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

Abstract

This work guarantees the existence of a positive instant t = T and a unique solution ( u , w ) ∈ [ C ( [ 0 , T ] ; H a , σ s ( R 2 ) ) ] 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: lim t → ∞ t s 2 ‖ ( u , w ) ( t ) ‖ H ˙ a , σ s ( R 2 ) 2 = lim t → ∞ t s + 1 2 ‖ w ( t ) ‖ H ˙ a , σ s ( R 2 ) 2 = lim t → ∞ ‖ ( u , w ) ( t ) ‖ H a , σ λ ( R 2 ) = 0 , ∀ λ ⩽ s . These limits are established by applying the estimate ‖ F − 1 ( e T | · | ( u ˆ , w ˆ ) ( t ) ) ‖ H s ( R 2 ) ⩽ [ 1 + 2 M 2 ] 1 2 , ∀ 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 ) ‖ H s ( R 2 ) (for all t ⩾ 0) which results from the limits lim t → ∞ t s 2 ‖ ( u , w ) ( t ) ‖ H ˙ s ( R 2 ) = lim t → ∞ ‖ ( u , w ) ( t ) ‖ L 2 ( R 2 ) = 0.