Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T19:21:02.775Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic solution for the perturbed Stokes problem in a bounded domain in two and three dimensions

Published online by Cambridge University Press:  14 November 2011

Dialla Konate
Dialla Konate
Affiliation:
LICIA, 37 Rue de la République, 92800 Puteaux, France ([email protected])
Get access Rights & Permissions [Opens in a new window]

Extract

We consider the Stokes problem with a small viscosity. When the viscosity goes to zero, the boundary-layer phenomenon can appear. In this case, the solution of the given perturbed Stokes equation cannot be properly approximated by the solution of its limiting equation ‘near’ the boundary Γ of the domain of study, say Ω To overcome this problem, we need to construct a corrector term in the neighbourhood of Γ Lions has studied this problem and has constructed a corrector for the case where Ω is a half space in ℝ2. The case where Ω is an open and bounded domain of ℝ2 or ℝ3, which remained unsolved, is the concern of this paper. The construction of the corrector to the perturbed Stokes equation depends heavily on the geometry of Ω In two dimensions, we construct the corrector in the form of a stream function, while in ℝ3 we construct it in the form of a potential vector. The corrector acts effectively in a neighbourhood of Γ that is the boundary layer. Using similar methods to those of Baranger and Tartar, we define the thickness of the boundary layer in a natural way. In addition, in this paper we study the behaviour of the corrected solution in some Hölder spaces.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 4 , 1999 , pp. 811 - 824
Copyright
Copyright © Royal Society of Edinburgh 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Baranger, J.. Estimations d'erreur à l'intérieur pour un problème de couche limite. In Singular perturbation and boundary layer theory. Lecture Notes in Mathematics, no. 594 (Berlin: Springer, 1977).Google Scholar
2Baranger, J.. On the thickness of boundary layer in elliptic singular perturbation problems. In Numerical analysis of singular perturbation problems (ed. Hemker, P. W. and Miller, J. J. H.) (Academic, 1979).Google Scholar
3Galdi, G. P.. An introduction to the mathematical theory of the Navier–Stokes equations, vol. 1. Linearized steady problems. Springer Tracts in Natural Philosophy, no. 38 (Springer, 1994).Google Scholar
4Lions, J.-L.. Perturbations singulières dans les problémes aux limites et en contrôle optimal. Lecture Notes in Mathematics, no. 323 (Springer, 1973).CrossRefGoogle Scholar
5Lions, J.-L. and Magenes, E.. Problèmes aux limites non homogènes et applications, vol. 1 (Paris: Dunod, 1970).Google Scholar
6Temam, R.. Navier–Stokes equations (North-Holland, 1984)Google Scholar