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Zbl 1155.37043
Chen, Jianwen; Chen, Zhi-Min; Dong, Bo-Qing
Uniform attractors of non-homogeneous micropolar fluid flows in non-smooth domains. (English)
[J] Nonlinearity 20, No. 7, 1619-1635 (2007). ISSN 0951-7715; ISSN 1361-6544/e

The authors deal with two-dimensional nonautonomous micropolar fluid flows, that is for velocity vector field $v= (v_1,v_2)$ $$\gathered \text{div\,}v= 0,\\ {\partial v\over\partial t}- (\nu+ k)\Delta v- 2k\nabla xw+ \nabla\pi+ v\cdot\nabla v= f_1(x, t),\\ {\partial w\over\partial t}- \gamma\Delta w+ 4kw- 2k\nabla x v+ v\cdot\nabla w= f_2(x, t),\\ (v(\tau), w(t))= (v_\tau, w_\tau).\endgathered\tag1$$ The fluid motion is specified by the following non-homogeneous boundary condition $$v= \varphi(x),\quad w= 0\quad\text{on }\partial\Omega.\tag2$$ Here $\Omega\subset\bbfR^2$ is a bounded, simply connected, $\nu$ is the Newtonian kinetic viscosity, $k\ge 0$ and $\gamma> 0$ 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:\par a) $\Omega$ is a simply connected Lipschitz domain,\par b) $\varphi\in L^\infty(\partial\Omega)$, $\varphi\cdot n= 0$ on $\partial\Omega$,\par c) $f= (f_1,f_2)$ is normal in the space $L^{\text{loc}}_2(\bbfR, D(A^{-1/4}))$ where $A(w,w):= ((\nu+ k)P\Delta v,\gamma\Delta w)$ for $v|_{\partial\Omega}= w|_{\partial\Omega}= 0$ with $P: v\mapsto Pv$ is the projection operator satisfying $\nabla(Pv)= 0$.
[Messoud A. Efendiev (Berlin)]

MSC 2000:: *37L30 Attractors and their dimensions
35B41 Attractors
35Q35 Other equations arising in fluid mechanics
76D05 Navier-Stokes equations (fluid dynamics)

Keywords: micropolar fluid flow; normal external function; uniform attractor

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Zentralblatt MATH Berlin [Germany]