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 $(u,m)$ into its Lagrangian dynamics for characteristics $X(\xi,t)$, where $\xi\in\mathbb{R}$ is the Lagrangian label. When $X_\xi(\xi,t)>0$, we use the solutions to the Lagrangian dynamics to recover the classical solutions with $m(\cdot,t)\in C_0^k(\mathbb{R})$ ($k\in\mathbb{N},~~k\geq1$) to the gmCH equation. The classical solutions $(u,m)$ to the gmCH equation will blow up if $\inf_{\xi\in\mathbb{R}}X_\xi(\cdot,T_{\max})=0$ for some $T_{\max}>0$. After the blow-up time $T_{\max}$, 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.