The Gevrey analyticity and decay for the micropolar system in the critical Besov space

Abstract

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.