Global Weak Solutions to the Generalized mCH Equation via Characteristics

Fanqin Zeng, Yu Gao, Xiaoping Xue

Research output: Journal article publicationJournal articleAcademic researchpeer-review

Abstract

In this paper, we study the generalized modified Camassa-Holm (gmCH) equation via characteristics. We first change the gmCH equation for unknowns (u, m) into its Lagrangian dynamics for characteristics X(ξ, t), where ξ ∈ R is the Lagrangian label. When X ξ(ξ, t) > 0, we use the solutions to the Lagrangian dynamics to recover the classical solutions with m(·, t) ∈ C 0 k(R) (k ∈ N, k ≥ 1) to the gmCH equation. The classical solutions (u, m) to the gmCH equation will blow up if inf ξR X ξ(·, Tmax) = 0 for some Tmax > 0. After the blow-up time Tmax, we use a double mollification method to mollify the Lagrangian dynamics and construct global weak solutions (with m in space-time Radon measure space) to the gmCH equation by some space-time BV compactness arguments.

Original languageEnglish
Pages (from-to)4317-4329
Number of pages13
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume27
Issue number8
DOIs
Publication statusPublished - Aug 2022

Keywords

  • Lagrangian dynamics
  • double mollification method
  • global weak solutions
  • local classical solutions

ASJC Scopus subject areas

  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

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