Skip to main content
SpringerLink
Log in

Exact boundary zero controllability of three-dimensional Navier-Stokes equations

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

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,

$$\parallel v(t, \cdot )\parallel _{H^3 (\Omega )} \leqslant c\exp \left( { - k/(T - t)^2 } \right) as t \to T$$

, wherec>0,k>0 are certain constants.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 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.

    Google Scholar 

  2. —, On exact boundary zero controllability of the two-dimensional Navier-Stokes equations.Acta Appl. Math.,36 (1994), 67–76.

    Google Scholar 

  3. J.-L. Lions and E. Magenes, Problemes aux limites non homogènes et applications.Dunod, Paris 1 (1968).

    Google Scholar 

  4. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and quasilinear parabolic equations. (Russian)Moscow, Nauka, 1967, 1–736.

    Google Scholar 

  5. 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.

    Google Scholar 

  6. V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal control.Consultants Bureau, New York London, 1987.

    Google Scholar 

  7. O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow.Gordon and Breach, New York, 1963.

    Google Scholar 

  8. R. Temam, Navier-Stokes equations, theory and numerical analysis.North-Holland Publishing Company, Amsterdam, 1979.

    Google Scholar 

  9. 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.

    Google Scholar 

Download references

Author information

Author notes

  1. A. V. Fursikov

    Present address: Department of Mechanics and Mathematics, Moscow State University, 119899, Moscow, Russia

Authors and Affiliations

    Authors
    1. A. V. Fursikov
      View author publications

      You can also search for this author in PubMed Google Scholar

    Additional information

    Supported by Paderborn University-GH while visiting there and by Grant No. M76000 from the International Science Foundation.

    Rights and permissions

    Reprints and permissions

    About this article

    Cite this article

    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

    Download citation

    • Received:

    • Issue Date:

    • DOI: https://doi.org/10.1007/BF02269373

    1991 Mathematics Subject Classification

    Search

    Navigation