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A remark on regularity criterion for the dissipative quasi-geostrophic equations. (English) Zbl 1154.76339
Summary: This paper concerns with a regularity criterion of solutions to the 2D dissipative quasi-geostrophic equations. Based on a logarithmic Sobolev inequality in Besov spaces, the absence of singularities of \(\theta \) in \([0,T]\) is derived for \(\theta \) a solution on the interval [\(0,T\)) satisfying the condition \[ \nabla^{\perp}\theta \in L^r(0,T; \dot B^0_{p,\infty})\quad \text{for} \frac{2}{p}+\frac{\alpha}{r}=\alpha,\frac{4}{\alpha} \leqslant p \leqslant \infty \] This is an extension of earlier regularity results in the Serrin’s type space \(L^r(0,T;L^p)\).

MSC:
76D50 Stratification effects in viscous fluids
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
76U05 General theory of rotating fluids
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