Regularizing rate estimates for the 3-D incompressible micropolar fluid system in critical Besov spaces

In this paper, we are concerned with regularizing rates of the higher order derivatives of solutions for the initial value problem of a three-dimensional incompressible micropolar fluid system with initial data (u0,.0).. B -1+3/p p,8 (R3) for 1 = p < 6. More precisely, we prove that there exist two positive constants K1 and K2 such that for any ss. N30, the. B sq,8norm of ss-order derivatives of solutions obeys the following regularizing rates: (. ss x u(center dot, t),.ss x.(center dot, t)). B s q,8 = K1(K2|ss|)|ss| t -|ss|/2-(1/2)(s+1-3/q) for some suitable choices of s and q. These estimates particularly imply the spatial analyticity and the temporal decay of global solutions.