Strichartz Estimates for the Water-Wave Problem with Surface Tension

Strichartz-type estimates for one-dimensional surface water-waves under surface tension are studied, based on the formulation of the problem as a nonlinear dispersive equation. We establish a family of dispersion estimates on time scales depending on the size of the frequencies. We infer that a solution u of the dispersive equation we introduce satisfies local-in-time Strichartz estimates with loss in derivative: [image omitted] where C depends on T and on the norms of the Hs-norm of the initial data. The proof uses the frequency analysis and semiclassical Strichartz estimates for the linealized water-wave operator.