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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 \(1 \leq p < 2\), 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 \(q \geq {\frac{2}{2\alpha-1}}\), when \(\theta_{0}\) is in \(L^2 \cap L^{\frac{2}{2\alpha-1}}\).

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
35B40 Asymptotic behavior of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
76U05 General theory of rotating fluids
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
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