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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:
76D50Stratification effects in viscous fluids
76D03Existence, uniqueness, and regularity theory
35Q35PDEs in connection with fluid mechanics
76U05Rotating fluids
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References:
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