Large time behavior for MHD micropolar fluids in [formula omitted

Abstract

We investigate the large time behavior of solutions to the equations of the MHD micropolar fluids in Sobolev spaces Hm(Rn), with n=2 or 3. More precisely, we show that ‖(Dmu,Dmw,Dmb)(⋅,t)‖L2(Rn)≤C(t+1)−m2−n4 for all t≫1. Furthermore, we prove a faster decay estimate for the micro-rotation, namely ‖Dmw(⋅,t)‖L2(Rn)≤C(t+1)−m2−n4−12 for every t≫1. It is also shown that ‖(u,w,b)(⋅,t)−(u‾,w‾,b‾)(⋅,t)‖L2(Rn)≤C(t+1)−n4−12 and ‖w(⋅,t)−w‾(⋅,t)‖L2(Rn)≤C(t+1)−n4−1 for any t≥0, where (u‾,w‾,b‾) is the solution to the related linear system with the same initial data. We also present some related results, e.g., decay rates for the total pressure of the fluid and space-time derivatives estimates.