Propagation of singularities for gravity-capillary water waves

Abstract

We obtain two results of propagation for the system of gravity-capillary water waves --- first the propagation of oscillations and decay at the spatial infinity and second a microlocal smoothing effect when the initial surface is non-trapping --- extending the results of Craig, Kappeler and Strauss, Wunsch and Nakamura to quasilinear dispersive equations. We also prove the existence of water waves with an asymptotically Euclidean surface and an asymptotically stationary velocity field. To obtain these results, we extend the paradifferential calculus to weighted Sobolev spaces and develop a semiclassical paradifferential calculus, we also define a family of wavefront sets --- the quasi-homogeneous wavefront sets which, at least in the Euclidean geometry, generalize the wavefront set of H\"{o}rmander, the scattering wavefront set of Melrose, the quadratic scattering wavefront set of Wunsch and the homogeneous wavefront set of Nakamura.Comment: 44 page