Capillary-gravity water waves with vorticity : steady wind-driven waves and waves with a submerged dipole
In this thesis, we study two mathematical problems on water waves in the setting of the incompressible Euler equations with vorticity, gravity, and surface tension. We investigate the existence of small-amplitude steady wind-driven water waves in finite depth, using the Crandall Rabinowitz theorem. As part of the result, elliptic equations with transmission and Wentzell boundary conditons are also examined, and Schauder type estimates on classical solutions are established. The second chapter considers the existence and instability of solitary water waves with a nite dipole in in nite depth. We construct waves of this type using an Implicit Function Theorem argument. Then we establish orbital instability. This is proved using a modi cation of the classical Grillaks Shatash Strauss method.