On a multi-dimensional transport equation with nonlocal velocity

We study a multi-dimensional nonlocal active scalar equation of the form ut+v⋅∇u=0ut+v⋅∇u=0 in R+×RdR+×Rd, where v=Λ−2+α∇uv=Λ−2+α∇u with Λ=(−Δ)1/2Λ=(−Δ)1/2. We show that when α∈(0,2]α∈(0,2] certain radial solutions develop gradient blowup in finite time. In the case when α=0α=0, the equations are globally well-posed with arbitrary initial data in suitable Sobolev spaces.