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Averaging of a 3D Navier-Stokes-Voight equation with singularly oscillating forces. (English) Zbl 1239.35128

Summary: For $\rho \in \left[0,1\right)$ and $\epsilon \in \left(0,1\right)$, we investigate the uniform attractors of a 3D non-autonomous Navier-Stokes-Voight equation with singularly oscillating forces

$\begin{array}{cc}& {u}_{t}-\nu {\Delta }u-{\alpha }^{2}{\Delta }{u}_{t}+\left(u·\nabla \right)u+\nabla p={f}_{0}\left(t,x\right)+{\epsilon }^{-\rho }{f}_{1}\left(t/\epsilon ,x\right),\phantom{\rule{4pt}{0ex}}x\in {\Omega },\hfill \\ & \nabla ·u=0,\phantom{\rule{4pt}{0ex}}x\in {\Omega },\hfill \\ & {u\left(t,x\right)|}_{\partial {\Omega }}=0,\hfill \\ & u\left(\tau ,x\right)={u}_{\tau }\left(x\right),\phantom{\rule{4pt}{0ex}}\tau \in ℝ,\hfill \end{array}$

together with the averaged equations (corresponding to the limiting case $\epsilon =0\right)$

$\begin{array}{cc}& {u}_{t}-\nu {\Delta }u-{\alpha }^{2}{\Delta }{u}_{t}+\left(u·\nabla \right)u+\nabla p={f}_{0}\left(t,x\right),\phantom{\rule{4pt}{0ex}}x\in {\Omega },\hfill \\ & \nabla ·u=0,\phantom{\rule{4pt}{0ex}}x\in {\Omega },\hfill \\ & {u\left(t,x\right)|}_{\partial {\Omega }}=0,\hfill \\ & u\left(\tau ,x\right)={u}_{\tau }\left(x\right),\phantom{\rule{4pt}{0ex}}\tau \in ℝ·\hfill \end{array}$

Under suitable assumptions on the external forces, we obtain the uniform boundedness of the related uniform attractor ${𝒜}^{\epsilon }$ of the first system, and the convergence of ${𝒜}^{\epsilon }$ to the attractor ${𝒜}^{\epsilon }$ of the second system as $\epsilon \to {0}^{+}$.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B41 Attractors (PDE) 76D05 Navier-Stokes equations (fluid dynamics)
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