alberto Spectral theory of non-commutative harmonic oscillators: an introduction, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1992 (2010), pp. xii+254 | Article | MR 2650633 | Zbl 1200.35346

[30] Robert, Didier Autour de l’approximation semi-classique, Birkhäuser Boston Inc., Boston, MA, Progress in Mathematics, Tome 68 (1987), pp. x+329 | MR 897108 | Zbl 0621.35001

[31] Rybin, A. V.; Varzugin, G. G.; Lindberg, M.; Timonen, J.; Bullough, R. K. Similarity solutions and collapse in the attractive Gross-Pitaevskii equation, Phys. Rev. E (3), Tome 62 (2000) no. 5, part A, pp. 6224-6228 | Article | MR 1796440

[32] Tao, Terence Nonlinear dispersive equations, Published for the Conference Board of the Mathematical Sciences, Washington, DC, CBMS Regional Conference Series in Mathematics, Tome 106 (2006), pp. xvi+373 (Local and global analysis) | MR 2233925 | Zbl 1106.35001

[33] Tao, Terence A pseudoconformal compactification of the nonlinear Schrödinger equation and applications, New York J. Math., Tome 15 (2009), pp. 265-282 http://nyjm.albany.edu:8000/j/2009/15_265.html | MR 2530148 | Zbl 1184.35296

[34] Thomann, Laurent Random data Cauchy problem for supercritical Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Tome 26 (2009) no. 6, pp. 2385-2402 | Article | Numdam | MR 2569900 | Zbl 1180.35491

[35] Thomann, Laurent A remark on the Schrödinger smoothing effect, Asymptot. Anal., Tome 69 (2010) no. 1-2, pp. 117-123 | MR 2732195 | Zbl 1208.35026

[36] Tzvetkov, N. Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation, Probab. Theory Related Fields, Tome 146 (2010) no. 3-4, pp. 481-514 | Article | MR 2574736 | Zbl 1188.35183

[37] Tzvetkov, Nikolay Invariant measures for the nonlinear Schrödinger equation on the disc, Dyn. Partial Differ. Equ., Tome 3 (2006) no. 2, pp. 111-160 | Article | MR 2227040 | Zbl 1142.35090

[38] Tzvetkov, Nikolay Invariant measures for the defocusing nonlinear Schrödinger equation, Ann. Inst. Fourier (Grenoble), Tome 58 (2008) no. 7, pp. 2543-2604 http://aif.cedram.org/item?id=AIF_2008__58_7_2543_0 | Article | Numdam | MR 2498359 | Zbl 1171.35116

[39] Yajima, Kenji; Zhang, Guoping Smoothing property for Schrödinger equations with potential superquadratic at infinity, Comm. Math. Phys., Tome 221 (2001) no. 3, pp. 573-590 | Article | MR 1852054 | Zbl 1102.35320

[40] Yajima, Kenji; Zhang, Guoping Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity, Sūrikaisekikenkyūsho Kōkyūroku (2002) no. 1255, pp. 183-204 (Spectral and scattering theory and related topics (Japanese) (Kyoto, 2001)) | MR 1927174 | Zbl 1060.35121

[41] Zhang, Jian Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential, Comm. Partial Differential Equations, Tome 30 (2005) no. 10-12, pp. 1429-1443 | Article | MR 2182299 | Zbl 1081.35109

[42] Zhidkov, Peter E. Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1756 (2001), pp. vi+147 | MR 1831831 | Zbl 0987.35001