abstract
In a bounded three-dimensional domain Ω a solenoidal initial vector fieldv 0(x)∈H 3 (Ω) is given. We construct a vector fieldz(t, x) defined on the lateral surface [0,T]×ϖΩ of the cylinder [0,T]×Ω which possesses the following property: the solutionv(t, x) of the boundary value problem for the Navier-Stokes equation with the initial valuev 0(x) and the boundary Dirichlet conditionz(t, x) satisfies the relationv(T, x)≡0 at the instantT. Moreover,
, wherec>0,k>0 are certain constants.
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A. V. Fursikov and O. Yu. Imanuvilov, On controllability of certain systems simulating a fluid flow.IMA vol. in Math. and Appl. “Flow Control,” Ed. M. D. Gunzburger,Springer Verlag, New York,68 (1994) 149–184.
—, On exact boundary zero controllability of the two-dimensional Navier-Stokes equations.Acta Appl. Math.,36 (1994), 67–76.
J.-L. Lions and E. Magenes, Problemes aux limites non homogènes et applications.Dunod, Paris 1 (1968).
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and quasilinear parabolic equations. (Russian)Moscow, Nauka, 1967, 1–736.
M. S. Agranovich and M. I. Vishik, Elliptic boundary value problems with parameter and general parabolic boundary value problems.Russian Math. Surveys 19 (1964), No. 3, 53–161.
V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal control.Consultants Bureau, New York London, 1987.
O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow.Gordon and Breach, New York, 1963.
R. Temam, Navier-Stokes equations, theory and numerical analysis.North-Holland Publishing Company, Amsterdam, 1979.
V. A. Solonnikov, Overdetermined elliptic boundary value problems.Notes of Sci. Semin. of the Leningrad Dept. of Steklov Math. Inst.,21 (1977), No. 5, 112–158.
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Supported by Paderborn University-GH while visiting there and by Grant No. M76000 from the International Science Foundation.
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Fursikov, A.V. Exact boundary zero controllability of three-dimensional Navier-Stokes equations. Journal of Dynamical and Control Systems 1, 325–350 (1995). https://doi.org/10.1007/BF02269373
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DOI: https://doi.org/10.1007/BF02269373