Local well-posedness for the generalized surface quasi-geostrophic equation with singular velocity in optimal space

For standard inviscid surface quasi-geostrophic (SQG) equation, it is well-known that H2(R2)H2(R2) is the borderline space when we consider the corresponding local well-posedness. In this paper, we study a new generalized SQG equation with the singular velocity u=∇⊥Λ−2+α(log⁡(I−Δ))μθu=∇⊥Λ−2+α(log⁡(I−Δ))μθ, 1<α<2,μ>01<α<2,μ>0. We find the borderline space is H1+α(R2)H1+α(R2), which is consistent with the standard SQG equation when α=1α=1, μ=0μ=0. This result can be seen as an extensive work of [3], which depends on a new observation.