We study the inverse of the divergence operator on a domain perforated by a system of tiny holes. We show that such inverse can be constructed on the Lebesgue space for any , with a norm independent of perforation, provided the holes are suitably small and their mutual distance suitably large. Applications are given to problems arising in homogenization of steady compressible fluid flows.
DOI : 10.1051/cocv/2016016
Mots-clés : Perforated domains, Bogovskii type operators, homogenization, compressible Navier–Stokes system
@article{COCV_2017__23_3_851_0,
author = {Diening, Lars and Feireisl, Eduard and Lu, Yong},
title = {The inverse of the divergence operator on perforated domains with applications to homogenization problems for the compressible {Navier{\textendash}Stokes} system},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {851--868},
publisher = {EDP-Sciences},
volume = {23},
number = {3},
year = {2017},
doi = {10.1051/cocv/2016016},
mrnumber = {3660451},
zbl = {1375.35026},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2016016/}
}
TY - JOUR AU - Diening, Lars AU - Feireisl, Eduard AU - Lu, Yong TI - The inverse of the divergence operator on perforated domains with applications to homogenization problems for the compressible Navier–Stokes system JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 851 EP - 868 VL - 23 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2016016/ DO - 10.1051/cocv/2016016 LA - en ID - COCV_2017__23_3_851_0 ER -
%0 Journal Article %A Diening, Lars %A Feireisl, Eduard %A Lu, Yong %T The inverse of the divergence operator on perforated domains with applications to homogenization problems for the compressible Navier–Stokes system %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 851-868 %V 23 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2016016/ %R 10.1051/cocv/2016016 %G en %F COCV_2017__23_3_851_0
Diening, Lars; Feireisl, Eduard; Lu, Yong. The inverse of the divergence operator on perforated domains with applications to homogenization problems for the compressible Navier–Stokes system. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 851-868. doi : 10.1051/cocv/2016016. http://www.numdam.org/articles/10.1051/cocv/2016016/
, and , Solutions of the divergence operator on John domains. Adv. Math. 206 (2006) 373–401. | DOI | MR | Zbl
, Homogenization of the Stokes flow in a connected porous medium. Asymptot. Anal. 2 (1989) 203–222. | MR | Zbl
, 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 (1990) 209–259. | DOI | MR | Zbl
, Homogenization of the Navier−Stokes equations in open sets perforated with tiny holes. II. Noncritical sizes of the holes for a volume distribution and a surface distribution of holes. Arch. Ration. Mech. Anal. 113 (1990) 261–298. | DOI | MR | Zbl
, Solution of some vector analysis problems connected with operators div and grad. Tr. Sem. S.L. Soboleva 80 (1980) 5–40. In Russian. | MR | Zbl
and , On weak solutions of steady Navier−Stokes equations for monatomic gas. Comment. Math. Univ. Carolin. 49 (2008) 611–632. | MR | Zbl
, and , A decomposition technique for John domains. Ann. Acad. Sci. Fenn. 35 (2010) 87–114. | DOI | MR | Zbl
and , Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989) 511–547. | DOI | MR | Zbl
and , Homogenization of stationary Navier−Stokes equations in domains with tiny holes. J. Math. Fluid Mech. 17 (2015) 381–392. | DOI | MR | Zbl
, and , On the existence of globally defined weak solutions to the Navier−Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech. 3 (2001) 358–392. | DOI | MR | Zbl
E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids. Birkhäuser Verlag, Basel (2009). | MR | Zbl
, and , Homogenization and singular limits for the complete Navier−Stokes−Fourier system. J. Math. Pures Appl. 94 (2010) 33–57. | DOI | MR | Zbl
, and , Homogenization of the evolutionary Navier−-Stokes system. Manusc. Math. 149 (2016) 251–274. | DOI | MR | Zbl
, and , The Dirichlet problem for steady viscous compressible flow in three dimensions. Journal de Mathématiques Pures et Appliquées 97 (2012) 85–97. | DOI | MR | Zbl
G.P. Galdi, An Introduction to the Mathematical Theory of the Navier−Stokes Equations: Steady-State Problems. Springer Science and Business Media (2011). | MR | Zbl
P.-L. Lions,Mathematical topics in fluid dynamics, Compressible models. Oxford Science Publication, Oxford (1998). Vol. 2. | MR | Zbl
, Homogenization of the compressible Navier−Stokes equations in a porous medium. ESAIM: COCV 8 (2002) 885–906. | Numdam | MR | Zbl
, Homogenization of nonstationary Navier−Stokes equations in a domain with a grained boundary. Ann. Mat. Pura Appl. 158 (1991) 167–179. | DOI | MR | Zbl
A. Novotný and I. Straskraba,Introduction to the mathematical theory of compressible flow. Oxford University Press, Oxford (2004). | MR | Zbl
P. Plotnikov and J. Sokolowski, Compressible Navier−Stokes equations. Vol. 73 of Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)]. Birkhäuser/Springer Basel AG, Basel (2012). Theory and shape optimization. | MR | Zbl
and , Steady 3D viscous compressible flows with adiabatic exponent . J. Math. Pures Appl. 104 (2015) 58–82. | DOI | MR | Zbl
E. Sánchez-Palencia, Non homogeneous media and vibration theory. Vol. 127 of Lect. Notes Phys. Springer-Verlag (1980). | MR | Zbl
L. Tartar, Incompressible fluid flow in a porous medium: convergence of the homogenization process, in Nonhomogeneous media and vibration theory, edited by E. Sánchez-Palencia (1980) 368–377.
Cité par Sources :
