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.

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Acknowledgement

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.

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Correspondence to Wei Yan.

Additional information

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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Cite this article

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

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Keywords

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

2000 MR Subject Classification

  • 35G25
  • 37K05