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 ξ∈ R 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 language | English |
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Pages (from-to) | 4317-4329 |
Number of pages | 13 |
Journal | Discrete and Continuous Dynamical Systems - Series B |
Volume | 27 |
Issue number | 8 |
DOIs | |
Publication status | Published - Aug 2022 |
Keywords
- Lagrangian dynamics
- double mollification method
- global weak solutions
- local classical solutions
ASJC Scopus subject areas
- Applied Mathematics
- Discrete Mathematics and Combinatorics