Global dynamics of Kato's solutions for the 3D incompressible micropolar system

Abstract: We consider the global well-posedness and decay rates for solutions of 3D incompressible micropolar equation in the critical Besov space. Spectrum analysis allows us to find not only parabolic behaviors of solutions, but also damping effect of angular velocity in the low frequencies. Based on this observation, we establish the global well-posedness with more general regularity on the initial data. The approach concerns large time behaviors is so-called the Gevrey method which bases on the parabolic mechanics of the micropolar system. This method enables us even to derive decay of \textit{arbitrary} higher order derivatives by transforming it into the control of the radius of analyticity under a particular regularity. To bound the growth of the radius of analyticity in general $L^p$ Besov spaces, we shall develop some new techniques concern Gevrey estimates, especially an extended Coifman-Meyer theory, to cover those endpoint Lebesgue framework.