Two Dimensional Gravity Water Waves with Constant Vorticity: I. Cubic Lifespan

The motion of water in contact with air is well described by the incompressible Euler equations in the fluid domain, combined with two boundary conditions on the free surface, i.e., the interface with air. In special, but still physically relevant cases, the equations of motion can be viewed as evolution equations for the free surface. These equations are commonly referred to as the water wave equations. Most notably, this is the case for irrotational flows. However, in two space dimensions there is a natural extension of these equations to flows with constant vorticity. In previous work [12], [15] we have considered the local and long time behaviour for the irrotational gravity wave equations with infinite depth in two space dimensions. In this article we take a first step toward understanding the more difficult question of the long time behavior of gravity waves with infinite depth and constant vorticity, either in R × R or in the periodic case R × T. We begin by establishing a local well-posedness result. Then we consider the lifespan of small data solutions, where, like in the zero vorticity case [12], we are able to prove cubic lifespan bounds. To the best of our knowledge, this is the first long time well-posedness result for this problem. We remark that it is of further interest to consider small localized data, and establish an almost global in time result, as it was done in the irrotational case in [12], improving an earlier result of Wu [33]. However, in the constant vorticity case this problem presents some interesting new challenges. We hope to consider this in subsequent work. The motivation to study the constant vorticity problem comes from multiple sources. On one hand, from a mathematical perspective, it provides us with the possibility to consider vorticity effects in a framework where the equations of motion can still be described in terms of the water/air interface, while allowing for a larger range of dynamic behavior, which is particularly interesting over large time scales. On the other hand, from a practical perspective, constant vorticity flows are good models for the water motion in the presence of countercurrents. An interesting example of this type is provided by tidal effects.