Propagation of Singularities for Gravity-Capillary Water Waves

Hui Zhu

Published 2018-10-22Version 1

We generalize the wavefront set of H\"ormander and the homogeneous wavefront set of Nakamura to the quasi-homogeneous wavefront set, which enables us to obtain the propagation of singularities for gravity-capillary water waves of finite depth. Consequences of this study include firstly the existence of solutions to the gravity-capillary water wave equation in weighted Sobolev spaces, secondly a microlocal smoothing effect for gravity-capillary water waves, and thirdly the propagation of singularities and microlocal smoothing effects for linear models such as fractional Schr\"odinger equations, the fourth order Schr\"odinger equation, etc. Our proof is based on the paradifferential calculus on weighted Sobolev spaces and the semiclassical paradifferential calculus.