Author:
Zeng Fanqin,Gao Yu,Xue Xiaoping
Abstract
<p style='text-indent:20px;'>In this paper, we study the generalized modified Camassa-Holm (gmCH) equation via characteristics. We first change the gmCH equation for unknowns <inline-formula><tex-math id="M1">\begin{document}$ (u,m) $\end{document}</tex-math></inline-formula> into its Lagrangian dynamics for characteristics <inline-formula><tex-math id="M2">\begin{document}$ X(\xi,t) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \xi\in\mathbb{R} $\end{document}</tex-math></inline-formula> is the Lagrangian label. When <inline-formula><tex-math id="M4">\begin{document}$ X_\xi(\xi,t)>0 $\end{document}</tex-math></inline-formula>, we use the solutions to the Lagrangian dynamics to recover the classical solutions with <inline-formula><tex-math id="M5">\begin{document}$ m(\cdot,t)\in C_0^k(\mathbb{R}) $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M6">\begin{document}$ k\in\mathbb{N},\; \; k\geq1 $\end{document}</tex-math></inline-formula>) to the gmCH equation. The classical solutions <inline-formula><tex-math id="M7">\begin{document}$ (u,m) $\end{document}</tex-math></inline-formula> to the gmCH equation will blow up if <inline-formula><tex-math id="M8">\begin{document}$ \inf_{\xi\in\mathbb{R}}X_\xi(\cdot,T_{\max}) = 0 $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M9">\begin{document}$ T_{\max}>0 $\end{document}</tex-math></inline-formula>. After the blow-up time <inline-formula><tex-math id="M10">\begin{document}$ T_{\max} $\end{document}</tex-math></inline-formula>, we use a double mollification method to mollify the Lagrangian dynamics and construct global weak solutions (with <inline-formula><tex-math id="M11">\begin{document}$ m $\end{document}</tex-math></inline-formula> in space-time Radon measure space) to the gmCH equation by some space-time BV compactness arguments.</p>
Publisher
American Institute of Mathematical Sciences (AIMS)
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics
Reference14 articles.
1. S. C. Anco and E. Recio, A general family of multi-peakon equations and their properties, J. Phys. A, 52 (2019), 125203.
2. A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem, Oxford University Press on Demand, 2000.
3. A. S. Fokas.The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305.
4. B. Fuchssteiner.Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Phys. D, 95 (1996), 229-243.
5. Y. Gao, H. Liu.Global $N$-peakon weak solutions to a family of nonlinear equations, J. Differential Equations, 271 (2021), 343-355.