Geodesic
Stopping a viscous fluid by a feedback dissipative field: II. The stationary Navier-Stokes problem
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 3-4, pp. 257-270.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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. 
@article{RLIN_2004_9_15_3-4_a8,
     author = {Antontsev, Stanislav Nikolaevich and D{\'\i}az, Jes\'us Ildefonso and de Oliveira, Hermenegildo Borges},
     title = {Stopping a viscous fluid by a feedback dissipative field: {II.} {The} stationary {Navier-Stokes} problem},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
     pages = {257--270},
     publisher = {mathdoc},
     volume = {Ser. 9, 15},
     number = {3-4},
     year = {2004},
     zbl = {1105.35074},
     mrnumber = {1858749},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RLIN_2004_9_15_3-4_a8/}
}
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. Rendiconti Lincei. Matematica e applicazioni
PY  - 2004
SP  - 257
EP  - 270
VL  - 15
IS  - 3-4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/RLIN_2004_9_15_3-4_a8/
LA  - en
ID  - RLIN_2004_9_15_3-4_a8
ER  - 
%0 Journal Article
%A Antontsev, Stanislav Nikolaevich
%A Díaz, Jesús Ildefonso
%A de Oliveira, Hermenegildo Borges
%T Stopping a viscous fluid by a feedback dissipative field: II. The stationary Navier-Stokes problem
%J Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
%D 2004
%P 257-270
%V 15
%N 3-4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/RLIN_2004_9_15_3-4_a8/
%G en
%F RLIN_2004_9_15_3-4_a8
Antontsev, Stanislav Nikolaevich; Díaz, Jesús Ildefonso; 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. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 3-4, pp. 257-270. http://geodesic.mathdoc.fr/item/RLIN_2004_9_15_3-4_a8/

[1] S.N. Antontsev - J.I. Díaz - H.B. De Oliveira, Stopping a viscous fluid by a feedback dissipative external field: I. The stationary Stokes equations. Book of abstratcs of NSEC8, Euler International Mathematical Institute, St. Petersburg 2002. | Zbl

[2] S.N. Antontsev - J.I. Díaz - H.B. De Oliveira, On the confinement of a viscous fluid by means of a feedback external field. C.R. Mécanique, 330, 2002, 797-802. | Zbl

[3] S.N. Antontsev - J.I. Díaz - H.B. De Oliveira, Stopping a viscous fluid by a feedback dissipative field: I. The stationary Stokes problem. J. Math. Fluid Mech., to appear. | Zbl

[4] S.N. Antontsev - J.I. Díaz - S.I. Shmarev, Energy Methods for Free Boundary Problems, Applications to non-linear PDEs and fluid mechanics. Birkhäuser, Boston 2002. | DOI | MR | Zbl

[5] F. Bernis, Extinction of the solutions of some quasilinear elliptic problems of arbitrary order: Part 1. Proc. Symp. Pure Math., 45, 1986, 125-132. | MR | Zbl

[6] F. Bernis, Qualitative properties for some non-linear higher order degenerate parabolic equations. Houston J. Math., 14, 1988, 319-352. | MR | Zbl

[7] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Nonlinear Steady Problems. Springer-Verlag, New York 1994. | DOI | MR | Zbl

[8] D. Gilbarg - N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin-Heidelberg 1998. | Zbl

[9] C.O. Horgan, Plane entry flows and energy estimates for the Navier-Stokes equations. Arch. Rat. Mech. and Analysis, 68, 1978, 359-381. | MR | Zbl

[10] J.K. Knowles, On Saint-Venant’s Principle in the Two-Dimensional Linear Theory of Elasticity. Arch. Ration. Mech. Anal., 21, 1966, 1-22. | MR | Zbl

[11] O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow. Mathematics and its Applications, 2, Gordon and Breach, New York 1969. | MR | Zbl

[12] R.A. Toupin, Saint-Venant’s Principle. Arch. Ration. Mech. Anal., 18, 1965, 83-96. | MR | Zbl