Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D

We study the chemotaxis-fluid system <INF>{</INF><INF>n<INF>t</INF>+u .del n = Delta n -del. (n/c del c), x epsilon Omega, t > 0,</INF> c(t) + u .del c = Delta c - nc, x epsilon Omega, t > 0, u(t) + del P = Delta u + n del phi, x epsilon Omega, t > 0, del . u = 0, x epsilon Omega, t > 0, under homogeneous Neumann boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u, where Omega subset of R-2 is a bounded domain with smooth boundary and phi epsilon C-2 ((Omega)over-bar). From recent results it is known that for suitable regular initial data, the corresponding initial-boundary value problem possesses a global generalized solution. We will show that for small initial mass f(Omega) n(0) these generalized solutions will eventually become classical solutions of the system and obey certain asymptotic properties. Moreover, from the analysis of certain energy-type inequalities arising during the investigation of the eventual regularity, we will also derive a result on global existence of classical solutions under assumption of certain smallness conditions on the size of no in L-1(Omega) and in L log L(Omega), up in L-4(Omega), and of del c(0) in L-2(Omega). (C) 2018 Elsevier Inc. All rights reserved.