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Zbl 1206.35048
Kapustyan, O.V.; Kasyanov, P.O.; Valero, J.
Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system.
(English)
[J] J. Math. Anal. Appl. 373, No. 2, 535-547 (2011). ISSN 0022-247X

The authors consider an optimal control problem associated with the 3D Navier-Stokes system \aligned &\frac{\partial v}{\partial t}-\nu\Delta v +u\cdot\nabla v +\nabla p=f,\quad \operatorname{div}v=0,\quad x\in \Omega\\ &v|_{\partial \Omega}=0,\quad v(x,\tau)=v_\tau(x), \endaligned\tag1 where $\Omega\subset {\Bbb R}^3$ is a bounded domain with smooth boundary, $u$ is a control function. It is necessary to find a pair $\{u,v\}$ such that $v$ is the solution of (1) associated to $u$ and $$J_\tau=\int_\tau^{+\infty}\Vert v(\cdot,y)-u(\cdot,y)\Vert\, e^{-\delta y}\,dy\,\rightarrow\inf$$ 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.
[Il'ya Sh. Mogilevskij (Tver')]
MSC 2000:
*35B41 Attractors
35Q35 Other equations arising in fluid mechanics
49J20 Optimal control problems with PDE (existence)
35Q93

Keywords: optimal problem; multivalued process

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