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.
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Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equations, Geom. Funct. Anal., 3, 1993, 107–156.
Bourgain, J., Periodic Schrödinger equation and invariant measures, Comm. Math. Phys., 166, 1994, 1–26.
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.
Bressan, A. and Constantin, A., Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183, 2007, 215–239.
Burq, N., Thomann, L. and Tzvetkov, N., Remarks on the Gibbs measures for nonlinear dispersive equations, Ann. Fac. Sci. Toulouse Math., 27, 2018, 527–597.
Burq, N. and Tzvetkov, N., Random data Cauchy theory for supercritical wave equations, I, Local theory, Invent. Math., 173, 2008, 449–475.
Burq, N. and Tzvetkov, N., Random data Cauchy theory for supercritical wave equations, II, A global existence result, Invent. Math., 173, 2008, 477–496.
Camassa, R. and Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1993, 1661–1664.
Chen, Y. and Gao, H. J., The Cauchy problem for the Hartree equations under random influences, J. Differential Equations, 259, 2015, 5192–5219.
Colliander, J. and Oh, T., Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L2(T), Duke Math. J., 161, 2012, 367–414.
Constantin, A. and Mckean, H. P., A shallow water equation on the circle, Comm. Pure Appl. Math., 52, 2000, 949–982.
Danchin, R., A note on well-posedness of the Camassa-Holm equation, J. Differential Equations, 192, 2003, 429–444.
Deng, C. and Cui, S. B., Random-data Cauchy problem for the Navier-Stokes equations on T3, J. Differential Equations, 251, 2011, 902–917.
Federico, C. and Suzzoni de, A. S., Invariant measure for the Schrödinger equation on the real line, J. Funct. Anal., 269, 2015, 271–324.
Gorsky, J., On the Cauchy problem for a KdV-type equation on the circle, Ph. D. Thesis, University of Notre Dame, 2004, 128 pages.
Himonas, A. A. and Misiolek, C., Well-posedness of the Cauchy problem for a shallow water equation on the circle, J. Differential Equations, 161, 2000, 479–495.
Lebowitz, J., Rose, H. and Speer, E., Statistical mechanics of the nonlinear Schrödinger equation, J. Stat. Phys., 50, 1988, 657–687.
Li, Y., Yan, W. and Yang, X., Well-posedness of a higher order modified Camassa-Holm equation in spaces of low regularity, J. Evol. Equ., 10, 2010, 465–486.
Oh, T., Invariance of the Gibbs measure for the Schrödinger-Benjamin-Ono system, SIAM J. Math. Anal., 41, 2009, 2207–2225.
Oh, T., Invariance of the white noise for KdV, Comm. Math. Phys., 292, 2009, 217–236.
Oh, T. and Quastel, J., On invariant Gibbs measures conditioned on mass and momentum, J. Math. Soc. Japan, 65, 2013, 13–35.
Thomann, L. and Tzvetkov, N., Gibbs measure for the periodic derivative nonlinear Schrödinger equation, Nonlinearity, 23, 2010, 2771–2791.
Tzvetkov, N., Construction of a Gibbs measures associated to the periodic Benjamin-Ono equation, Probab. Theory Relat. Fileds, 146, 2010, 481–514.
Tzvetkov, N. and Visciglia, N., Gaussian measures associated to the higher order conservation laws of the Benjamin-Ono equation, Ann. Sci. Éc. Norm. Supér., 46, 2013, 249–299.
Wang, H. and Cui, S. B., Global well-posedness of the Cauchy problem of the fifth order shallow water equation, J. Differential Equations, 230, 2006, 600–613.
Xin, Z. and Zhang, P., On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53, 2000, 1411–1433.
Yan, W., Li, Y. S., Zhai, X. P. and Zhang, Y. M., The Cauchy problem for the shallow water type equations in low regularity spaces on the circle, Adv. Diff. Eqns., 22, 2017, 363–402.
Zhang, T. and Fang, D. Y., Random data Cauchy theory for the generalized incompressible Navier-Stokes equations, J. Math. Fluid Mech., 14, 2012, 311–324.
Part of this work was completed while Lin Lin and Wei Yan were visiting Center for Mathematical Sciences (mathcenter.hust.edu.cn), Huazhong University of Science and Technology, Wuhan, China, as the members of a Research Team on stochastic partial differential equations, whose support is greatly appreciated.
This work was supported by the National Natural Science Foundation of China (Nos. 11901302, 11401180), the Natural Science Foundation from Jiangsu province BK20171029 and the Academic Discipline Project of Shanghai Dianji University (No. 16JCXK02).
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Lin, L., Yan, W. & Duan, J. Gibbs Measure for the Higher Order Modified Camassa-Holm Equation. Chin. Ann. Math. Ser. B 42, 105–120 (2021). https://doi.org/10.1007/s11401-021-0247-8
- Higher-order modified Camassa-Holm equation
- The randomization of the initial value
- Gibbs measure
- Global solution
2000 MR Subject Classification