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Vanishing viscosity limit and long-time behavior for 2D quasi-geostrophic equations. (English) Zbl 1044.35055
The quasi-geostrophic equation arises in the study of fast rotating fluids and in geophysical flows. The author studies the 2D surface quasi-geostrophic equation for thermal active scalars: \[ \frac{\partial \theta \left( x,t\right) }{\partial t}+u\left( x,t\right) \cdot \nabla \theta \left( x,t\right) =f\left( x,t\right) , \]
\[ u=\left( -\frac{\partial \psi }{\partial x_1},\frac{\partial \psi }{\partial x_2}\right) ,\;\psi =-\left( -\triangle \right) ^{-1/2}\theta , \] which is a model problem for the 3D Euler equation. One considers the vanishing viscosity limits for the equation perturbed by some fractional powers of the Laplace operators, the long time behaviour and the various kinds of attractors that make sense for the quasi-geostrophic equation.

MSC:
35Q35 PDEs in connection with fluid mechanics
76U05 General theory of rotating fluids
86A05 Hydrology, hydrography, oceanography
35B41 Attractors
35B40 Asymptotic behavior of solutions to PDEs
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
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