Global weak solutions to the generalized mCH equation via characteristics

In this paper, we study the generalized modified Camassa-Holm (gmCH) equation via characteristics. We first change the gmCH equation for unknowns \begin{document}$ (u,m) $\end{document} into its Lagrangian dynamics for characteristics \begin{document}$ X(\xi,t) $\end{document} , where \begin{document}$ \xi\in\mathbb{R} $\end{document} is the Lagrangian label. When \begin{document}$ X_\xi(\xi,t)>0 $\end{document} , we use the solutions to the Lagrangian dynamics to recover the classical solutions with \begin{document}$ m(\cdot,t)\in C_0^k(\mathbb{R}) $\end{document} ( \begin{document}$ k\in\mathbb{N},\; \; k\geq1 $\end{document} ) to the gmCH equation. The classical solutions \begin{document}$ (u,m) $\end{document} to the gmCH equation will blow up if \begin{document}$ \inf_{\xi\in\mathbb{R}}X_\xi(\cdot,T_{\max}) = 0 $\end{document} for some \begin{document}$ T_{\max}>0 $\end{document} . After the blow-up time \begin{document}$ T_{\max} $\end{document} , we use a double mollification method to mollify the Lagrangian dynamics and construct global weak solutions (with \begin{document}$ m $\end{document} in space-time Radon measure space) to the gmCH equation by some space-time BV compactness arguments.