Global Existence and Asymptotic Behavior of Solutions to a Chemotaxis-Fluid System on General Bounded Domain

Abstract

In this paper, we investigate an initial-boundary value problem for a chemotaxis-fluid system in a general bounded regular domain $\Omega \subset {\mathbb{R}}^N$ ($N\in \{2,3\}$), not necessarily being convex. Thanks to the elementary lemma given by Mizoguchi & Souplet [10], 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 $(n_{\infty },0,0)$ as $t\to +\infty$, 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 [15,16], in which the domain $\Omega$ is essentially assumed to be convex.

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