Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces

The two-dimensional (2D) quasi-geostrophic (QG) equation is a 2D model of the 3D incompressible Euler equations, and its dissipative version includes an extra term bearing the operator $(-\Delta)^\alpha$ with $\alpha\in [0,1]$. Existing research appears to indicate the criticality of $\alpha=\frac12$ in the sense that the issue of global existence for the 2D dissipative QG equation becomes extremely difficult when $\alpha\le \frac12$. It is shown here that for any $\alpha\le \frac12$ the 2D dissipative QG equation with an initial datum in the Besov space $B^r_{2,\infty}$ or $B^r_{p,\infty}$ $(p>2)$ possesses a unique global solution if the norm of the datum in these spaces is comparable to $\kappa$, the diffusion coefficient. Since the Sobolev space $H^r$ is embedded in $B^r_{2,\infty}$, a special consequence is the global existence of small data solutions in $H^r$ for any $r>2-2\alpha$.