@article{077b90df219945a7b15c3190a2b8847b,

title = "A dispersive regularization for the modified camassa–holm equation",

abstract = "In this paper, we present a dispersive regularization approach to construct a global N-peakon weak solution to the modified Camassa–Holm equation (mCH) in one dimension. In particular, we perform a double mollification for the system of ODEs describing trajectories of N-peakon solutions and obtain N smoothed peakons without collisions. Though the smoothed peakons do not give a solution to the mCH equation, the weak consistency allows us to take the smoothing parameter to zero and the limiting function is a global N-peakon weak solution. The trajectories of the peakons in the constructed solution are globally Lipschitz continuous and do not cross each other. When N = 2, the solution is a sticky peakon weak solution. At last, using the N-peakon solutions and through a mean field limit process, we obtain global weak solutions for general initial data m0 in Radon measure space.",

keywords = "Correct speed of singularity, Dispersive limit, Nonuniqueness, Peakon interaction, Selection principle, Weak solutions",

author = "Yu Gao and Lei Li and Liu, {Jian Guo}",

note = "Funding Information: ∗Received by the editors June 1, 2017; accepted for publication (in revised form) February 14, 2018; published electronically June 5, 2018. http://www.siam.org/journals/sima/50-3/M113275.html Funding: The work of the authors was supported by the National Science Foundation through the research network KI-Net RNMS11-07444. The work of the third author was supported by the National Science Foundation under grant DMS-1514826. †Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, People{\textquoteright}s Republic of China, and Department of Physics and Mathematics, Duke University, Durham, NC 27708 (yugao@hit.edu.cn). ‡Department of Mathematics, Duke University, Durham, NC 27708 (leili@math.duke.edu). §Department of Physics and Mathematics, Duke University, Durham, NC 27708 (jliu@phy. duke.edu). Funding Information: The work of the authors was supported by the National Science Foundation through the research network KI-Net RNMS11-07444. The work of the third author was supported by the National Science Foundation under grant DMS-1514826. Publisher Copyright: {\textcopyright} 2018 Society for Industrial and Applied Mathematics",

year = "2018",

month = jun,

day = "5",

doi = "10.1137/17M1132756",

language = "English",

volume = "50",

pages = "2807--2838",

journal = "SIAM Journal on Mathematical Analysis",

issn = "0036-1410",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "3",

}