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