Partial regularity of suitable weak solution to a three-dimensional fractional parabolic-elliptic chemotaxis-Navier–Stokes system

Abstract

In this paper, we investigate a fractional parabolic-elliptic chemotaxis-Navier–Stokes system in spatial dimensions three and obtain the global existence of the suitable weak solution by a contraction mapping theorem. Furthermore, we improve the regularity of the solution through a local maximal L p regularity estimate for the fractional heat equation such that the suitable weak solution is smooth away from a closed set whose one-dimensional parabolic Hausdorff measure is zero, which extends the partial regularity theory of Caffarelli, Kohn and Nirenberg [] on the Navier–Stokes equation to the fractional parabolic-elliptic chemotaxis-Navier–Stokes system.