Asymptotic behavior of solutions for the 2D micropolar equations in Sobolev-Gevrey spaces

This work guarantees the existence of a positive instant t = T and a unique solution (u, w) is an element of [C ([0, T]; H-a,sigma(s) (R-2))](3) (with a > 0, sigma > 1, s > 0 and s not equal 1) for the micropolar equations. Furthermore, we consider the global existence in time of this solution in order to prove the following decay rates: lim(t ->infinity) t(s/2 )parallel to(u,w)(t)parallel to( )(2 )((H)over dota,sigma s)((R2))= lim(t ->infinity) t(s+1/2 )parallel to w(t)parallel to(2)((H)over dota,sigma s (R2))( )= lim(t ->infinity )parallel to(u,w)(t)parallel to(Ha,sigma lambda (R2) )= 0, for all lambda <= s. These limits are established by applying the estimate parallel to F-1 (e(T vertical bar.vertical bar)((u) over cap,(w) over cap (t))parallel to(Hs(R2) )<= [1 + 2M(2)](1/2), for all t >= T, where T relies only on s,mu,nu and M (the inequality above is also demonstrated in this paper). Here M is a bound for parallel to(u, w)(t)parallel to(Hs(R2)) (for all t >= 0) which results from the limits lim(t) -> infinity t(s/2 )parallel to(u,w)(t)parallel to(Hs(R2)) = lim(t ->infinity )parallel to(u, w)(t)parallel to(L2(R2)) = 0.