A dispersive regularization for the modified camassa–holm equation

Yu Gao, Lei Li, Jian Guo Liu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

3 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)2807-2838
Number of pages32
JournalSIAM Journal on Mathematical Analysis
Volume50
Issue number3
DOIs
Publication statusE-pub ahead of print - 5 Jun 2018

Keywords

  • Correct speed of singularity
  • Dispersive limit
  • Nonuniqueness
  • Peakon interaction
  • Selection principle
  • Weak solutions

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

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