Lower bounds of decay rates for the MHD micropolar equations

Abstract

We derive lower bounds for the decay rates of solutions to the 3D equations describing the motion of a micropolar fluid under the influence of a magnetic field. To accomplish this, we establish a lower bound for the decay of the solution $(\overline{u},\overline{w},\overline{b})$ of the linearized system, as well as an upper bound for the difference $(u-\overline{u},w-\overline{w},b-\overline{b})$, where $(u,w,b)$ represents the solution of the full nonlinear system. More specifically, for a certain class of initial data, we prove that $\Vert u(\cdot ,t){\Vert }_{L^2\left(R^3\right)}^2+\Vert w(\cdot ,t){\Vert }_{L^2\left(R^3\right)}^2+\Vert b(\cdot ,t){\Vert }_{L^2\left(R^3\right)}^2\ge C{(t+1)}^{-\frac{3}{2}}$, for all $t\ge 0$.