The global well-posedness for the 3-D compressible micropolar system in the critical Besov space

We are concerned with 3-D compressible micropolar fluid system in the critical Besov space. We will focus on the global well-posedness, which is based on the results for the incompressible case given by Chen and Miao (J Differ Equ 252:2698–2724, 2012). To deal with the linear system which is a couple system with (a,u,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a,u,\omega )$$\end{document}, inspired by Wu and Wang (J Differ Equ 265:2544–2576, 2018), we find the linear system for the compressible micropolar equations could be decomposed into a compressible Navier–Stokes equation and an incompressible micropolar system. We underline that instead of establishing estimates similar to the heat equation for the angular velocity ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} in Chen and Miao (2012), we find ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} is dominated by damping effect in the low frequency. By borrowing the idea from Haspot (Arch Ration Mech Anal 202:427–460, 2011) named the effective velocity, we are able to decouple the linear system for the incompressible part and reach a damping regularity in the low frequency which is necessary to close the uniform estimates for nonlinear terms.