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

摘要

In this paper, we investigate an initial-boundary value problem for a chemotaxis-fluid system in a general bounded regular domain Omega subset of R-N (N is an element of{2, 3}), not necessarily being convex. Thanks to the elementary lemma given by Mizoguchi and Souplet [Ann. Inst. H. Poincare - 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(infinity), 0, 0) as t -> +infinity, 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 Omega is essentially assumed to be convex.