Note on the well-posedness of a slightly supercritical surface quasi-geostrophic equation

For the following slightly supercritical surface quasi-geostrophic equation∂tθ+u⋅∇θ+|D|βθ=0,u=∇⊥|D|β−2m(D)θ,β∈]0,1], where m∈C∞(R2∖{0})m∈C∞(R2∖{0}) is a radial non-decreasing positive function which roughly has a logarithmic growth near infinity, we apply the method of nonlocal maximum principle to show the global well-posedness of smooth solutions.