Vol. 11, No. 7, 2018

Download this article
Download this article For screen
For printing
Recent Issues

Volume 16
Issue 2, 309–612
Issue 1, 1–308

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
 
Other MSP Journals
On the global stability of a beta-plane equation

Fabio Pusateri and Klaus Widmayer

Vol. 11 (2018), No. 7, 1587–1624
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 the β-plane approximation. In vorticity formulation, the model we consider is then given by the Euler equation with the addition of a linear anisotropic, nondegenerate, 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.

Keywords
nonlinear dispersive equations, Euler equation, Coriolis, global behavior, dispersive decay, beta-plane, rotating Euler
Mathematical Subject Classification 2010
Primary: 35B34, 35Q35, 76B03, 76B15
Milestones
Received: 21 October 2016
Accepted: 4 March 2018
Published: 20 May 2018
Authors
Fabio Pusateri
Department of Mathematics
Princeton University
Princeton, NJ 08544
United States
Klaus Widmayer
Institute of Mathematics
EPFL
Bâtiment des Mathématiques
Lausanne
Switzerland