Asymptotic behavior of solutions for the 2D micropolar equations in Sobolev–Gevrey spaces

Article type: Research Article

Authors: Melo, Wilberclay G.^{a; *} | Rocha, Natã F.^b | Zingano, Paulo R.^c

Affiliations: [a] Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão SE 49100-000, Brazil. E-mail: wilberclay@gmail.com | [b] Campus Clóvis Moura, Universidade Estadual do Piauí, Teresina PI 64078-213, Brazil. E-mail: natafirmino@ccm.uespi.br | [c] Departamento de Matemática Pura e Aplicada, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS 91509-900, Brazil. E-mail: zingano@mat.ufrgs.br

Correspondence: [*] Corresponding author. E-mail: wilberclay@gmail.com.

Abstract: 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.

Keywords: Micropolar equations, decay rates, Sobolev–Gevrey spaces

DOI: 10.3233/ASY-201630

Journal: Asymptotic Analysis, vol. 123, no. 1-2, pp. 157-179, 2021