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QUASIINVARIANT GAUSSIAN MEASURES FOR ONE-DIMENSIONAL HAMILTONIAN PARTIAL DIFFERENTIAL EQUATIONS

Part of: Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  02 December 2015

NIKOLAY TZVETKOV
NIKOLAY TZVETKOV*
Affiliation:
Université de Cergy-Pontoise, Cergy-Pontoise, F-95000, UMR 8088 du CNRS, France; nikolay.tzvetkov@u-cergy.fr

Abstract

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We prove the quasiinvariance of Gaussian measures (supported by functions of increasing Sobolev regularity) under the flow of one-dimensional Hamiltonian partial differential equations such as the regularized long wave, also known as the Benjamin–Bona–Mahony (BBM) equation.

MSC classification

Primary: 35Q53: KdV-like equations (Korteweg-de Vries)
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e28
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2015

References

Ambrosio, L. and Figalli, A., ‘On flows associated to Sobolev vector fields in Wiener spaces: an approach a la DiPerna–Lions’, J. Funct. Anal. 256 (2009), 179–214.Google Scholar
Babin, A., Ilyin, A. and Titi, E., ‘On the regularization mechanism for the periodic Korteweg–de Vries equation’, Comm. Pure Appl. Math. 64 (2011), 591–648.Google Scholar
Benjamin, B., Bona, J. and Mahony, J., ‘Model equations for long waves in nonlinear dispersive systems’, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 272 (1972), 47–78.Google Scholar
Bogachev, V., Gaussian Measures, Mathematical Surveys and Monographs, 62 (American Mathematical Society, Providence, RI, 1998).Google Scholar
Bogachev, V. and Mayer-Wolf, E., ‘Absolutely continuous flows generated by Sobolev class vector fields in finite and infinite dimensions’, J. Funct. Anal. 167 (1999), 1–68.Google Scholar
Bourgain, J., ‘Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part II’, GAFA 3 (1993), 209–262.Google Scholar
Bourgain, J., ‘Periodic nonlinear Schrödinger equation and invariant measures’, Comm. Math. Phys. 166 (1994), 1–26.Google Scholar
Bourgain, J., ‘Invariant measures for the 2d-defocusing nonlinear Schrödinger equation’, Comm. Math. Phys. 176 (1996), 421–445.Google Scholar
Bourgain, J. and Bulut, A., ‘Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3D ball’, J. Funct. Anal. 266 (2014), 2319–2340.Google Scholar
Bourgain, J. and Bulut, A., ‘Almost sure global well posedness for the radial nonlinear Schrodinger equation on the unit ball I: the 2D case’, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2013), 1267–1288.Google Scholar
Bourgain, J. and Bulut, A., ‘Almost sure global well posedness for the radial nonlinear Schrodinger equation on the unit ball II: the 3D case’, J. Eur. Math. Soc. 16 (2014), 1289–1325.Google Scholar
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
Burq, N. and Tzvetkov, N., ‘Invariant measure for a three dimensional nonlinear wave equation’, Int. Math. Res. Not. IMRN (2007), 1–26.Google Scholar
Burq, N. and Tzvetkov, N., ‘Random data Cauchy theory for supercritical wave equations I. Local theory’, Invent. Math. 173 (2008), 449–475.Google Scholar
Burq, N. and Tzvetkov, N., ‘Random data Cauchy theory for supercritical wave equations II. A global existence result’, Invent. Math. 173 (2008), 477–496.Google Scholar
Burq, N., Thomann, L. and Tzvetkov, N., ‘Long time dynamics for the one dimensional non linear Schrödinger equation’, Ann. Inst. Fourier 63 (2013), 2137–2198.Google Scholar
Cameron, R. H. and Martin, W. T., ‘Transformation of Wiener integrals under translations’, Ann. of Math. (2) 45 (1944), 386–396.Google Scholar
Cruzeiro, A. B., ‘Equations différentielles ordinaire: non explosition et mesures quasi-invariantes’, J. Funct. Anal. 54 (1983), 193–206.Google Scholar
Cruzeiro, A. B., ‘Equations différentielles sur l’espace de Wiener et formules de Cameron–Martin non linéaires’, J. Funct. Anal. 54 (1983), 206–227.Google Scholar
Deng, Y., ‘Two dimensional NLS equation with random radial data’, Anal. PDE 5 (2012), 913–960.Google Scholar
Deng, Y., ‘Invariance of the Gibbs measure for the Benjamin–Ono equation’, J. Eur. Math. Soc. 17 (2015), 1107–1198.Google Scholar
de Suzzoni, A. S., ‘Wave turbulence for the BBM equation: stability of a Gaussian statistics under the flow of BBM’, Comm. Math. Phys. 326 (2014), 773–813.Google Scholar
de Suzzoni, A. S., ‘Invariant mesure for the cubic non linear wave equation on the unit ball of R 3 ’, Dyn. Partial Differ. Equ. 8 (2011), 127–147.Google Scholar
Erdogan, M. and Tzirakis, N., ‘Global smoothing for the periodic KdV evolution’, Int. Math. Res. Not. IMRN (2012), 4589–4614.Google Scholar
Lannes, D., The Water Waves Problem: Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs (American Mathematical Society, 2013).Google Scholar
Lebowitz, J., Rose, R. and Speer, E., ‘Statistical dynamics of the nonlinear Schrödinger equation’, J. Stat. Phys. V 50 (1988), 657–687.Google Scholar
Nahmod, A., Oh, T., Rey-Bellet, L. and Staffilani, G., ‘Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS’, J. Eur. Math. Soc. 14 (2012), 1275–1330.Google Scholar
Nahmod, A., Rey-Bellet, L., Sheffield, S. and Staffilani, G., ‘Absolute continuity of Brownian bridges under certain gauge transformations’, Math. Res. Lett. 18(5) (2011), 875–887.Google Scholar
Oh, T., ‘Invariance of the Gibbs measure for the Schrödinger–Benjamin–Ono system’, SIAM J. Math. Anal. 41 (2009), 2207–2225.Google Scholar
Oh, T., ‘Invariance of the white noise for KdV’, Comm. Math. Phys. 292(1) (2009), 217–236.Google Scholar
Olver, P. J., ‘Euler operators and conservation laws of the BBM equation’, Math. Proc. Cambridge Philos. Soc. 85 (1979), 143–160.Google Scholar
Ramer, R., ‘On nonlinear transformations of Gaussian measures’, J. Funct. Anal. 15 (1974), 166–187.Google Scholar
Richards, G., ‘Invariance of the Gibbs measure for the periodic quartic gKdV’, Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear.Google Scholar
Thomann, L. and Tzvetkov, N., ‘Gibbs measure for the periodic derivative non linear Schrödinger equation’, Nonlinearity 23 (2010), 2771–2791.Google Scholar
Tzvetkov, N., ‘Invariant measures for the defocusing NLS’, Ann. Inst. Fourier 58 (2008), 2543–2604.Google Scholar
Tzvetkov, N. and Visciglia, N., ‘Invariant measures and long-time behavior for the Benjamin–Ono equation’, Int. Math. Res. Not. IMRN (2014), 4679–4714.Google Scholar
Tzvetkov, N. and Visciglia, N., ‘Invariant measures and long time behaviour for the Benjamin–Ono equation II’, J. Math. Pures Appl. 103 (2015), 102–141.Google Scholar
Quastel, J. and Valko, B., ‘KdV preserves white noise’, Comm. Math. Phys. 277 (2008), 707–714.Google Scholar
Yudovich, V., ‘Non-stationary flows of an ideal incompressible fluid’, Zh. Vychisl. Mat. Mat. Fiz. (1963), 1032–1066. (in Russian).Google Scholar
Zhidkov, P., KdV and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Mathematics, 1756 (Springer, 2001).Google Scholar