Global well-posedness to a chemotaxis-Stokes system

This paper concerns the chemotaxis-Stokes system {n(t) + u . del n = Delta n(m) - del . (n del c) + mu n(1- n), c(t) + u . del c = Delta c - cn(alpha), u(t) = Delta u - del pi + n del phi, del . u = 0 in a three dimensional bounded domain under no-flux boundary conditions for n, c and no-slip boundary conditions for u. The purpose of this paper is to study the global solvability and large time asymptotic behavior of solutions. Here, it is worth mentioning that the nonlinear consumption term cn(alpha) (when alpha > 1) will lead to some higher order nonlinear terms in the proof of some uniformly bounded prior estimates of the approximation solutions, which brings great difficulties to the study of the problem. To overcome these difficulties, we make some very precise analysis, combined with some iterative techniques, and finally establish the uniform boundedness of weak solutions for m > 1, 0 < alpha < 2m - 1. Then, the global solvability of weak solutions is derived for any large initial data. Furthermore, we focus on the convergence of weak solutions, and prove that the solutions will converge to the constant steady state (1, 0, 0) in the large time limit. (C) 2021 Elsevier Ltd. All rights reserved.