Global Weak Solutions for a Shallow Water Equation

where α, γ, ω are given real constants. Equation (1) was first introduced as a model describing propagation of unidirectional gravitational waves in a shallow water approximation over a flat bottom, with u representing the fluid velocity [DGH01]. For α = 0 and for α = 1, γ = 0 we obtain the Korteweg–de Vries and the Camassa–Holm [CH93, J02] equations, respectively. Both of them describe unidirectional shallow water waves. Moreover, all these three equations have a bi-Hamiltonian structure, they are completely integrable, they have infintely many conserved quantities. From a mathematical point of view the Camassa–Holm equation is well studied, see [CHK05.1] for an extensive list of references. In particular, we recall that existence and uniqueness results for global weak solutions have been proved by Coclite, Holden, and Karlsen [CHK05.1], Constantin and Escher [CE98], Constantin and Molinet [CM00], and Xin and Zhang [XZ00, XZ02], see also Danchin [D01, D03] as well as others.