On the Hölder 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 ∂tu+b·∇u-Δu+∇p=0,divu=0,u|t=0(x)=u0(x),where x∈ Rⁿ 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 u0∈(L2(Rn))n, we prove that there exists a weak solution u(t) to this system such that u(t) for any time t> 0 is α-Hölder continuous with α∈ (0 , 1). The proof of the Hölder regularity result utilizes a maximum-principle type method to improve the regularity of weak solution step by step.