PULLBACK EXPONENTIAL ATTRACTORS FOR THE NON-AUTONOMOUS MICROPOLAR FLUID FLOWS

Abstract

Abstract: This paper investigates the pullback asymptotic behaviors for the non-autonomous micropolar fluid flows in 2D bounded domains. We use the energy method, combining with some important properties of the generated processes, to prove the existence of pullback exponential attractors and global pullback attractors and show that they both with finite fractal dimension. Further, we give the relationship between global pullback attractors and pullback exponential attractors.

Key words: micropolar fluid flow, pullback exponential attractor, global pullback attractor, fractal dimension, enstrophy equality

Wenlong SUN, Yeping LI. PULLBACK EXPONENTIAL ATTRACTORS FOR THE NON-AUTONOMOUS MICROPOLAR FLUID FLOWS[J].Acta mathematica scientia,Series B, 2018, 38(4): 1370-1392.

Trendmd

References 32

[1]	Chepyzhov V V, Vishik M I. Attractors for Equations of Mathematical Physics. Providence:Amercian Mathematical Society, 2002
[2]	Chen J, Chen Z, Dong B. Existence of H2-global attractors of two-dimensional micropolar fluid flows. J Math Anal Appl, 2006, 322:512-522
[3]	Chen Z, Price W G. Decay estimates of linearized micropolar fluid flows in R3 space with applications to L3-strong solutions. Internat J Engrg Sci, 2006, 44:859-873
[4]	Chen J, Dong B, Chen Z. Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains. Nonlinearity, 2007, 20:1619-1635
[5]	Chen G. Pullback attractor for non-homogeneous micropolar fluid flows in non-smooth domains. Nonlinear Anal, 2009, 10:3018-3027
[6]	Dong B, Chen Z. Global attractors of two-dimensional micropolar fluid flows in some unbounded domains. Appl Math Comp, 2006, 182:610-620
[7]	Dong B, Chen Z. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows. Discrete Contin Dyn Syst, 2009, 23:765-784
[8]	Dong B, Zhang Z. Global regularity of the 2D micropolar fluid flows with zero angular viscosity. J Differential Equations, 2010, 249:200-213
[9]	Dong B, Zhang W. On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces. Nonlinear Analysis, 2010, 73:2334-2341
[10]	Dong B, Li J., Wu J. Global well-posedness and large-time decay for the 2D micropolar equations. J Differential Equations, 2017, 262:3488-3523
[11]	Eringen A C. Theory of micropolar fluids. J Math Mech, 1966, 16:1-18
[12]	Förste J. On the theory of micropolar fluids. Adv Mechanics, 1979, 2(2):81-100
[13]	Lucas C F Ferreira, Juliana C Precioso. Existence of solutions for the 3D-micropolar fluid system with initial data in Besov-Morrey spaces. Z Angew Math Phys, 2013, 64:1699-1710
[14]	Galdi P, Rionero S. A note on the existence and uniqueness of the micropolar fluid equations. Int J Engng Sci, 1977, 15:105-108
[15]	Giga Y, Sohr H. Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J Funct Anal, 1991, 102:71-94
[16]	Lukaszewicz G. Micropolar Fluids:Theory and Applications. Boston:Birkhäuser, 1999
[17]	Lukaszewicz G. Long time behavior of 2D micropolar fluid flows Math Comput Modelling, 2001, 34:487-509
[18]	Lukaszewicz G, Sadowski W. Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains. Z Angew Math Phys, 2004, 55:247-257
[19]	Li Y R, Guo B L. Random attractors of boussinesq equations with multiplicative noise. Acta Math Sin, English Ser, 2009, 25:481-490
[20]	Lukaszewicz G, Tarasińska A. On H1-pullback attractors for nonautonomous micropolar fluid equations in a bounded domain. Nonlinear Anal, 2009, 71:782-788
[21]	Langa J A, Miranville A, Real J. Pullback exponential attractors. Discrete Contin Dynam Systems, 2010, 26:1329-1357
[22]	Liang Y Y, Li C J, Zhao C D. Compact kernel sections and Kolmogorov entropy for the lattice Zakharov equations with a quantum correction. Acta Math Sci, 2014, 34A(5):1203-1218
[23]	Nowakowski B. Long-time behavior of micropolar fluid equations in cylindrical domains. Nonlinear AnalRWA, 2013, 14:2166-2179
[24]	Sun W, Li Y. Trajectory attractor and global attractor for Keller-Segel-Stokes model with arbitrary porous medium diffusion. Topol Methods Nonlinear Anal, 2017, 50:581-602
[25]	Sun W. Micropolar fluid flows with delay on 2D unbounded domains. J Appl Anal Comput, 2018, 8:356-378
[26]	Sun W, Li Y. Asymptotic behavior of pullback attractors for non-autonomous micropolar fluid flows in 2D unbounded domains. Electronic Journal of Differential Equations, 2018, 2018:1-21
[27]	Xue L. Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations. Math Methods Appl Sci, 2011, 34:1760-1777
[28]	Zhao C, Zhou S, Lian X. H1-uniform attractor and asymptotic smoothing effect of solutions for a nonautonomous micropolar fluid flow in 2D unbounded domains. Nonlinear Anal-RWA, 2008, 9:608-627
[29]	Zhou S, Lou J. Uniform exponential attractor for nonautonomous partly dissipative lattice dynamical system. Acta Math Sin, English Ser, 2014, 30:1381-1394
[30]	Zhao C, Sun W, Hsu C. Pullback dynamical behaviors of the non-autonomous micropolar fluid flows. Dynamics of PDE, 2015, 12:265-288
[31]	Zhao C, Sun W. Global well-posedness and pullback attractors for a two-dimensional non-autonomous micropolar fluid flows with infinite delays. Commun Math Sci, 2017, 15:97-121
[32]	Zhou G, Liu G, Sun W. H2-boundedness of the pullback attractor of the micropolar fluid flows with infinite delays. Boundary Value Problem, 2017, DOI:10.1186/s13661-017-0866-x