An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation
[Régularité optimale pour la quasi-invariance de mesures gaussiennes par le flot de l’équation de Schrödinger non linéaire d’ordre 4]
Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 793-841.

Nous étudions le transport de mesures gaussiennes par le flot de l’équation de Schrödinger non linéaire d’ordre 4. La nouveauté principale est une estimation d’énergie améliorée faisant appel à un nombre infini de transformations de forme normale sur la fonctionnelle d’énergie. De plus, nous démontrons que la dispersion est essentielle dans cette problématique en prouvant qu’en son absence le même résultat de quasi-invariance ne peut être vrai.

We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation on the circle. In particular, we establish an optimal regularity result for quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces. The main new ingredient is an improved energy estimate established by performing an infinite iteration of normal form reductions on the energy functional. Furthermore, we show that the dispersion is essential for such a quasi-invariance result by proving non quasi-invariance of the Gaussian measures under the dynamics of the dispersionless model.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.83
Classification : 35Q55
Mots clés : Équation de Schrödinger non linéaire, mesures quasi-invariantes, méthode de forme normale
@article{JEP_2018__5__793_0,
     author = {Oh, Tadahiro and Sosoe, Philippe and Tzvetkov, Nikolay},
     title = {An optimal regularity result on the~quasi-invariant {Gaussian} measures for the~cubic fourth order nonlinear {Schr\"odinger~equation}},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {793--841},
     publisher = {Ecole polytechnique},
     volume = {5},
     year = {2018},
     doi = {10.5802/jep.83},
     zbl = {06988593},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jep.83/}
}
TY  - JOUR
AU  - Oh, Tadahiro
AU  - Sosoe, Philippe
AU  - Tzvetkov, Nikolay
TI  - An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2018
DA  - 2018///
SP  - 793
EP  - 841
VL  - 5
PB  - Ecole polytechnique
UR  - http://archive.numdam.org/articles/10.5802/jep.83/
UR  - https://zbmath.org/?q=an%3A06988593
UR  - https://doi.org/10.5802/jep.83
DO  - 10.5802/jep.83
LA  - en
ID  - JEP_2018__5__793_0
ER  - 
Oh, Tadahiro; Sosoe, Philippe; Tzvetkov, Nikolay. An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 793-841. doi : 10.5802/jep.83. http://archive.numdam.org/articles/10.5802/jep.83/

[1] Babin, A. V.; Ilyin, A. A.; Titi, E. S. On the regularization mechanism for the periodic Korteweg-de Vries equation, Comm. Pure Appl. Math., Volume 64 (2011) no. 5, pp. 591-648 | Article | MR 2789490 | Zbl 1284.35365

[2] Ben-Artzi, M.; Koch, H.; Saut, J.-C. Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math., Volume 330 (2000) no. 2, pp. 87-92 | Article | Zbl 0942.35160

[3] Benassi, A.; Jaffard, S.; Roux, D. Elliptic Gaussian random processes, Rev. Mat. Iberoamericana, Volume 13 (1997) no. 1, pp. 19-90 | Article | MR 1462329 | Zbl 0880.60053

[4] Bourgain, J. Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., Volume 166 (1994) no. 1, pp. 1-26 | Zbl 0822.35126

[5] Bourgain, J. Gibbs measures and quasi-periodic solutions for nonlinear Hamiltonian partial differential equations, The Gelfand Mathematical Seminars, 1993–1995, Birkhäuser Boston, Boston, MA, 1996, pp. 23-43 | Article | Zbl 0866.35109

[6] Bourgain, J. Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., Volume 176 (1996) no. 2, pp. 421-445

[7] Cameron, R. H.; Martin, W. T. Transformations of Wiener integrals under translations, Ann. of Math. (2), Volume 45 (1944), pp. 386-396 | Article | MR 10346 | Zbl 0063.00696

[8] Chung, J.; Guo, Z.; Kwon, S.; Oh, T. Normal form approach to global well-posedness of the quadratic derivative nonlinear Schrödinger equation on the circle, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 34 (2017) no. 5, pp. 1273-1297 | Article | Zbl 1386.35376

[9] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. A refined global well-posedness result for Schrödinger equations with derivative, SIAM J. Math. Anal., Volume 34 (2002) no. 1, pp. 64-86 | Article | Zbl 1034.35120

[10] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Sharp global well-posedness for KdV and modified KdV on and 𝕋, J. Amer. Math. Soc., Volume 16 (2003) no. 3, pp. 705-749 | Article | MR 1969209

[11] Cruzeiro, A. B. Équations différentielles ordinaires: non explosion et mesures quasi-invariantes, J. Funct. Anal., Volume 54 (1983) no. 2, pp. 193-205 | Article | Zbl 0523.28020

[12] Cruzeiro, A. B. Équations différentielles sur l’espace de Wiener et formules de Cameron-Martin non-linéaires, J. Funct. Anal., Volume 54 (1983) no. 2, pp. 206-227 | Article | Zbl 0524.47028

[13] Fibich, G.; Ilan, B.; Papanicolaou, G. Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., Volume 62 (2002) no. 4, pp. 1437-1462 | MR 1898529 | Zbl 1003.35112

[14] Forlano, J.; Trenberth, W. On the transport property of Gaussian measures under the one-dimensional fractional nonlinear Schrödinger equations (preprint)

[15] Gérard, P.; Grellier, S. Effective integrable dynamics for a certain nonlinear wave equation, Anal. PDE, Volume 5 (2012) no. 5, pp. 1139-1155 | Article | MR 3022852 | Zbl 1268.35013

[16] Gérard, P.; Lenzmann, E.; Pocovnicu, O.; Raphaël, P. A two-soliton with transient turbulent regime for the cubic half-wave equation on the real line, Ann. PDE, Volume 4 (2018) no. 1 (Art. no. 7) | Article | MR 3747579 | Zbl 1397.35062

[17] Gross, L. Abstract Wiener spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1, Univ. California Press, Berkeley, Calif., 1967, pp. 31-42 | Zbl 0187.40903

[18] Guo, Z.; Kwon, S.; Oh, T. Poincaré-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS, Comm. Math. Phys., Volume 322 (2013) no. 1, pp. 19-48 | Zbl 1308.35270

[19] Guo, Z.; Oh, T. Non-existence of solutions for the periodic cubic NLS below L 2 , Internat. Math. Res. Notices (2018) no. 6, pp. 1656-1729 | MR 3801473

[20] Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers, The Clarendon Press, Oxford University Press, New York, 1979 | Zbl 0423.10001

[21] Ivanov, B. A.; Kosevich, A. M. Stable three-dimensional small-amplitude soliton in magnetic materials, So. J. Low Temp. Phys., Volume 9 (1983), pp. 439-442

[22] Judovič, V. I. Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., Volume 3 (1963), pp. 1032-1066 (in Russian) | MR 158189

[23] Kakutani, S. On equivalence of infinite product measures, Ann. of Math. (2), Volume 49 (1948), pp. 214-224 | Article | MR 23331 | Zbl 0030.02303

[24] Karpman, V. I. Stabilization of soliton instabilities by higher-order dispersion: Fourth-order nonlinear Schrödinger-type equations, Phys. Rev. E (3), Volume 53 (1996), p. R1336-R1339

[25] Karpman, V. I.; Shagalov, A. G. Solitons and their stability in high dispersive systems. I. Fourth-order nonlinear Schrödinger-type equations with power-law nonlinearities, Phys. Lett. A, Volume 228 (1997) no. 1-2, pp. 59-65 | Article | Zbl 0962.35512

[26] Kuo, H. H. Integration theory on infinite-dimensional manifolds, Trans. Amer. Math. Soc., Volume 159 (1971), pp. 57-78 | Article | MR 295393

[27] Kuo, H. H. Gaussian measures in Banach spaces, Lect. Notes in Math., 463, Springer-Verlag, Berlin-New York, 1975, vi+224 pages | MR 461643

[28] Kwon, S.; Oh, T. On unconditional well-posedness of modified KdV, Internat. Math. Res. Notices (2012) no. 15, pp. 3509-3534 | Article | MR 2959040 | Zbl 1248.35186

[29] 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 | Article | MR 1431687 | Zbl 0882.76035

[30] Marcus, M. B.; Rosen, J. Markov processes, Gaussian processes, and local times, Cambridge Studies in Advanced Mathematics, 100, Cambridge University Press, Cambridge, 2006 | MR 2250510 | Zbl 1129.60002

[31] Nahmod, A. R.; Rey-Bellet, L.; Sheffield, S.; Staffilani, G. Absolute continuity of Brownian bridges under certain gauge transformations, Math. Res. Lett., Volume 18 (2011) no. 5, pp. 875-887 | Article | MR 2875861 | Zbl 1250.60018

[32] Oh, T.; Tsutsumi, Y.; Tzvetkov, N. Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third order dispersion (2018) (arXiv:1805.08409)

[33] Oh, T.; Tzvetkov, N. On the transport of Gaussian measures under the flow of Hamiltonian PDEs, Séminaire Laurent Schwartz—Équations aux dérivées partielles et applications. Année 2015–2016, Ed. Éc. Polytech., Palaiseau, 2017 (Exp. No. VI)

[34] Oh, T.; Tzvetkov, N. Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation, Probab. Theory Related Fields, Volume 169 (2017) no. 3-4, pp. 1121-1168 | Article | Zbl 1382.35274

[35] Oh, T.; Tzvetkov, N. Quasi-invariant Gaussian measures for the two-dimensional cubic nonlinear wave equation, J. Eur. Math. Soc. (JEMS) (to appear) (arXiv:1703.10718)

[36] Oh, T.; Tzvetkov, N.; Wang, Y. Solving the 4NLS with white noise initial data (preprint)

[37] Oh, T.; Wang, Y. Global well-posedness of the periodic cubic fourth order NLS in negative Sobolev spaces, Forum Math. Sigma, Volume 6 (2018) (article no. e5) | MR 3800620 | Zbl 06872448

[38] Oh, T.; Wang, Y. On the ill-posedness of the cubic nonlinear Schrödinger equation on the circle, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) (to appear) (arXiv:1508.00827)

[39] Picard, J. Representation formulae for the fractional Brownian motion, Séminaire de Probabilités XLIII (Lect. Notes in Math.), Volume 2006, Springer, Berlin, 2011, pp. 3-70 | Article | MR 2790367 | Zbl 1221.60050

[40] Pocovnicu, O. First and second order approximations for a nonlinear wave equation, J. Dynam. Differential Equations, Volume 25 (2013) no. 2, pp. 305-333 | MR 3054639 | Zbl 1270.65060

[41] Ramer, R. On nonlinear transformations of Gaussian measures, J. Funct. Anal., Volume 15 (1974), pp. 166-187 | Article | MR 349945 | Zbl 0288.28011

[42] Rogers, L. C. G.; Williams, D. Diffusions, Markov processes, and martingales, Vol. 1. Foundations, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000 Reprint of the second (1994) edition | Zbl 0949.60003

[43] Turitsyn, S. K. Three-dimensional dispersion of nonlinearity and stability of multidimensional solitons, Teoret. Mat. Fiz., Volume 64 (1985) no. 2, pp. 226-232 | MR 826520

[44] Tzvetkov, N. Quasiinvariant Gaussian measures for one-dimensional Hamiltonian partial differential equations, Forum Math. Sigma, Volume 3 (2015) (article no. e28) | Article | MR 3482274 | Zbl 1333.35243

[45] Tzvetkov, N.; Visciglia, N. Invariant measures and long-time behavior for the Benjamin-Ono equation, Internat. Math. Res. Notices (2014) no. 17, pp. 4679-4714 | Article | MR 3257548 | Zbl 1301.35141

[46] Tzvetkov, N.; Visciglia, N. Invariant measures and long time behaviour for the Benjamin-Ono equation II, J. Math. Pures Appl. (9), Volume 103 (2015) no. 1, pp. 102-141 | Article | MR 3281949 | Zbl 1315.37051

Cité par Sources :