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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

v t-νΔv+u·v+p=f,divv=0,xΩv| Ω =0,v(x,τ)=v τ (x),(1)

where Ω 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 τ = τ + v(·,y)-u(·,y)e -δy dyinf

with δ>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:
35B41Attractors (PDE)
35Q35PDEs in connection with fluid mechanics
49J20Optimal control problems with PDE (existence)
35Q93PDEs in connection with control and optimization
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