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

Abstract

In this paper the long time behavior of the micropolar fluid equations energy on three dimensional space are studied. We show that $\Vert (u,w)(\cdot ,t){\Vert }_{L^2({\mathbb{R}}^3)}\to 0$ 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: $\Vert w(\cdot ,t){\Vert }_{L^2({\mathbb{R}}^3)}=o(t^{-1/2})$.

References in corpus (4)

• Global well-posedness for the micropolar fluid system in the critical Besov spaces
• Existence theorem and blow-up criterion of the strong solutions to the Magneto-micropolar fluid equations
• Sharp pointwise estimates for functions in the Sobolev spaces Hs(Rn)
• Decay Rates of global weak solutions for the MHD equations in $\dot{\mbox{\boldmath{$H$}}}^{s}(\mathbb{R}^n)$