Solutions of the 2D quasi-geostrophic equation in Hölder spaces

The 2D quasi-geostrophic equation∂tθ+u·∇θ+κ(-Δ)αθ=0,u=R⊥(θ)is a two-dimensional model of the 3D hydrodynamics equations. When α⩽12, the issue of existence and uniqueness concerning this equation becomes difficult. It is shown here that this equation with either κ=0 or κ>0 and 0⩽α⩽12 has a unique local in time solution corresponding to any initial datum in the space Cr∩Lq for r>1 and q>1.