A Remark on the Global Well-posedness of a Modified Critical Quasi-geostrophic Equation

The $β$-generalized quasi-geostrophic equation is studied in the range of $α\in (0, 1), β\in (1/2, 1), 1/2 < α+ β< 3/2$. When $α\in (1/2, 1), β\in (1/2, 1)$ such that $1 \leq α+ β< 3/2$, using the method introduced in [12] and [9], we prove global regularity of the unique and analytic solution and when $α\in (0, 1/2), β\in (1/2, 1)$ such that $1/2 < α+ β< 1$, that there exists a constant such that $\lVert \nablaθ_{0}\rVert_{L^{\infty}}^{2-2α-2β}\lVertθ_{0}\rVert_{L^{\infty}}^{2α+ 2β- 1} \leq c_{α,β}$ implies global regularity.

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