2006
DOI: 10.1007/s00220-006-0007-3
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Approximate Controllability of Three-Dimensional Navier–Stokes Equations

Abstract: We formulate two results on controllability properties of the 3D NavierStokes (NS) system. They concern the approximate controllability and exact controllability in finite-dimensional projections of the problem in question. As a consequence, we obtain the existence of a strong solution of the Cauchy problem for the 3D NS system with an arbitrary initial function and a large class of right-hand sides. We also discuss some qualitative properties of admissible weak solutions for randomly forced NS equations. AMS … Show more

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Cited by 53 publications
(95 citation statements)
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“…Let us recall some notation introduced in [1,2,12]. For any finite-dimensional subspace E ⊂ H k+2 σ , we denote by F (E) the largest vector space F ⊂ H k+2 σ such that for any η 1 ∈ F there are vectors η, ζ 1 , .…”
Section: Controllability Of the Velocitymentioning
confidence: 99%
See 1 more Smart Citation
“…Let us recall some notation introduced in [1,2,12]. For any finite-dimensional subspace E ⊂ H k+2 σ , we denote by F (E) the largest vector space F ⊂ H k+2 σ such that for any η 1 ∈ F there are vectors η, ζ 1 , .…”
Section: Controllability Of the Velocitymentioning
confidence: 99%
“…From equation (1.1) it follows that the pressure can be expressed in terms of the velocity, so we can not expect to control approximately the pressure and the velocity simultaneously. The proofs of these results are based on a development of some ideas from [1,2,12,13].…”
Section: Introductionmentioning
confidence: 99%
“…In [3] by Agrachev and Sarychev and [41] by Shirikyan, the authors prove exact controllability results for dissipative equations. In [19] by Chambrion, Mason, Sigalotti and Boscain, the authors prove approximate controllability results for Schrödinger equations.…”
Section: Iterated Lie Brackets For General Bilinear Systemsmentioning
confidence: 98%
“…They studied the 2D NavierStokes equations on a torus controlled by a finite-dimensional external force and proved the properties of approximate controllability and exact controllability in observed projections. These results were later extended to the Euler and Navier-Stokes systems on various 2D and 3D manifolds; see [AS06,Rod06,Shi06,Rod07,AS07,Shi07].…”
Section: Introductionmentioning
confidence: 99%