Decay rates for the magneto-micropolar system in $\varvec{L^2(}\pmb {\mathbb {R}}^{\varvec{n)}}$

Abstract

In this paper, the large time decay of the magneto-micropolar fluid equations on $\mathbb {R}^n$ ($n=2,3$) is studied. We show, for Leray global solutions, that $\Vert ({\varvec{u}},{\varvec{w}},{\varvec{b}})(\cdot ,t) \Vert _{{L^2(\mathbb {R}^n)}} \rightarrow 0$ as $t \rightarrow \infty$ with arbitrary initial data in $L^2(\mathbb {R}^n)$. When the vortex viscosity is present, we obtain a (faster) decay for the micro-rotational field: $\Vert {\varvec{w}}(\cdot ,t) \Vert _{{L^2(\mathbb {R}^n)}} = o(t^{-1/2})$. Some related results are also included.