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 H-1(R) x H-1(R) and some a priori estimates on the first-order derivatives of approximation solutions. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim