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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 θ 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 θ 0 in L p L 2 , with 1p<2, we are able to obtain a uniform decay rate in L 2 . We also prove that when the L 2 2α-1 norm of θ 0 is small enough, the L q norms, for q>2 2α-1, have uniform decay rates. This result allows us to prove decay for the L q norms, for q2 2α-1, when θ 0 is in L 2 L 2 2α-1 .

MSC:
76D05Navier-Stokes equations (fluid dynamics)
35B40Asymptotic behavior of solutions of PDE
35Q35PDEs in connection with fluid mechanics
76U05Rotating fluids
86A05Hydrology, hydrography, oceanography
86A10Meteorology and atmospheric physics
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