Global solvability and boundedness to a coupled chemotaxis-fluid model with arbitrary porous medium diffusion

In this paper, we deal with the following coupled chemotaxis-fluid model {n(t) + u . del n = del n(m) - del . (n(1 + n)(-alpha)del c) + gamma n - mu n(2), c(t) + u . del c = Delta c - c + n, u(t) = Delta u - del pi + n del phi, del . u = 0 in a bounded domain Omega subset of R-3 with zero-flux boundary for n, c and no-slip boundary for u. It is shown that for any large initial datum, for any m > 0, alpha > 0, the problem admits a global weak solution, which is uniformly bounded. On the basis of this, the stability of the steady states also be discussed. The study of this paper improve the results in [15], in which, the global existence and boundedness of weak solutions are established for m > 1/3, alpha > 6/5 - m. (c) 2018 Elsevier Inc. All rights reserved.