Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 7 (2018), 1587-1624.

On the global stability of a beta-plane equation

Fabio Pusateri and Klaus Widmayer

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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.

Article information

Source
Anal. PDE, Volume 11, Number 7 (2018), 1587-1624.

Dates
Received: 21 October 2016
Accepted: 4 March 2018
First available in Project Euclid: 15 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.apde/1547521406

Digital Object Identifier
doi:10.2140/apde.2018.11.1587

Mathematical Reviews number (MathSciNet)
MR3810467

Zentralblatt MATH identifier
1393.35180

Subjects
Primary: 35B34: Resonances 35Q35: PDEs in connection with fluid mechanics 76B03: Existence, uniqueness, and regularity theory [See also 35Q35] 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]

Keywords
nonlinear dispersive equations Euler equation Coriolis global behavior dispersive decay beta-plane rotating Euler

Citation

Pusateri, Fabio; Widmayer, Klaus. On the global stability of a beta-plane equation. Anal. PDE 11 (2018), no. 7, 1587--1624. doi:10.2140/apde.2018.11.1587. https://projecteuclid.org/euclid.apde/1547521406


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