On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim x → ±∞ u 0 ( x ) = u ± . By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u − ≤ u + . The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^1(\mathbb {R})\times H^1(\mathbb {R})$\end{document} and some a priori estimates on the first‐order derivatives of approximation solutions.