Global existence and asymptotic behavior of solutions to a chemotaxis–fluid system on general bounded domains

Abstract

In this paper, we investigate an initial-boundary value problem for a chemotaxis–fluid system in a general bounded regular domain Ω⊂R^N (N∈{2,3}), not necessarily being convex. Thanks to the elementary lemma given by Mizoguchi and Souplet [Ann. Inst. H. Poincaré – AN 31 (2014), 851–875], we can derive a new type of entropy–energy estimate, which enables us to prove the following: (1) for N=2, there exists a unique global classical solution to the full chemotaxis–Navier–Stokes system, which converges to a constant steady state (n_∞,0,0) as t→+∞, and (2) for N=3, the existence of a global weak solution to the simplified chemotaxis–Stokes system. Our results generalize the recent work due to Winkler [Commun. Partial Diff. Equ. 37 (2012), 319–351; Arch. Rational Mech. Anal. 211 (2014), 455–487], in which the domain Ω is essentially assumed to be convex.