On the Holder regularity of the weak solution to a drift-diffusion system with pressure

In this paper we address the regularity issue of weak solution for the following linear drift-diffusion system with pressure partial derivative(t)u + b center dot del u - del u +. p = 0, div u = 0, u| t= 0(x) = u0(x), where and b is a given divergence-free vector field. Under some assumptions of the drift field b in the critical sense, and for the initial data , we prove that there exists a weak solution u(t) to this system such that u(t) for any time is -Holder continuous with . The proof of the Holder regularity result utilizes a maximum-principle type method to improve the regularity of weak solution step by step.