Holder continuity for a drift-diffusion equation with pressure

Abstract: We address the persistence of H\"older continuity for weak solutions of the linear drift-diffusion equation with nonlocal pressure [ u_t + b \cdot \grad u - \lap u = \grad p,\qquad \grad\cdot u =0 ] on $[0,\infty) \times \R^{n}$, with $n \geq 2$. 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 H\"older 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.