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Averaging of 2D Navier-Stokes equations with singularly oscillating forces. (English) Zbl 1170.35345
The authors consider the non-autonomous 2D Navier-Stokes equations with nonslip boundary condition for $\rho\in[0,1)$, $\varepsilon>0$ $$\aligned &\frac{\partial v}{\partial t}-\nu\Delta v+(v\cdot\nabla)v+\nabla p=g_0(x,t)+\varepsilon^{-\rho}g_1(x,t/\varepsilon),\quad \text{div}\,v=0,\quad x\in\Omega,\\ &v|_{\partial\Omega}=0, \endaligned\tag1$$ where $v(x,t)=(v_1,v_2)$ is the velocity vector field, $p$ is the pressure, $\nu= \text{const}>0$ is the kinematic viscosity. Along with (1) the averaged Navier-Stokes equations are considered $$\aligned &\frac{\partial v}{\partial t}-\nu\Delta v+(v\cdot\nabla)v+\nabla p=g_0(x,t),\quad \text{div}\,v=0,\quad x\in\Omega,\\ &v|_{\partial\Omega}=0. \endaligned\tag2$$ Under suitable assumptions on the external force, the uniform boundedness of the related uniform global attractors $A^\varepsilon$ is established for the equations (1). $A^0$ is the attractor to the equations (2). It is proved that $A^\varepsilon$ converge to $A^0$ as $\varepsilon\rightarrow0^+$ in the standard Hausdorff semidistance in the space $H(\Omega)$.

35B41Attractors (PDE)
35Q30Stokes and Navier-Stokes equations
35B40Asymptotic behavior of solutions of PDE
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