On the weak solutions to a shallow water equation. (Q2711840)

The authors obtain the existence of global-in-time weak solutions to the Cauchy problem for a one-dimensional formally integrable shallow-water equation of Camassa-Holm type. First, for an approximate viscous problem they establish uniform in viscosity energy estimates for the solution and its gradient. This allows to derive a priori one-sided supernorm and space-time higher-norm estimates for first-order derivatives for viscous solutions, and thus, as the solution of the original problem is obtained as the limit of viscous approximation, these estimates are also valid for the solution of the original problem. The results are applied to the construction of large-time asymptotics.