A dispersive regularization for the modified camassa–holm equation


Journal Article

© 2018 Society for Industrial and Applied Mathematics 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.

Full Text

Duke Authors

Cited Authors

  • Gao, Y; Li, L; Liu, JG

Published Date

  • January 1, 2018

Published In

Volume / Issue

  • 50 / 3

Start / End Page

  • 2807 - 2838

Electronic International Standard Serial Number (EISSN)

  • 1095-7154

International Standard Serial Number (ISSN)

  • 0036-1410

Digital Object Identifier (DOI)

  • 10.1137/17M1132756

Citation Source

  • Scopus