Gibbs Measure for the Higher Order Modified Camassa-Holm Equation


Lin LIN,Wei YAN,Jinqiao DUAN.Gibbs Measure for the Higher Order Modified Camassa-Holm Equation[J].Chinese Annals of Mathematics B,2021,42(1):105~120
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Lin LIN; Wei YAN;Jinqiao DUAN


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 linebreak Discipline Project of Shanghai Dianji University (No.,16JCXK02).
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.


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


35G25, 37K05
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