On a Periodic 2-Component Camassa-Holm Equation with Vorticity

We consider a periodic 2-component Camassa—Holm equation with vorticity in the paper. We first give the local well-posedness and the blow-up criterion for strong solutions to the equation in the Sobolev space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H^s},s > {3 \over 2}$$\end{document} We then present a global existence result for strong solutions to the equation. We finally obtain several blow-up results and the blow-up rate of strong solutions to the equation. The obtained results cover and improve the earlier results for a periodic 2-component Camssa–Holm equation without vorticity.