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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.

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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