Journal of Mathematical Fluid Mechanics

, Volume 6, Issue 4, pp 439–461 | Cite as

Stopping a Viscous Fluid by a Feedback Dissipative Field: I. The Stationary Stokes Problem

  • S. N. Antontsev
  • J. I. Díaz
  • H. B. de Oliveira
Original paper

Abstract.

We consider a planar stationary flow of an incompressible viscous fluid in a semi-infinite strip governed by the Stokes system with a body forces field. We show how this fluid can be stopped at a finite distance of the entrance of the semi-infinite strip by means of a feedback field depending in a sub-linear way on the velocity field. This localization effect is proved reducing the problem to a non-linear bi-harmonic type one for which the localization of solutions is obtained by means of the application of a suitable energy method. Since the presence of the non-linear terms defined through the body forces field is not standard in the fluid mechanics literature, we establish also some results about the existence and uniqueness of weak solutions for this problem.

Mathematics Subject Classification (2000).

35Q30 76D07 76D03 35J30 53R35 76E30 35B99 35G30 35J60 

Keywords.

Stokes system feedback dissipative field non-linear higher order equation energy method localization effect 

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Copyright information

© Birkhäuser Verlag, Basel 2004

Authors and Affiliations

  • S. N. Antontsev
    • 1
  • J. I. Díaz
    • 2
  • H. B. de Oliveira
    • 3
  1. 1.Departamento de MatemáticaUniversidade da Beira InteriorCovilhãPortugal
  2. 2.Facultad de MatematicasUniversidad ComplutenseMadridSpain
  3. 3.Faculdade de Ciências e TecnologiaUniversidade do AlgarveFaroPortugal

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