Liouville-type results for three-dimensional equatorial water waves with surface tension and an interface

In this paper, we investigate the dynamics of geophysically stratified water waves with constant vorticity near the equator. Firstly, we prove that the bounded solutions of the three-dimensional inviscid gravity water wave governed by the f-plane approximation equation are two-dimensional. Moreover, we establish the two-dimensional characteristics of bounded solutions for gravity-capillary water waves and gravity water waves with rigid lids. Secondly, we focus on rigid lid case of gravity and gravity-capillary stratified water waves controlled by the \(\beta \) $\beta$ -plane approximation. Here, we find that the only flow with a constant vorticity vector is a stationary flow with a flat interface and a vanishing velocity field.