Affiliations: [a] Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão SE 49100-000, Brazil. E-mail: [email protected] | [b] Campus Clóvis Moura, Universidade Estadual do Piauí, Teresina PI 64078-213, Brazil. E-mail: [email protected] | [c] Departamento de Matemática Pura e Aplicada, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS 91509-900, Brazil. E-mail: [email protected]
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.
