Asymptotic behavior of global weak solutions for the micropolar dynamics in $L^{2}(\mathbb{R}^{3})$

TL;DR

This paper investigates the long-term decay of solutions to the micropolar fluid equations in three dimensions, showing solutions tend to zero over time and decay rates improve with vortex viscosity.

Contribution

It establishes the asymptotic decay rates of global weak solutions for the micropolar fluid equations in (3), including the effects of vortex viscosity on decay speed.

Findings

Solutions decay to zero as time approaches infinity.

Decay rate of micro-rotational field improves with positive vortex viscosity.

Established decay behavior for both inviscid and viscous micropolar fluids.

Abstract

In this paper the long time behavior of the micropolar fluid equations energy on three dimensional space are studied. We show that $ \| (u,w)(\cdot,t) \|_{{L^{2}(\mathbb{R}^{3})}} \to 0 $ as $t \to \infty$ for Leray-Hopf's global weak solutions in inviscid vortex case. Moreover, when the vortex viscosity are considered, i.e., $\chi >0$, we obtain a (faster) decay for micro-rotational field: $ \| w (\cdot,t) \|_{{L^{2}(\mathbb{R}^{3})}} = o(t^{-1/2})$.