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Decay of weak solutions to the 2D dissipative quasi-geostrophic equation. (English) Zbl 1194.76040
Summary: We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data ${\theta }_{0}$ is in ${L}^{2}$ only, we prove that the ${L}^{2}$ norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For ${\theta }_{0}$ in ${L}^{p}\cap {L}^{2}$, with $1\le p<2$, we are able to obtain a uniform decay rate in ${L}^{2}$. We also prove that when the ${L}^{\frac{2}{2\alpha -1}}$ norm of ${\theta }_{0}$ is small enough, the ${L}^{q}$ norms, for $q>\frac{2}{2\alpha -1}$, have uniform decay rates. This result allows us to prove decay for the ${L}^{q}$ norms, for $q\ge \frac{2}{2\alpha -1}$, when ${\theta }_{0}$ is in ${L}^{2}\cap {L}^{\frac{2}{2\alpha -1}}$.

##### MSC:
 76D05 Navier-Stokes equations (fluid dynamics) 35B40 Asymptotic behavior of solutions of PDE 35Q35 PDEs in connection with fluid mechanics 76U05 Rotating fluids 86A05 Hydrology, hydrography, oceanography 86A10 Meteorology and atmospheric physics
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