Hölder continuity for a drift-diffusion equation with pressure

We address the persistence of Hölder continuity for weak solutions of the linear drift-diffusion equation with nonlocal pressureu _t+b·∇ u-Δu=∇p, Δu=0 on [0,∞) × ℝⁿ, with n≥2. The drift velocity b is assumed to be at the critical regularity level, with respect to the natural scaling of the equations. The proof draws on Campanato's characterization of Hölder spaces, and uses a maximum-principle-type argument by which we control the growth in time of certain local averages of u. We provide an estimate that does not depend on any local smallness condition on the vector field b, but only on scale invariant quantities.