Long Time Stability for Solutions of a Beta-Plane Equation

Abstract

We prove stability for arbitrarily long times of the zero solution for the so-called $\beta$-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 2d incompressible Euler equations, thus making this problem amenable to an analysis from the point of view of nonlinear dispersive equations. The dispersive operator, $L_1:=\frac{{\partial }_1}{|\nabla {|}^2}$, exhibits good decay, but has numerous unfavorable properties, chief among which are its anisotropy and its behavior at small frequencies.

43 pages; revised with referee's comments

References in corpus (1)

• Sharp decay estimates for an anisotropic linear semigroup and applications to the SQG and inviscid Boussinesq systems