In this paper, we consider the logarithmically improved regularity criterion for the supercritical quasi-geostrophic equation in Besov space $$\dot B_{\infty ,\infty }^{ - r}\left( {{\mathbb{R}^2}} \right)$$ B ˙ ∞ , ∞ − r ( ℝ 2 ) . The result shows that if θ is a weak solutions satisfies $$\int_0^T {\frac{{\left\| {\nabla \theta ( \cdot ,s)} \right\|_{\dot B_{\infty ,\infty }^{ - r} }^{\tfrac{\alpha } {{\alpha - r}}} }} {{1 + \ln \left( {e + \left\| {\nabla ^ \bot \theta ( \cdot ,s)} \right\|_{L^{\tfrac{2} {r}} } } \right)!}}ds < \infty for some 0 < r < \alpha and 0 < \alpha < 1,}$$ ∫ 0 T ∥ ∇ θ ( ⋅ , s ) ∥ B ˙ ∞ , ∞ − r α α − r 1 + ln ( e + ∥ ∇ ⊥ θ ( ⋅ , s ) ∥ L 2 r ) ! d s < ∞ f o r s o m e 0 < r < α a n d 0 < α < 1 , then θ is regular at t = T. In view of the embedding $${L^{\frac{2}{r}}} \subset M_{\frac{2}{r}}^p \subset \dot B_{\infty ,\infty }^{ - r}$$ L 2 r ⊂ M 2 r p ⊂ B ˙ ∞ , ∞ − r with $$2 \leqslant p < \frac{2}{r}$$ 2 ≤ p < 2 r and 0 ≤ r < 1, we see that our result extends the results due to [20] and [31].
Acta Mathematicae Applicatae Sinica – Springer Journals
Published: Aug 7, 2017
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