Comptes Rendus
On the confinement of a viscous fluid by means of a feedback external field
Comptes Rendus. Mécanique, Volume 330 (2002) no. 12, pp. 797-802.

In this work we consider a planar stationary flow of an incompressible viscous fluid in a semi-infinite strip governed by the standard Stokes system. We show how this fluid can be stopped at a finite distance from the entrance of the semi-infinite strip by means of a feedback source depending in a sublinear way on the velocity field. This localization effect is proved by reducing the problem to a non-linear biharmonic type one for which the localization of solutions is obtained through the application of an energy method, in the spirit of the monograph by S.N. Antontsev, J.I. Díaz and S.I. Shmarev (Energy Methods for Free Boundary Problems: Applications to Non-Linear PDEs and Fluid Mechanics, Birkäuser, Boston, 2002). Since the presence of the non-linear terms defined by the source is not standard in fluid mechanics literature, we give also some results about the existence and uniqueness of weak solutions for this problem.

Dans ce travail nous considérons un écoulement plan stationaire incompressible d'un fluide visqueux situé dans un milieu semi-infini régi par le système de Stokes standard. Nous montrons comment ce fluide peut être arrêté à une distance finie de l'entrée du milieu, utilisant une source avec mémoire dépendant d'une manière sous-linéaire du champ de vitesses. Cet effet de localisation est atteint en réduisant le problème à un problème non linéaire du type biharmonique, où la localisation des solutions est obtenue par l'application d'une méthode d'énergie dans l'esprit de la monographie de S.N. Antontsev, J.I. Dı́az and S.I. Shmarev (Energy Methods for Free Boundary Problems: Applications to Non-Linear PDEs and Fluid Mechanics, Birkäuser, Boston, 2002). En outre, du fait que la présence de terms non linéaires definis par la source est non fréquente dans la littérature de la mécanique des fluides, nous donnons aussi des résultats sur l'existence et l'unicité des solutions faibles de ce problème.

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DOI: 10.1016/S1631-0721(02)01536-X
Keywords: computational fluid mechanics, Stokes system, feedback dissipative field, non-linear higher order equation, energy method, localization effect
Mot clés : mécanique des fluides numérique, systéme de Stokes, champ avec mémoire dissipatif, equations non-linèaires d'ordres superieures, méthode d'énergie, effet de localisation

S.N. Antontsev 1; J.I. Dı́az 2; H.B. de Oliveira 3

1 Departamento de Matemática, Universidade da Beira Interior, 6201-001 Covilhã, Portugal
2 Facultad de Matematicas, Universidad Complutense, 28040 Madrid, Spain
3 Faculdade de Ciências e Tecnologia, Universidade do Algarve, 8000-062 Faro, Portugal
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S.N. Antontsev; J.I. Dı́az; H.B. de Oliveira. On the confinement of a viscous fluid by means of a feedback external field. Comptes Rendus. Mécanique, Volume 330 (2002) no. 12, pp. 797-802. doi : 10.1016/S1631-0721(02)01536-X. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(02)01536-X/

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