On existence and regularity of solutions for 2‐D micropolar fluid equations with periodic boundary conditions

Abstract

SUMMARYIn this paper, we prove the existence and uniqueness of a global solution for 2-D micropolar fluid equation with periodic boundary conditions. Then we restrict ourselves to the autonomous case and show the existence of a global attractor.

Gevrey class regularity for solutions of micropolar fluid equations

“…The former was proved in [11] and the latter in [12]. The fractal dimension of the global attractor A ν r is estimated by…”

“…The existence of the global attractor A ν r is shown in [11], the fractal dimension of A ν r is estimated in [12] as follows d f (A ν r ) c(| f | 2 L 2 (Q ) + |g| 2 ) 1/2 -this is a finite-dimensional compact subset of [L 2 (Q )] 3 . In the course of proof of Theorem 2 we have shown that the global attractor A ν r is bounded in D(e τ A 1/2 ) (step 3 of the proof).…”

Ladder estimates for micropolar fluid equations and regularity of global attractor

“…Before we present the theorem concerning the existence of a global attractor for MFE, we shall introduce some function spaces. Denote Theorem 2 (Szopa [9]) Let Q = (0,L) 2 , let external forces and moments are time independent and (f, g) ∈ H. Then there exists a unique global attractor A for the semigroup {S(t)} t 0 in H associated with the system of equations of micropolar fluids. The attractor is bounded in V and compact in H.…”

“…e.g. [8,9]) implies that it vanishes; therefore, we will start the summation from l = 1. Applying the generalized Hölder inequality to (22), we obtain…”

“…e.g. [8,9]). Moreover, it is assumed that the velocity component u 3 = 0 and the axes of rotation of particles are perpendicular to the x 1 ,x 2 plane.…”