An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation
Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 793-841.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.83
Classification: 35Q55
Keywords: Fourth order nonlinear Schrödinger equation, biharmonic nonlinear Schrödinger equation, quasi-invariant measure, normal form method
Mot clés : Équation de Schrödinger non linéaire, mesures quasi-invariantes, méthode de forme normale
Oh, Tadahiro 1; Sosoe, Philippe 2; Tzvetkov, Nikolay 3

1 School of Mathematics, The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
2 Department of Mathematics, Cornell University 584 Malott Hall, Ithaca, New York 14853, USA
3 Université de Cergy-Pontoise 2, av. Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France
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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, Volume 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 | DOI | MR | Zbl

[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 | DOI | Zbl

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

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

[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 | DOI | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | Zbl

[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 | DOI | Zbl

[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 | DOI | MR

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

[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 | DOI | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

[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) | DOI | MR | Zbl

[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

[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

[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

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

[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

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

[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 | DOI | Zbl

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

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

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

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

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

[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

[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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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