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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 \(v= (v_1,v_2)\) \[ \begin{gathered} \text{div\,}v= 0,\\ {\partial v\over\partial t}- (\nu+ k)\Delta v- 2k\nabla xw+ \nabla\pi+ v\cdot\nabla v= f_1(x, t),\\ {\partial w\over\partial t}- \gamma\Delta w+ 4kw- 2k\nabla x v+ v\cdot\nabla w= f_2(x, t),\\ (v(\tau), w(t))= (v_\tau, w_\tau).\end{gathered}\tag{1} \] 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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