Long Time Stability for Solutions of a β-Plane Equation

We prove stability for arbitrarily long times of the zero solution for the so-called β-plane equation, which describes the motion of a two-dimensional inviscid, ideal fluid under the influence of the Coriolis effect. The Coriolis force introduces a linear dispersive operator into the two-dimensional incompressible Euler equations, thus making this problem amenable to an analysis from the point of view of nonlinear dispersive equations. The dispersive operator, L₁:(Formula presented.), exhibits good decay, but has numerous unfavorable properties, chief among which are its anisotropy and its behavior at small frequencies.