The global well-posedness of the modified quasi-geostrophic equation in frequency spaces

We show that the solution, θ(x,t)θ(x,t), of the modified quasi-geostrophic equations exists globally (i.e.,  θ(x,t)∈L∞(R+;FL1r,0)∩L1(R+;FL1r,α)), if the initial data ‖θ0(x)‖FL1r,0