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Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system. (English) Zbl 1206.35048

The authors consider an optimal control problem associated with the 3D Navier-Stokes system

$\begin{array}{cc}& \frac{\partial v}{\partial t}-\nu {\Delta }v+u·\nabla v+\nabla p=f,\phantom{\rule{1.em}{0ex}}divv=0,\phantom{\rule{1.em}{0ex}}x\in {\Omega }\hfill \\ & {v|}_{\partial {\Omega }}=0,\phantom{\rule{1.em}{0ex}}v\left(x,\tau \right)={v}_{\tau }\left(x\right),\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

where ${\Omega }\subset {ℝ}^{3}$ is a bounded domain with smooth boundary, $u$ is a control function. It is necessary to find a pair $\left\{u,v\right\}$ such that $v$ is the solution of (1) associated to $u$ and

${J}_{\tau }={\int }_{\tau }^{+\infty }\parallel v\left(·,y\right)-u\left(·,y\right)\parallel \phantom{\rule{0.166667em}{0ex}}{e}^{-\delta y}\phantom{\rule{0.166667em}{0ex}}dy\phantom{\rule{0.166667em}{0ex}}\to inf$

with $\delta >0$. Two results are obtained in the paper. First, the solutions of the optimal control problem generate a multivalued process which has a pullback attractor. Second, under the unproved assumption of strong global solvability of the 3D Navier-Stokes system the pullback attractor of the process coincides with the global attractor of the semiflow.

##### MSC:
 35B41 Attractors (PDE) 35Q35 PDEs in connection with fluid mechanics 49J20 Optimal control problems with PDE (existence) 35Q93 PDEs in connection with control and optimization
##### Keywords:
optimal problem; multivalued process
##### References:
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