Gibbs Measure for the Higher Order Modified Camassa-Holm Equation

Abstract

This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition. Motivated by the works of Thomann and Tzvetkov (Nonlinearity, 23 (2010), 2771–2791), Tzvetkov (Probab. Theory Relat. Fields, 146 (2010), 4679–4714), Burq, Thomann and Tzvetkov (Ann. Fac. Sci. Toulouse Math., 27 (2018), 527–597), the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity, and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem, Skorokhod convergence theorem and Gibbs measure.

Keywords

Higher-order modified Camassa-Holm equation / The randomization of the initial value / Gibbs measure / Global solution

References

[1] Bourgain J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equations. Geom. Funct. Anal., 1993, 3: 107-156

[2] Bourgain J. Periodic Schrödinger equation and invariant measures. Comm. Math. Phys., 1994, 166: 1-26

[3] Bourgain J, Bulut A. Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3d ball. J. Funct. Anal., 2014, 266: 2319-2340

[4] Bressan A, Constantin A. Global conservative solutions of the Camassa-Holm equation. Arch. Ration. Mech. Anal., 2007, 183: 215-239

[5] Burq N, Thomann L, Tzvetkov N. Remarks on the Gibbs measures for nonlinear dispersive equations. Ann. Fac. Sci. Toulouse Math., 2018, 27: 527-597

[6] Burq N, Tzvetkov N. Random data Cauchy theory for supercritical wave equations, I, Local theory. Invent. Math., 2008, 173: 449-475

[7] Burq N, Tzvetkov N. Random data Cauchy theory for supercritical wave equations, II, A global existence result. Invent. Math., 2008, 173: 477-496

[8] Camassa R, Holm D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 1993, 71: 1661-1664

[9] Chen Y, Gao H J. The Cauchy problem for the Hartree equations under random influences. J. Differential Equations, 2015, 259: 5192-5219

[10] Colliander J, Oh T. Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L²(T). Duke Math. J., 2012, 161: 367-414

[11] Constantin A, Mckean H P. A shallow water equation on the circle. Comm. Pure Appl. Math., 2000, 52: 949-982

[12] Danchin R. A note on well-posedness of the Camassa-Holm equation. J. Differential Equations, 2003, 192: 429-444

[13] Deng C, Cui S B. Random-data Cauchy problem for the Navier-Stokes equations on T ³. J. Differential Equations, 2011, 251: 902-917

[14] Federico C, Suzzoni de A S. Invariant measure for the Schrödinger equation on the real line. J. Funct. Anal., 2015, 269: 271-324

[15] Gorsky, J., On the Cauchy problem for a KdV-type equation on the circle, Ph. D. Thesis, University of Notre Dame, 2004, 128 pages.

[16] Himonas A A, Misiolek C. Well-posedness of the Cauchy problem for a shallow water equation on the circle. J. Differential Equations, 2000, 161: 479-495

[17] Lebowitz J, Rose H, Speer E. Statistical mechanics of the nonlinear Schrödinger equation. J. Stat. Phys., 1988, 50: 657-687

[18] Li Y, Yan W, Yang X. Well-posedness of a higher order modified Camassa-Holm equation in spaces of low regularity. J. Evol. Equ., 2010, 10: 465-486

[19] Oh T. Invariance of the Gibbs measure for the Schrödinger-Benjamin-Ono system. SIAM J. Math. Anal., 2009, 41: 2207-2225

[20] Oh T. Invariance of the white noise for KdV. Comm. Math. Phys., 2009, 292: 217-236

[21] Oh T, Quastel J. On invariant Gibbs measures conditioned on mass and momentum. J. Math. Soc. Japan, 2013, 65: 13-35

[22] Thomann L, Tzvetkov N. Gibbs measure for the periodic derivative nonlinear Schrödinger equation. Nonlinearity, 2010, 23: 2771-2791

[23] Tzvetkov N. Construction of a Gibbs measures associated to the periodic Benjamin-Ono equation. Probab. Theory Relat. Fileds, 2010, 146: 481-514

[24] Tzvetkov N, Visciglia N. Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation. Ann. Sci. Éc. Norm. Supér., 2013, 46: 249-299

[25] Wang H, Cui S B. Global well-posedness of the Cauchy problem of the fifth order shallow water equation. J. Differential Equations, 2006, 230: 600-613

[26] Xin Z, Zhang P. On the weak solutions to a shallow water equation. Comm. Pure Appl. Math., 2000, 53: 1411-1433

[27] Yan W, Li Y S, Zhai X P, Zhang Y M. The Cauchy problem for the shallow water type equations in low regularity spaces on the circle. Adv. Diff. Eqns., 2017, 22: 363-402

[28] Zhang T, Fang D Y. Random data Cauchy theory for the generalized incompressible Navier-Stokes equations. J. Math. Fluid Mech., 2012, 14: 311-324