Global convergence of a sticky particle method for the modified Camassa-Holm equation

Yu Gao, Jian Guo Liu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

4 Citations (Scopus)

Abstract

In this paper, we prove convergence of a sticky particle method for the modified Camassa-Holm equation (mCH) with cubic nonlinearity in one dimension. As a byproduct, we prove global existence of weak solutions u with regularity: u and ux are space-time BV functions. The total variation of m(•, t) = u(•, t) - uxx(•, t) is bounded by the total variation of the initial data m0. We also obtain W1,1(ℝ)-stability of weak solutions when solutions are in L (0, ∞; W1,2(ℝ)). (Notice that peakon weak solutions are not in W1,2(ℝ).) Finally, we provide some examples of nonuniqueness of peakon weak solutions to the mCH equation.

Original languageEnglish
Pages (from-to)1267-1294
Number of pages28
JournalSIAM Journal on Mathematical Analysis
Volume49
Issue number2
DOIs
Publication statusE-pub ahead of print - 6 Apr 2017
Externally publishedYes

Keywords

  • Global existence
  • N-peakon solutions
  • Nonuniqueness
  • Space-time BV estimates
  • Sticky collisions

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

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