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Séminaire EDP-Analyse ICJ

Global Stability of Solutions to a Beta-Plane Equation

by Klaus Widmayer (Lausanne)

mardi 14 novembre 2017 de au (Europe/Paris)
at Université Claude Bernard Lyon 1 - Campus de la Doua, Bâtiment Braconnier ( Fokko Du Cloux )
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 waves with parallel frequencies and a Lemma for Fourier integral operators, which allows us to control a strong weighted norm and is based on a non-degeneracy property of the nonlinear phase function associated with the problem.

Joint work with Fabio Pusateri; prior work with Tarek Elgindi.