Invariance of the Gibbs measure for the periodic quartic gKdV

Richards, Geordie

doi:10.1016/j.anihpc.2015.01.003

Richards, Geordie

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 3, pp. 699-766.

Résumé

We prove invariance of the Gibbs measure for the (gauge transformed) periodic quartic gKdV. The Gibbs measure is supported on $H^{s} (T)$ for $s < \frac{1}{2}$ , and the quartic gKdV is analytically ill-posed in this range. In order to consider the flow in the support of the Gibbs measure, we combine a probabilistic argument with the second iteration and construct local-in-time solutions to the (gauge transformed) quartic gKdV almost surely in the support of the Gibbs measure. Then, we use Bourgain's idea to extend these local solutions to global solutions, and prove the invariance of the Gibbs measure under the flow. Finally, inverting the gauge, we construct almost sure global solutions to the (ungauged) quartic gKdV below $H^{\frac{1}{2}} (T)$ .

Zbl

DOI : 10.1016/j.anihpc.2015.01.003

Mots clés : gKdV, Gibbs measure, Nonlinear smoothing, A.s. global well-posedness

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     author = {Richards, Geordie},
     title = {Invariance of the {Gibbs} measure for the periodic quartic {gKdV}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {699--766},
     publisher = {Elsevier},
     volume = {33},
     number = {3},
     year = {2016},
     doi = {10.1016/j.anihpc.2015.01.003},
     zbl = {1342.35310},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2015.01.003/}
}

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Richards, Geordie. Invariance of the Gibbs measure for the periodic quartic gKdV. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 3, pp. 699-766. doi : 10.1016/j.anihpc.2015.01.003. http://www.numdam.org/articles/10.1016/j.anihpc.2015.01.003/

Bibliographie
Cité par

[1] Bourgain, J. Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geom. Funct. Anal., Volume 3 (1993) no. 2, pp. 107–156 (Parts I, II Geom. Funct. Anal., 3, 3, 1993, 209–262) | DOI | Zbl

[2] Bourgain, J. Periodic Korteweg–de Vries equation with measures as initial data, Sel. Math. New Ser., Volume 3 (1997), pp. 115–159 | DOI | Zbl

[3] Bourgain, J. Periodic nonlinear Schrödinger equation and invariant measures, Commun. Math. Phys., Volume 166 (1994), pp. 1–26 | DOI | Zbl

[4] Bourgain, J. Invariant measures for the 2D defocusing nonlinear Schrödinger equation, Commun. Math. Phys., Volume 176 (1996), pp. 421–445 | DOI | Zbl

[5] Bourgain, J.; Bulut, A. Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case, J. Eur. Math. Soc., Volume 16 (2014) no. 6, pp. 1289–1325 | DOI | Zbl

[6] Bourgain, J.; Bulut, A. Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3d ball, J. Funct. Anal., Volume 266 (2014) no. 4, pp. 2319–2340 | DOI | Zbl

[7] Burq, N.; Thomann, L.; Tzvetkov, N. Long time dynamics for the one-dimensional nonlinear Schrödinger equation, Ann. Inst. Fourier, Volume 63 (2013) no. 6, pp. 2137–2198 | DOI

[8] Burq, N.; Tzvetkov, N. Invariant measure for a three dimensional nonlinear wave equation, Int. Math. Res. Not., Volume 2007 (2007) no. 22 | DOI | Zbl

[9] Burq, N.; Tzvetkov, N. Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math., Volume 173 (2008) no. 3, pp. 449–475 | Zbl

[10] Burq, N.; Tzvetkov, N. Random data Cauchy theory for supercritical wave equations. II. Global theory, Invent. Math., Volume 173 (2008) no. 3, pp. 477–496 | Zbl

[11] Christ, M.; Colliander, J.; Tao, T. Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Am. J. Math., Volume 125 (2003) no. 6, pp. 1235–1293 | DOI | Zbl

[12] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Multilinear estimates for periodic gKdV equations, and applications, J. Funct. Anal., Volume 211 (2004), pp. 173–218 | DOI | Zbl

[13] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Sharp global well-posedness for KdV and modified KdV on $R$ and $T$ , J. Am. Math. Soc., Volume 16 (2003), pp. 705–749 | DOI | Zbl

[14] Colliander, J.; Oh, T. Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^{2}$ , Duke Math. J., Volume 161 (2012) no. 3, pp. 367–414 | DOI | Zbl

[15] Deng, Y. Two-dimensional nonlinear Schrödinger equation with random radial data, Anal. PDE, Volume 5 (2012) no. 5, pp. 913–960 | DOI | Zbl

[16] Deng, Y. Invariance of the Gibbs measure for the Benjamin–Ono equation, J. Eur. Math. Soc. (2015) (in press) | arXiv | DOI | Zbl

[17] Deng, C.; Cui, S. Random-data Cauchy problem for the Navier–Stokes equations on $T^{3}$ , J. Differ. Equ., Volume 251 (2011) no. 4–5, pp. 902–917 | Zbl

[18] Deng, C.; Cui, S. Random-data Cauchy problem for the periodic Navier–Stokes equations with initial data in negative-order Sobolev spaces | arXiv

[19] Fernique, X. Intégrabilité des vecteurs gaussiens, C. R. Acad. Sci. Paris Sér. A–B, Volume 270 (1970), pp. A1698–A1699 | Zbl

[20] Ginibre, J.; Tsutsumi, Y.; Velo, G. On the Cauchy problem for the Zakharov system, J. Funct. Anal., Volume 151 (1997) no. 2, pp. 384–436 | DOI | Zbl

[21] Kappeler, T.; Topalov, P. Global wellposedness of KdV in $H^{- 1} (T, R)$ , Duke Math. J., Volume 135 (2006) no. 2, pp. 327–360 | DOI | Zbl

[22] Kenig, C.; Ponce, G.; Vega, L. Well-posedness and Scattering results for the generalized Korteweg–de Vries equation via the contraction principle, Commun. Pure Appl. Math., Volume XLVI (1993), pp. 527–620 | Zbl

[23] Kenig, C.; Ponce, G.; Vega, L. A bilinear estimate with applications to the KdV equation, J. Am. Math. Soc., Volume 9 (1996) no. 2, pp. 573–603 | DOI | Zbl

[24] Klainerman, S.; Selberg, S. Bilinear estimates and applications to nonlinear wave equations, Commun. Contemp. Math., Volume 4 (2002) no. 2, pp. 223–295 | DOI | Zbl

[25] Kuksin, S. Analysis of Hamiltonian PDEs, Oxford Lecture Ser. Math. Appl., vol. 19, Clarendon Press, 2000 (224 pp) | Zbl

[26] Kuo, H. Gaussian Measures in Banach Spaces, Lect. Notes Math., vol. 463, Springer-Verlag, New York, 1975 | DOI | Zbl

[27] Lebowitz, J.; Rose, H.; Speer, E. Statistical mechanics of the nonlinear Schrödinger equation, J. Stat. Phys., Volume 50 (1988) no. 3–4, pp. 657–687 | Zbl

[28] Luhrmann, J.; Mendelson, D. Random data Cauchy theory for nonlinear wave equations of power-type on $R^{3}$ | arXiv | Zbl

[29] Molinet, L. Sharp ill-posedness results for the KdV and mKdV equations on the torus, Adv. Math., Volume 230 (2012) no. 4–6, pp. 895–1930 | Zbl

[30] Nahmod, A.; Oh, T.; Rey-Bellet, L.; Staffilani, G. Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc., Volume 14 (2012), pp. 1275–1330 | Zbl

[31] Nahmod, A.; Pavlović, N.; Staffilani, G. Almost sure existence of global weak solutions for super-critical Navier–Stokes equations | arXiv | Zbl

[32] Nahmod, A.; Rey-Bellet, L.; Sheffield, Scott; Staffilani, G. Absolute continuity of Brownian bridges under certain gauge transformations, Math. Res. Lett., Volume 18 (2011) no. 5, pp. 875–887 | DOI | Zbl

[33] Nahmod, A.; Staffilani, G. Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space | arXiv | DOI | Zbl

[34] Oh, T. Invariant Gibbs measures and a.s. global well-posedness for coupled KdV systems, Differ. Integral Equ., Volume 22 (2009) no. 7–8, pp. 637–668 | Zbl

[35] Oh, T. Invariance of the Gibbs measure for the Schrödinger–Benjamin–Ono system, SIAM J. Math. Anal., Volume 41 (2009) no. 6, pp. 2207–2225 | Zbl

[36] Oh, T. White noise for KdV and mKdV on the circle, RIMS Kôkyûroku Bessatsu, Volume B18 (2010), pp. 99–124 | Zbl

[37] Oh, T. Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szego equation, Funkc. Ekvacioj, Volume 54 (2011), pp. 335–365 | Zbl

[38] Poiret, A. Solutions globales pour l'équation de Schrödinger cubique en dimension 3 | arXiv

[39] Quastel, J.; Valko, B. KdV preserves white noise, Commun. Math. Phys., Volume 277 (2008) no. 3, pp. 707–714 | DOI | Zbl

[40] Richards, G. Maximal-in-time behavior of deterministic and stochastic dispersive PDEs, University of Toronto, 2012 http://hdl.handle.net/1807/32973 (Ph.D. thesis)

[41] de Suzzoni, A.S. Invariant measure for the cubic wave equation on the unit ball of $R^{3}$ , Dyn. Partial Differ. Equ., Volume 8 (2011) no. 2, pp. 127–147 | DOI | Zbl

[42] de Suzzoni, A.S. Large data low regularity scattering results for the wave equation on the Euclidean space, Commun. Partial Differ. Equ., Volume 38 (2013) no. 1, pp. 1–49 | DOI | Zbl

[43] de Suzzoni, A.S.; Tzvetkov, N. On the propagation of weakly nonlinear random dispersive waves, Arch. Ration. Mech. Anal., Volume 212 (2014) no. 3, pp. 849–874 | DOI | Zbl

[44] Staffilani, G. On solutions for periodic generalized gKdV equations, Int. Math. Res. Not., Volume 1997 (1997) no. 18, pp. 899–917 | DOI | Zbl

[45] Thomann, L.; Tzvetkov, N. Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity, Volume 23 (2010) no. 11, pp. 2771–2791 | DOI | Zbl

[46] Tzvetkov, N. Invariant measures for the nonlinear Schrödinger equation on the disc, Dyn. Partial Differ. Equ., Volume 3 (2006) no. 2, pp. 111–160 | DOI | Zbl

[47] Tzvetkov, N. Invariant measures for the defocusing nonlinear Schrödinger equation, Ann. Inst. Fourier, Volume 58 (2008) no. 7, pp. 2543–2604 | DOI | Numdam | Zbl

[48] Tzvetkov, N. Construction of a Gibbs measure associated to the periodic Benjamin–Ono equation, Probab. Theory Relat. Fields, Volume 146 (2010) no. 3–4, pp. 481–514 | Zbl

[49] Tzvetkov, N.; Visciglia, N. Invariant measures and long time behavior for the Benjamin–Ono equation, Int. Math. Res. Not., Volume 2014 (2014) no. 17, pp. 4679–4714 | DOI | Zbl

[50] Zhidkov, P. Korteweg–de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lect. Notes Math., vol. 1756, Springer-Verlag, 2001 | Zbl

[51] Zhang, T.; Fang, D. Random data Cauchy theory for the incompressible three dimensional Navier–Stokes equations, Proc. Am. Math. Soc., Volume 139 (2011) no. 8, pp. 2827–2837 | DOI | Zbl

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