# Stopping a viscous fluid by a feedback dissipative field: II. The stationary Navier-Stokes problem

Stanislav Nikolaevich Antontsev; Jesús Ildefonso Díaz; Hermenegildo Borges de Oliveira

- Volume: 15, Issue: 3-4, page 257-270
- ISSN: 1120-6330

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topAntontsev, Stanislav Nikolaevich, Díaz, Jesús Ildefonso, and de Oliveira, Hermenegildo Borges. "Stopping a viscous fluid by a feedback dissipative field: II. The stationary Navier-Stokes problem." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 15.3-4 (2004): 257-270. <http://eudml.org/doc/252413>.

@article{Antontsev2004,

abstract = {We consider a planar stationary flow of an incompressible viscous fluid in a semi-infinite strip governed by the Navier-Stokes system with a feed-back body forces field which depends on the velocity field. Since the presence of this type of non-linear terms is not standard in the fluid mechanics literature, we start by establishing some results about existence and uniqueness of weak solutions. Then, we prove how this fluid can be stopped at a finite distance of the semi-infinite strip entrance by means of this body forces field which depends in a sub-linear way on the velocity field. This localization effect is proved by reducing the problem to a fourth order non-linear one for which the localization of solutions is obtained by means of a suitable energy method.},

author = {Antontsev, Stanislav Nikolaevich, Díaz, Jesús Ildefonso, de Oliveira, Hermenegildo Borges},

journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},

keywords = {Navier-Stokes system; Body forces field; Non-linear fourth order equation; Energy method; Localization effect; body forces field; energy method; localization},

language = {eng},

month = {12},

number = {3-4},

pages = {257-270},

publisher = {Accademia Nazionale dei Lincei},

title = {Stopping a viscous fluid by a feedback dissipative field: II. The stationary Navier-Stokes problem},

url = {http://eudml.org/doc/252413},

volume = {15},

year = {2004},

}

TY - JOUR

AU - Antontsev, Stanislav Nikolaevich

AU - Díaz, Jesús Ildefonso

AU - de Oliveira, Hermenegildo Borges

TI - Stopping a viscous fluid by a feedback dissipative field: II. The stationary Navier-Stokes problem

JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

DA - 2004/12//

PB - Accademia Nazionale dei Lincei

VL - 15

IS - 3-4

SP - 257

EP - 270

AB - We consider a planar stationary flow of an incompressible viscous fluid in a semi-infinite strip governed by the Navier-Stokes system with a feed-back body forces field which depends on the velocity field. Since the presence of this type of non-linear terms is not standard in the fluid mechanics literature, we start by establishing some results about existence and uniqueness of weak solutions. Then, we prove how this fluid can be stopped at a finite distance of the semi-infinite strip entrance by means of this body forces field which depends in a sub-linear way on the velocity field. This localization effect is proved by reducing the problem to a fourth order non-linear one for which the localization of solutions is obtained by means of a suitable energy method.

LA - eng

KW - Navier-Stokes system; Body forces field; Non-linear fourth order equation; Energy method; Localization effect; body forces field; energy method; localization

UR - http://eudml.org/doc/252413

ER -

## References

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- GILBARG, D. - TRUDINGER, N.S., Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin-Heidelberg1998. Zbl0361.35003
- HORGAN, C.O., Plane entry flows and energy estimates for the Navier-Stokes equations. Arch. Rat. Mech. and Analysis, 68, 1978, 359-381. Zbl0397.76025MR521600
- KNOWLES, J.K., On Saint-Venant’s Principle in the Two-Dimensional Linear Theory of Elasticity. Arch. Ration. Mech. Anal., 21, 1966, 1-22. Zbl0283.73005MR187480
- LADYZHENSKAYA, O.A., The mathematical theory of viscous incompressible flow. Mathematics and its Applications, 2, Gordon and Breach, New York1969. Zbl0121.42701MR254401
- TOUPIN, R.A., Saint-Venant’s Principle. Arch. Ration. Mech. Anal., 18, 1965, 83-96. Zbl0203.26803MR172506

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