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.

Solutions for the Navier-Stokes equations with critical and subcritical fractional dissipation in Lei-Lin and Lei-Lin-Gevrey spaces

“…The fractional Laplacian (−∆) α has been studied in many works in the literature (see, for instance, [32,34] and references therein). To cite some models involving this kind of operator, we refer: Diffusion-reaction, Quasi-geostrophic, Cahn-Hilliard, Porous medium, Schrödinger, Ultrasound, Magnetohydrodynamics (MHD), Magnetohydrodynamics-α (MHD-α) and Navier-Stokes itself (see [1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,17,18,21,22,23,24,25,26,27,28,29,30,31,33,35] and references therein). It is important to recall that, by applying the Spectral Theorem, (−∆) α assumes the diagonal form in the Fourier variable, i.e., this is a Fourier multiplier operator with symbol |ξ| 2α (which extends Fourier multiplier property of −∆).…”

“…In fact, Wu [35] showed that the generalized Navier-Stokes equations (1.1) admit global classical solution provided that the initial data u 0 is smooth and α ≥ 5 4 . More precisely, [35] assumes that α ≥ 5/4 and u 0 ∈ H s (R 3 ), with s > 2α, to obtain a unique global classical solution for (1.1) (see also [1,4,5,7,8,9,15,16,17,18,27,28,29,31,35] and references therein).…”