In this paper, we are concerned with the 3-D incompressible micropolar fluid system, which is a non-Newtonian fluid exhibiting micro-rotational effects and micro-rotational inertia. We aim at establishing the global Gevrey analyticity in the critical Besov space. As a first step, inspired by Chemin’s work (J Anal Math 77:27–50, 1999), we construct the global-in-time existence of the strong solutions in a more general Besov space \(\dot{B}^{\frac{3}{p}-1}_{p,q}\) with \(1\le p<\infty , 1\le q\le \infty \). A new effective variable \(R=\nabla \times \omega +\frac{1}{2}\Delta u\) is introduced at the low frequencies, which allows to eliminate the linear coupling terms \(\nabla \times u\) and \(\nabla \times \omega \), and obtain a global priori estimate. Secondly, observing the parabolic behaviors for u and \(\omega \), we would establish Gevrey analyticity based on the work by Bae, Biswas and Tadmor (Arch Ration Mech Anal 205:963–991, 2012) for the incompressible Navier–Stokes equations. The idea of effective velocity is also essential for establishing the Gevrey analyticity. As a by-product, the time-decay estimates on any derivative of solutions are also available for large time.

在本文中，我们关注的是 3-D 不可压缩微极流体系统，它是一种表现出微旋转效应和微旋转惯性的非牛顿流体。我们的目标是在临界 Besov 空间中建立全局 Gevrey 分析。作为第一步，受 Chemin 的工作 (J Anal Math 77:27–50, 1999) 的启发，我们在更一般的 Besov 空间中构建了强解的全局时间存在性\(\dot{B}^{ \frac{3}{p}-1}_{p,q}\)与\(1\le p<\infty , 1\le q\le \infty \)。在低频引入了一个新的有效变量\(R=\nabla \times \omega +\frac{1}{2}\Delta u\)，它允许消除线性耦合项\(\nabla \times u \)和\(\nabla \times \omega \)，并获得全局先验估计。其次，观察u和\(\omega \)的抛物线行为，我们将根据 Bae、Biswas 和 Tadmor (Arch Ration Mech Anal 205:963–991, 2012) 对不可压缩 Navier-Stokes 的工作建立 Gevrey 分析方程。有效速度的概念对于建立 Gevrey 分析也是必不可少的。作为副产品，对解的任何导数的时间衰减估计也可在很长一段时间内使用。