## 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 $$v= (v_1,v_2)$$ $\begin{gathered} \text{div\,}v= 0,\\ {\partial v\over\partial t}- (\nu+ k)\Delta v- 2k\nabla xw+ \nabla\pi+ v\cdot\nabla v= f_1(x, t),\\ {\partial w\over\partial t}- \gamma\Delta w+ 4kw- 2k\nabla x v+ v\cdot\nabla w= f_2(x, t),\\ (v(\tau), w(t))= (v_\tau, w_\tau).\end{gathered}\tag{1}$ The fluid motion is specified by the following non-homogeneous boundary condition $v= \varphi(x),\quad w= 0\quad\text{on }\partial\Omega.\tag{2}$ Here $$\Omega\subset\mathbb{R}^2$$ is a bounded, simply connected, $$\nu$$ is the Newtonian kinetic viscosity, $$k\geq 0$$ 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:
a) $$\Omega$$ is a simply connected Lipschitz domain,
b) $$\varphi\in L^\infty(\partial\Omega)$$, $$\varphi\cdot n= 0$$ on $$\partial\Omega$$,
c) $$f= (f_1,f_2)$$ is normal in the space $$L^{\text{loc}}_2(\mathbb{R}, D(A^{-1/4}))$$ 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 $$\nabla(Pv)= 0$$.

### MSC:

 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 35B41 Attractors 35Q35 PDEs in connection with fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids
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