Global conservative and dissipative solutions of a coupled Camassa-Holm equations

Abstract

This paper develops a new approach in the analysis of a coupled Camassa-Holm equations. A continuous semigroup of global conservative solutions and a continuous semigroup of global dissipative solutions are obtained, respectively. The solutions are conservative, in the sense that the total energy equals to a constant, for almost every time. While the solutions are dissipative, in the sense that energy loss occurs through wave breaking. Compared to the approaches used by Bressan, A. and Constantin, A., “Global conservative solutions of the Camassa-Holm equation,” Arch. Ration. Mech. Anal. 183, 215–239 (2007)10.1007/s00205-006-0010-z and Bressan, A. and Constantin, A., “Global dissipative solutions of the Camassa–Holm equation,” Anal. Appl. (Singapore) 5, 1–27 (2007)10.1142/S0219530507000857 for the Camassa-Holm equation, two characteristics are introduced and a new set of independent and dependent variables are applied to the coupled Camassa-Holm equations.