Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T20:52:26.300Z Has data issue: false hasContentIssue false
  • English
  • Français

$L^{p}$ ESTIMATES FOR THE HOMOGENIZATION OF STOKES PROBLEM IN A PERFORATED DOMAIN

Part of: Incompressible viscous fluids Partial differential equations Basic methods in fluid mechanics

Published online by Cambridge University Press:  10 April 2018

Amina Mecherbet  and
Matthieu Hillairet
Amina Mecherbet
Affiliation:
Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, CNRS, Place Eugène Bataillon, 34090 Montpellier, France (amina.mecherbet@umontpellier.fr; matthieu.hillairet@umontpellier.fr)
Matthieu Hillairet
Affiliation:
Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, CNRS, Place Eugène Bataillon, 34090 Montpellier, France (amina.mecherbet@umontpellier.fr; matthieu.hillairet@umontpellier.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we consider the Stokes equations in a perforated domain. When the number of holes increases while their radius tends to 0, it is proven in Desvillettes et al. [J. Stat. Phys. 131 (2008) 941–967], under suitable dilution assumptions, that the solution is well approximated asymptotically by solving a Stokes–Brinkman equation. We provide here quantitative estimates in $L^{p}$-norms of this convergence.

Keywords

Stokes equationshomogenizationsuspension flows

MSC classification

Primary: 76D07: Stokes and related (Oseen, etc.) flows 76M50: Homogenization 35B27: Homogenization; equations in media with periodic structure
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 1 , January 2020 , pp. 231 - 258
Copyright
© Cambridge University Press 2018 

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

Allaire, G., Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Ration. Mech. Anal. 113(3) (1990), 209–259.Google Scholar
Beliaev, A. Yu. and Kozlov, S. M., Darcy equation for random porous media, Comm. Pure Appl. Math. 49(1) (1996), 1–34.Google Scholar
Boudin, L., Desvillettes, L., Grandmont, C. and Moussa, A., Global existence of solutions for the coupled Vlasov and Navier–Stokes equations, Differ. Integral Equ. 22(11–12) (2009), 1247–1271.Google Scholar
Cioranescu, D. and Murat, F., Un terme étrange venu d’ailleurs, in Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, Vol. II (Paris, 1979/1980), Research Notes in Mathematics, Volume 60, pp. 98–138, 389–390 (Pitman, Boston, MA, London, 1982).Google Scholar
Desvillettes, L., Golse, F. and Ricci, V., The mean field limit for solid particles in a Navier–Stokes flow, J. Stat. Phys. 131 (2008), 941–967.Google Scholar
Galdi, G. P., An Introduction to the Mathematical Theory of the Navier–Stokes Equations, 2nd edn, Springer Monographs in Mathematics, (Springer, New York, 2011).Google Scholar
Hamdache, K., Global existence and large time behaviour of solutions for the Vlasov–Stokes equations, Jpn. J. Ind. Appl. Math. 15(1) (1998), 51–74.Google Scholar
Hauray, M. and Jabin, P.-E., Particle approximation of Vlasov equations with singular forces: propagation of chaos, Ann. Sci. Éc. Norm. Supér. (4) 48(4) (2015), 891–940.Google Scholar
Hillairet, M., On the homogenization of the Stokes problem in a perforated domain,https://hal.archives-ouvertes.fr/hal-01302560, August 2016.Google Scholar
Landau, L. D. and Lifshitz, E. M., Fluid Mechanics, Course of Theoretical Physics, Volume 6, (Pergamon Press, London, 1959). Translated from the Russian by J. B. Sykes and W. H. Reid.Google Scholar
Mischler, S. and Mouhot, C., Kac’s program in kinetic theory, Invent. Math. 193(1) (2013), 1–147.Google Scholar
Raymond, J.-P., Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire 24(6) (2007), 921–951.Google Scholar
Rubinstein, J., On the macroscopic description of slow viscous flow past a random array of spheres, J. Stat. Phys. 44(5–6) (1986), 849–863.Google Scholar
Villani, C., Optimal Transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Volume 338, (Springer, Berlin, 2009). Old and new.Google Scholar