Vol. 11, No. 7, 2018

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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