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 (−∆)α with α ∈ [0, 1]. Existing research appears to indicate the criticality of α = 1 2 in the sense that the issue of global existence for the 2D dissipative QG equation becomes extremely difficult when α ≤ 1 2 . It is shown here that for any α ≤ 1 2 the 2D dissipative QG equation with an initial datum in the Besov space Br 2,∞ or B r p,∞ (p > 2) possesses a unique global solution if the norm of the datum in these spaces is comparable to κ, the diffusion coefficient. Since the Sobolev space Hr is embedded in Br 2,∞, a special consequence is the global existence of small data solutions in Hr for any r > 2− 2α.