Global weak solutions to the generalized mCH equation via characteristics

<p style='text-indent:20px;'>In this paper, we study the generalized modified Camassa-Holm (gmCH) equation via characteristics. We first change gmCH for unknowns <inline-formula><tex-math id="M1">\begin{document}$ (u,m) $\end{document}</tex-math></inline-formula> into its Lagrangian dynamics characteristics id="M2">\begin{document}$ X(\xi,t) $\end{document}</tex-math></inline-formula>, where id="M3">\begin{document}$ \xi\in\mathbb{R} is label. When id="M4">\begin{document}$ X_\xi(\xi,t)>0 use solutions to recover classical with id="M5">\begin{document}$ m(\cdot,t)\in C_0^k(\mathbb{R}) (<inline-formula><tex-math id="M6">\begin{document}$ k\in\mathbb{N},\; \; k\geq1 $\end{document}</tex-math></inline-formula>) equation. The id="M7">\begin{document}$ will blow up if id="M8">\begin{document}$ \inf_{\xi\in\mathbb{R}}X_\xi(\cdot,T_{\max}) = 0 some id="M9">\begin{document}$ T_{\max}>0 $\end{document}</tex-math></inline-formula>. After blow-up time id="M10">\begin{document}$ T_{\max} a double mollification method mollify and construct global weak (with id="M11">\begin{document}$ m in space-time Radon measure space) by BV compactness arguments.</p>