On the Global Stability of a Beta-Plane Equation

TL;DR

This paper proves the global stability and decay of small solutions to a dispersive Euler equation modeling geophysical fluid dynamics on a beta-plane, using null forms and Fourier analysis techniques.

Contribution

It introduces a novel analysis of the beta-plane Euler equation demonstrating global stability and decay, employing a double null form and Fourier integral operator estimates.

Findings

Global stability of small solutions established

Solutions decay to equilibrium over time

New analytical techniques for dispersive geophysical models

Abstract

We study the motion of an incompressible, inviscid two-dimensional fluid in a rotating frame of reference. There the fluid experiences a Coriolis force, which we assume to be linearly dependent on one of the coordinates. This is a common approximation in geophysical fluid dynamics and is referred to as beta-plane. In vorticity formulation the model we consider is then given by the Euler equation with the addition of a linear anisotropic, non-degenerate, dispersive term. This allows us to treat the problem as a quasilinear dispersive equation whose linear solutions exhibit decay in time at a critical rate. Our main result is the global stability and decay to equilibrium of sufficiently small and localized solutions. Key aspects of the proof are the exploitation of a "double null form" that annihilates interactions between spatially coherent waves and a lemma for Fourier integral operators which allows us to control a strong weighted norm.