Long Time Stability for Solutions of a Beta-Plane Equation

TL;DR

This paper demonstrates the long-term stability of solutions to the beta-plane equation, modeling 2D fluid flow with Coriolis effects, by analyzing the dispersive properties of the associated linear operator.

Contribution

It establishes stability results for the beta-plane equation over arbitrarily long times, leveraging dispersive analysis despite the operator's anisotropy and low-frequency challenges.

Findings

Proves long-time stability of zero solutions.

Analyzes dispersive decay properties of the linear operator.

Addresses anisotropy and low-frequency issues in the analysis.

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.