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 critical and super-critical NLS (without harmonic potential).
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 -critique et surcritique sur (sans potentiel harmonique) est globalement bien posée et diffuse pour ces données initiales.
Keywords: Nonlinear Schrödinger equation, potential, random data, Gibbs measure, invariant measure, global solutions
Mot clés : Equation de Schrödinger non linéaire, données aléatoires, mesure de Gibbs, mesures invariants, solutions globales
@article{AIF_2013__63_6_2137_0, author = {Burq, Nicolas and Thomann, Laurent and Tzvetkov, Nikolay}, title = {Long time dynamics for the one dimensional non linear {Schr\"odinger} equation}, journal = {Annales de l'Institut Fourier}, pages = {2137--2198}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {63}, number = {6}, year = {2013}, doi = {10.5802/aif.2825}, zbl = {06325429}, mrnumber = {3237443}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2825/} }
TY - JOUR AU - Burq, Nicolas AU - Thomann, Laurent AU - Tzvetkov, Nikolay TI - Long time dynamics for the one dimensional non linear Schrödinger equation JO - Annales de l'Institut Fourier PY - 2013 SP - 2137 EP - 2198 VL - 63 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2825/ DO - 10.5802/aif.2825 LA - en ID - AIF_2013__63_6_2137_0 ER -
%0 Journal Article %A Burq, Nicolas %A Thomann, Laurent %A Tzvetkov, Nikolay %T Long time dynamics for the one dimensional non linear Schrödinger equation %J Annales de l'Institut Fourier %D 2013 %P 2137-2198 %V 63 %N 6 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2825/ %R 10.5802/aif.2825 %G en %F AIF_2013__63_6_2137_0
Burq, Nicolas; Thomann, Laurent; Tzvetkov, Nikolay. Long time dynamics for the one dimensional non linear Schrödinger equation. Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2137-2198. doi : 10.5802/aif.2825. http://www.numdam.org/articles/10.5802/aif.2825/
[1] properties for Gaussian random series, Trans. Amer. Math. Soc., Volume 360 (2008) no. 8, pp. 4425-4439 | DOI | MR | Zbl
[2] Differential equations: theory and applications, Springer, New York, 2010, pp. xiv+626 | DOI | MR | Zbl
[3] Distributions spectrales pour des opérateurs perturbés (2000) (PhD Thesis, Nantes University)
[4] Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., Volume 166 (1994) no. 1, pp. 1-26 http://projecteuclid.org/getRecord?id=euclid.cmp/1104271501 | DOI | MR | Zbl
[5] Invariant measures for the D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., Volume 176 (1996) no. 2, pp. 421-445 http://projecteuclid.org/getRecord?id=euclid.cmp/1104286005 | DOI | MR | Zbl
[6] Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. Sci. École Norm. Sup. (4), Volume 38 (2005) no. 2, pp. 255-301 | DOI | Numdam | MR | Zbl
[7] Invariant measure for a three dimensional nonlinear wave equation, Int. Math. Res. Not. IMRN (2007) no. 22, pp. Art. ID rnm108, 26 | DOI | MR | Zbl
[8] Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., Volume 173 (2008) no. 3, pp. 449-475 | DOI | MR | Zbl
[9] Random data Cauchy theory for supercritical wave equations. II. A global existence result, Invent. Math., Volume 173 (2008) no. 3, pp. 477-496 | DOI | MR | Zbl
[10] Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Models Methods Appl. Sci., Volume 12 (2002) no. 10, pp. 1513-1523 | DOI | MR | Zbl
[11] Rotating points for the conformal NLS scattering operator, Dyn. Partial Differ. Equ., Volume 6 (2009) no. 1, pp. 35-51 | DOI | MR | Zbl
[12] Nonlinear Schrödinger equation with time dependent potential. 9 (2011), no. 4, 937–964., Commun. Math. Sci., Volume 9 (2011) no. 4, pp. 937-964 | DOI | MR | Zbl
[13] Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, New York University Courant Institute of Mathematical Sciences, New York, 2003, pp. xiv+323 | MR | Zbl
[14] Ill-posedness for nonlinear Schrödinger and wave equations (2011) (Annales IHP, to appear)
[15] Almost sure well-posedness of the cubic nonlinear Schrödinger equation below , Duke Math. Journal, Volume 161 (2012) no. 3, pp. 367-414 | DOI | MR | Zbl
[16] Global well-posedness and scattering for the defocusing, -critical, nonlinear Schrödinger equation when (Preprint, http://fr.arxiv.org/abs/1010.0040)
[17] Sobolev spaces related to Schrödinger operators with polynomial potentials, Math. Z., Volume 262 (2009) no. 4, pp. 881-894 | DOI | MR | Zbl
[18] Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential, Discrete Contin. Dynam. Systems, Volume 7 (2001) no. 3, pp. 525-544 | DOI | MR | Zbl
[19] The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 2 (1985) no. 4, pp. 309-327 | EuDML | Numdam | MR | Zbl
[20] The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 274, Springer-Verlag, Berlin, 1985, pp. viii+525 (Pseudodifferential operators) | MR | Zbl
[21] The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS), Volume 11 (2009) no. 6, pp. 1203-1258 | DOI | MR | Zbl
[22] eigenfunction bounds for the Hermite operator, Duke Math. J., Volume 128 (2005) no. 2, pp. 369-392 | DOI | MR | Zbl
[23] Statistical mechanics of the nonlinear Schrödinger equation, J. Statist. Phys., Volume 50 (1988) no. 3-4, pp. 657-687 | DOI | MR | Zbl
[24] Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions and , J. Funct. Anal., Volume 169 (1999) no. 1, pp. 201-225 | DOI | MR | Zbl
[25] The maximal kinematical invariance groups of the harmonic oscillator, Helv. Phys. Acta, Volume 46 (1973), pp. 191-200
[26] Invariant Gibbs measures and a.s. global well posedness for coupled KdV systems, Differential Integral Equations, Volume 22 (2009) no. 7-8, pp. 637-668 | MR | Zbl
[27] Invariance of the Gibbs measure for the Schrödinger-Benjamin-Ono system, SIAM J. Math. Anal., Volume 41 (2009/10) no. 6, pp. 2207-2225 | DOI | MR | Zbl
[28] Cauchy problem and Ehrenfest’s law of nonlinear Schrödinger equations with potentials, J. Differential Equations, Volume 81 (1989) no. 2, pp. 255-274 | DOI | MR | Zbl
[29] Spectral theory of non-commutative harmonic oscillators: an introduction, Lecture Notes in Mathematics, 1992, Springer-Verlag, Berlin, 2010, pp. xii+254 | DOI | MR | Zbl
[30] Autour de l’approximation semi-classique, Progress in Mathematics, 68, Birkhäuser Boston Inc., Boston, MA, 1987, pp. x+329 | MR | Zbl
[31] Similarity solutions and collapse in the attractive Gross-Pitaevskii equation, Phys. Rev. E (3), Volume 62 (2000) no. 5, part A, pp. 6224-6228 | DOI | MR
[32] Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006, pp. xvi+373 (Local and global analysis) | MR | Zbl
[33] A pseudoconformal compactification of the nonlinear Schrödinger equation and applications, New York J. Math., Volume 15 (2009), pp. 265-282 http://nyjm.albany.edu:8000/j/2009/15_265.html | EuDML | MR | Zbl
[34] Random data Cauchy problem for supercritical Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 26 (2009) no. 6, pp. 2385-2402 | DOI | EuDML | Numdam | MR | Zbl
[35] A remark on the Schrödinger smoothing effect, Asymptot. Anal., Volume 69 (2010) no. 1-2, pp. 117-123 | MR | Zbl
[36] Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation, Probab. Theory Related Fields, Volume 146 (2010) no. 3-4, pp. 481-514 | DOI | MR | Zbl
[37] Invariant measures for the nonlinear Schrödinger equation on the disc, Dyn. Partial Differ. Equ., Volume 3 (2006) no. 2, pp. 111-160 | DOI | MR | Zbl
[38] Invariant measures for the defocusing nonlinear Schrödinger equation, Ann. Inst. Fourier (Grenoble), Volume 58 (2008) no. 7, pp. 2543-2604 | DOI | EuDML | Numdam | MR | Zbl
[39] Smoothing property for Schrödinger equations with potential superquadratic at infinity, Comm. Math. Phys., Volume 221 (2001) no. 3, pp. 573-590 | DOI | MR | Zbl
[40] 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 | Zbl
[41] Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential, Comm. Partial Differential Equations, Volume 30 (2005) no. 10-12, pp. 1429-1443 | DOI | MR | Zbl
[42] Korteweg-de Vries and nonlinear Schrödinger equations: qualitative theory, Lecture Notes in Mathematics, 1756, Springer-Verlag, Berlin, 2001, pp. vi+147 | MR | Zbl
Cited by Sources: