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_257_0,
author = {Stanislav Nikolaevich Antontsev and Jes\'us Ildefonso D\'\i az and Hermenegildo Borges de Oliveira},
title = {Stopping a viscous fluid by a feedback dissipative field: II. The stationary Navier-Stokes problem},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
volume = {15},
year = {2004},
pages = {257-270},
zbl = {1105.35074},
mrnumber = {2148884},
language = {en},
url = {http://dml.mathdoc.fr/item/RLIN_2004_9_15_3-4_257_0}
}
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. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 15 (2004) pp. 257-270. http://gdmltest.u-ga.fr/item/RLIN_2004_9_15_3-4_257_0/
[1] - - , 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 1082.35117
[2] - - , On the confinement of a viscous fluid by means of a feedback external field. C.R. Mécanique, 330, 2002, 797-802. | Zbl pre05563590
[3] - - , Stopping a viscous fluid by a feedback dissipative field: I. The stationary Stokes problem. J. Math. Fluid Mech., to appear. | Zbl 1075.35029
[4] - - , Energy Methods for Free Boundary Problems, Applications to non-linear PDEs and fluid mechanics. Birkhäuser, Boston 2002. | MR 1858749 | Zbl 0988.35002
[5] , Extinction of the solutions of some quasilinear elliptic problems of arbitrary order: Part 1. Proc. Symp. Pure Math., 45, 1986, 125-132. | MR 843554 | Zbl 0604.35022
[6] , Qualitative properties for some non-linear higher order degenerate parabolic equations. Houston J. Math., 14, 1988, 319-352. | MR 985928 | Zbl 0682.35009
[7] , An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Nonlinear Steady Problems. Springer-Verlag, New York1994. | MR 1284206 | Zbl 0949.35005
[8] - , Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin-Heidelberg 1998. | Zbl 0361.35003
[9] , Plane entry flows and energy estimates for the Navier-Stokes equations. Arch. Rat. Mech. and Analysis, 68, 1978, 359-381. | MR 521600 | Zbl 0397.76025
[10] , On Saint-Venant’s Principle in the Two-Dimensional Linear Theory of Elasticity. Arch. Ration. Mech. Anal., 21, 1966, 1-22. | MR 187480 | Zbl 0283.73005
[11] , The mathematical theory of viscous incompressible flow. Mathematics and its Applications, 2, Gordon and Breach, New York1969. | MR 254401 | Zbl 0121.42701
[12] , Saint-Venant’s Principle. Arch. Ration. Mech. Anal., 18, 1965, 83-96. | MR 172506 | Zbl 0203.26803