Affiliations: Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, China. E-mail: jiangbryan@gmail.com | School of Mathematical Sciences, Fudan University, Shanghai, China and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, China. E-mail: haowufd@yahoo.com | School of Mathematical Sciences, Fudan University, Shanghai, China. E-mail: songmuzheng@yahoo.com
Note: [] Corresponding author: Jie Jiang, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, Hubei Province, China. E-mail: jiangbryan@gmail.com
Abstract: In this paper, we investigate an initial-boundary value problem for a chemotaxis–fluid system in a general bounded regular domain Ω⊂RN (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.
Keywords: chemotaxis, Navier–Stokes equation, global existence, general bounded domain
