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Global conservative and dissipative solutions of a coupled Camassa-Holm equations. (English) Zbl 1317.76022
Summary: 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 A. Bressan and A, Constantin [Arch. Ration. Mech. Anal. 183, No. 2, 215-239 (2007; Zbl 1105.76013); Anal. Appl., Singap. 5, No. 1, 1-27 (2007; Zbl 1139.35378)] 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.
©2011 American Institute of Physics

MSC:
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
37L20 Symmetries of infinite-dimensional dissipative dynamical systems
35Q35 PDEs in connection with fluid mechanics
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