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### Open Access Journals

DCDS

We consider the long time behavior of moments of solutions and of
the solutions itself to dissipative Quasi-Geostrophic flow (QG)
with sub-critical powers.
The flow under consideration
is described by the nonlinear scalar equation

$\frac{\partial \theta}{\partial t} + u\cdot \nabla \theta + \kappa (-\Delta)^{\alpha}\theta =f$, $\theta|_{t=0}=\theta_0 $

Rates of decay are obtained for moments of the solutions, and lower bounds of decay rates of the solutions are established.

DCDS-S

We continue the study initiated in [16] of dissipative differential equations governing fluid motion in the presence of an obstacle, in which the dissipative term is given by the Laplacian, or a fractional power of the Laplacian. Our main tools are the Ikebe-Ramm transform, and the localized version of the fractional Laplacian due to Caffarelli and Silvestre [5] as improved by Stinga and Torrea [21]. We give applications to the problem of existence of weak solutions of the two dimensional dissipative quasi-geostrophic equation and the decay of these solutions in the $L^2$-norm.

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