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Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains. (English) Zbl 1155.37043

The authors deal with two-dimensional nonautonomous micropolar fluid flows, that is for velocity vector field v=(v 1 ,v 2 )

divv=0,v t-(ν+k)Δv-2kxw+π+v·v=f 1 (x,t),w t-γΔw+4kw-2kxv+v·w=f 2 (x,t),(v(τ),w(t))=(v τ ,w τ )·(1)

The fluid motion is specified by the following non-homogeneous boundary condition


Here Ω 2 is a bounded, simply connected, ν is the Newtonian kinetic viscosity, k0 and γ>0 is the viscosity coefficient. The main goal of the authors is to show the existence of a uniform global attractor of (1)–(2) in the following situation:

a) Ω is a simply connected Lipschitz domain,

b) ϕL (Ω), ϕ·n=0 on Ω,

c) f=(f 1 ,f 2 ) is normal in the space L 2 loc (,D(A -1/4 )) where A(w,w):=((ν+k)PΔv,γΔw) for v| Ω =w| Ω =0 with P:vPv is the projection operator satisfying (Pv)=0.

37L30Attractors and their dimensions, Lyapunov exponents
35B41Attractors (PDE)
35Q35PDEs in connection with fluid mechanics
76D05Navier-Stokes equations (fluid dynamics)