On the weak solutions to a shallow water equation

We obtain the existence of global-in-time weak solutions to the Cauchy problem for a one-dimensional shallow-water equation that is formally integrable and can be obtained by approximating directly the Hamiltonian for Euler's equation in the shallow-water regime. The solution is obtained as a limit of viscous approximation. The key elements in our analysis are some new a priori one-sided supernorm and space-time higher-norm estimates on the first-order derivatives.