Global behavior for the classical solution of compressible viscous micropolar fluid with cylinder symmetry

Abstract

<p style='text-indent:20px;'>This paper is concerned with the global solutions of the 3D compressible micropolar fluid model in the domain to a subset of <inline-formula><tex-math id="M1">\begin{document}$ R^3 $\end{document}</tex-math></inline-formula> bounded with two coaxial cylinders that present the solid thermo-insulated walls, which is in a thermodynamical sense perfect and polytropic. Compared with the classical Navier-Stokes equations, the angular velocity <inline-formula><tex-math id="M2">\begin{document}$ w $\end{document}</tex-math></inline-formula> in this model brings benefit that is the damping term -<inline-formula><tex-math id="M3">\begin{document}$ uw $\end{document}</tex-math></inline-formula> can provide extra regularity of <inline-formula><tex-math id="M4">\begin{document}$ w $\end{document}</tex-math></inline-formula>. At the same time, the term <inline-formula><tex-math id="M5">\begin{document}$ uw^2 $\end{document}</tex-math></inline-formula> is bad, it increases the nonlinearity of our system. Moreover, the regularity and exponential stability in <inline-formula><tex-math id="M6">\begin{document}$ H^4 $\end{document}</tex-math></inline-formula> also are proved.</p>