Strichartz estimates for the water-wave problem with surface tension

Abstract

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: \[ \| u \|_{L^p([0,T]) W^{s-1/p,q}(\mathbb{R})} \leq C, \qquad \frac{2}{p} + \frac{1}{q} = {1/2}, \] where $C$ depends on $T$ and on the norms of the initial data in $H^s \times H^{s-3/2}$. The proof uses the frequency analysis and semiclassical Strichartz estimates for the linealized water-wave operator.