Communications in Mathematical Sciences

Volume 16 (2018)

Number 4

Pullback dynamical behaviors of the non-autonomous micropolar fluid flows with minimally regular force and moment

Pages: 1043 – 1065

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n4.a6

Authors

Wenlong Sun (Department of Mathematics, East China University of Science and Technology, Shanghai, China)

Yeping Li (Department of Mathematics, East China University of Science and Technology, Shanghai, China)

Abstract

In this paper, we investigate the pullback asymptotic behaviors of solutions for the nonautonomous micropolar fluid flows in 2D bounded domains. Firstly, when the force and the moment have a little additional regularity, we make use of the semigroup method and $\epsilon$-regularity method to obtain the existence of a compact pullback absorbing family in $\hat{H}$ and $\hat{V}$, respectively. Then, applying the global well-posedness and the estimates of the solutions, we verify the flattening property (also known as the “Condition (C)”) of the generated evolution process for the universe of fixed bounded sets and for another universe with a tempered condition in spaces $\hat{H}$ and $\hat{V}$, respectively. Further, we show the existence and regularity of the pullback attractors of the evolution process. Compared with the regularity of the force and the moment of “Pullback dynamical behaviors of the non-autonomous micropolar fluid flows” [C. Zhao, W. Sun, and C. Hsu, Dynamics of PDE, 12:265–288, 2015], here we only need the minimal regularity of the force and the moment.

Keywords

pullback attractor, flattening property, semigroup method, -regularity method, enstrophy equality

2010 Mathematics Subject Classification

35B40, 35B41, 76D07

Received 5 March 2017

Accepted 18 March 2018

Published 31 October 2018