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Asymptotic behavior of solutions for the 2D micropolar equations in Sobolev–Gevrey spaces
Asymptotic Analysis ( IF 1.1 ) Pub Date : 2020-07-27 , DOI: 10.3233/asy-201630
Wilberclay G. Melo 1 , Natã F. Rocha 2 , Paulo R. Zingano 3
Affiliation  

This work guarantees the existence of a positive instant t=T and a unique solution (u,w)∈[C([0,T];Ha,σs(R2))]3 (with a>0, σ>1, s>0 and s≠1) for the micropolar equations. Furthermore, we consider the global existence in time of this solution in order to prove the following decay rates: limt→∞ts2‖(u,w)(t)‖H˙a,σs(R2)2=limt→∞ts+12‖w(t)‖H˙a,σs(R2)2=limt→∞‖(u,w)(t)‖Ha,σλ(R2)=0,∀λ⩽s. These limits are established by applying the estimate ‖F−1(eT|·|(uˆ,wˆ)(t))‖Hs(R2)⩽[1+2M2]12,∀t⩾T, where T relies only on s,μ,ν and M (the inequality above is also demonstrated in this paper). Here M is a bound for ‖(u,w)(t)‖Hs(R2) (for all t⩾0) which results from the limits limt→∞ts2‖(u,w)(t)‖H˙s(R2)=limt→∞‖(u,w)(t)‖L2(R2)=0.

中文翻译:

Sobolev-Gevrey空间中二维微极性方程解的渐近行为

这项工作保证了存在正时刻t = T和唯一解(u,w)∈[C([0,T]; Ha,σs(R2))] 3(a> 0,σ> 1, s> 0且s≠1)。此外,为了证明以下衰减率,我们考虑了该解在时间上的整体存在性:limt→∞ts2‖(u,w)(t)‖H˙a,σs(R2)2 = limt→∞ts+ 12′w(t)′H˙a,σs(R2)2 = limt→∞′(u,w)(t)′Ha,σλ(R2)= 0,∀λ⩽s。通过应用估计值“ F-1(eT |·|(uˆ,wˆ)(t))” Hs(R2)⩽[1 + 2M2] 12,∀t⩾T来建立这些限制,其中T仅取决于s ,μ,ν和M(以上不等式也在本文中得到证明)。这里M是``(u,w)(t)''Hs(R2)(对于所有t⩾0)的界,这是由极限limt→∞ts2''(u,w)(t)''H˙s( R2)=极限→∞‖(u,w)(t)‖L2(R2)= 0。
更新日期:2020-07-29
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