Micropolar fluid system in a space of distributions and large time behavior

Abstract

We analyze the well-posedness of the initial value problem for the generalized micropolar fluid system in a space of tempered distributions and also prove the existence of the stationary solutions. The asymptotic stability of solutions is showed in this space, and as a consequence, a criterium for vanishing small perturbations of initial data (stationary solution) at large time is obtained. A fast decay of the solutions is obtained when we assume more regularity on the initial data.

Global existence of strong solutions to micropolar equations in cylindrical domains

“…For the past 40 years, a lot of results concerning the existence of global solutions have been obtained (see e.g. [15][16][17][18][19][20]). Let us now briefly discuss results, which are closely related to our work.…”

“…where the boundary integrals vanish due to the boundary conditions (5) 3,4 . The third term on the left-hand side in (18) is equal to…”

Existence and regularizing rate estimates of solutions to the 3‐D generalized micropolar system in Fourier‐Besov spaces

“…The existence and uniqueness of strong solutions to the micropolar flows and the magneto‐micropolar flows have been investigated in previous studies . Ferreira and Villamizar‐Roa proved the well‐posedness of the 3‐D generalized micropolar system in pseudo‐measure space P M a ( a is a given nonnegative parameter), where the pseudo‐measure space P M a is defined by (see previous works) Recently, Chen and Miao established global well‐posedness of the micropolar fluid system with small initial data in critical Besov spaces for 1≤ p <6.…”

“…When , the system is a generalization of the micropolar fluid system by replacing the Laplace operator −Δ with a general fractional Laplace operator , j =1,2(see Ferreira and Villamizar‐Roa). The purpose of this paper is to study the global existence and asymptotic stability of solutions (indeed, we generalize it to the β ‐th spatial derivatives of solutions) to the 3‐D generalized incompressible micropolar system in Fourier‐Besov spaces, as well as to establish some regularizing rate estimates for the β ‐th spatial derivatives of solutions, which imply that the solution is analytic in the spatial variables.…”

“…Before stating the main results, we introduce the notion of mild solution of . Inspired by Ferreira and Villamizar‐Roa, the corresponding linear system of is as follows: …”