Hölder estimates for fractional parabolic equations with critical divergence free drifts

This work focuses on drift-diffusion equations with fractional dissipation (−Δ)α in the regime α∈(1/2,1). Our main result is an a priori Hölder estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some β∈(0,1), the Cβ norm of the solution depends only on the size of the drift in critical spaces of the form Ltq(BMOx−γ) with q>2 and γ∈(0,2α−1], along with the Lx2 norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.