Vol. 11, No. 7, 2018

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

Volume 11
Issue 8, 1841–2148
Issue 7, 1587–1839
Issue 6, 1343–1586
Issue 5, 1083–1342
Issue 4, 813–1081
Issue 3, 555–812
Issue 2, 263–553
Issue 1, 1–261

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
Subscriptions
Editorial Board
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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