In this paper, we investigate an initial-boundary value problem for achemotaxis-fluid system in a general bounded regular domain $\Omega \subset\mathbb{R}^N$ ($N\in\{2,3\}$), not necessarily being convex. Thanks to theelementary lemma given by Mizoguchi & Souplet [10], we can derive a new type ofentropy-energy estimate, which enables us to prove the following: (1) for$N=2$, there exists a unique global classical solution to the fullchemotaxis-Navier-Stokes system, which converges to a constant steady state$(n_\infty, 0,0)$ as $t\to+\infty$, and (2) for $N=3$, the existence of aglobal weak solution to the simplified chemotaxis-Stokes system. Our resultsgeneralize the recent work due to Winkler [15,16], in which the domain $\Omega$is essentially assumed to be convex.
展开▼
机译：在本文中，我们研究了一般有界规则域$ \ Omega \ subset \ mathbb {R} ^ N $（$ N \ in \ {2,3 \} $）中的趋化流体系统的初边值问题。 ，不一定是凸的。得益于Mizoguchi＆Souplet [10]给出的基本引理，我们可以得出一种新型的熵能估计，这使我们可以证明以下几点：（1）对于$ N = 2 $，存在一个唯一的全局经典解fullchemotaxis-Navier-Stokes系统，收敛为一个恒定的稳态状态$（n_ \ infty，0,0）$作为$ t \ to + \ infty $，（2）对于$ N = 3 $，存在aglobal简化的趋化斯托克斯系统的弱解决方案。我们的结果概括了Winkler [15,16]的最新工作，其中域$ \ Omega $本质上被假定为凸的。
展开▼
