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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 spatially coherent waves and a lemma for Fourier integral operators which allows us to control a strong weighted norm.
The KPII equation is an integrable nonlinear PDE in 2+1 dimensions (two spatial and one temporal), which arises in several physical circumstances, including fluid mechanics where it describes waves in shallow water. It provides a multidimensional gen
A new method for the solution of initial-boundary value problems for textit{linear} and textit{integrable nonlinear} evolution PDEs in one spatial dimension was introduced by one of the authors in 1997 cite{F1997}. This approach was subsequently exte
We consider the Cauchy problem for the Gross-Pitaevskii (GP) equation. Using the DBAR generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of the GP in the solitonic regio
In the paper, we study the plane Couette flow of a rarefied gas between two parallel infinite plates at $y=pm L$ moving relative to each other with opposite velocities $(pm alpha L,0,0)$ along the $x$-direction. Assuming that the stationary state tak
In this paper we show global well-posedness near vacuum for the binary-ternary Boltzmann equation. The binary-ternary Boltzmann equation provides a correction term to the classical Boltzmann equation, taking into account both binary and ternary inter