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Convergence Rates and Fluctuations for the Stokes–Brinkman Equations as Homogenization Limit in Perforated Domains

Höfer, Richard M. and Jansen, Jonas LU (2024) In Archive for Rational Mechanics and Analysis 248(3).
Abstract

We study the homogenization of the Dirichlet problem for the Stokes equations in R3 perforated by m spherical particles. We assume the positions and velocities of the particles to be identically and independently distributed random variables. In the critical regime, when the radii of the particles are of order m-1, the homogenization limit u is given as the solution to the Brinkman equations. We provide optimal rates for the convergence um→u in L2, namely m-β for all β<1/2. Moreover, we consider the fluctuations. In the central limit scaling, we show that these converge to a Gaussian field, locally in L2(R3), with an explicit covariance. Our analysis is... (More)

We study the homogenization of the Dirichlet problem for the Stokes equations in R3 perforated by m spherical particles. We assume the positions and velocities of the particles to be identically and independently distributed random variables. In the critical regime, when the radii of the particles are of order m-1, the homogenization limit u is given as the solution to the Brinkman equations. We provide optimal rates for the convergence um→u in L2, namely m-β for all β<1/2. Moreover, we consider the fluctuations. In the central limit scaling, we show that these converge to a Gaussian field, locally in L2(R3), with an explicit covariance. Our analysis is based on explicit approximations for the solutions um in terms of u as well as the particle positions and their velocities. These are shown to be accurate in H˙1(R3) to order m-β for all β<1. Our results also apply to the analogous problem regarding the homogenization of the Poisson equations.

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Contribution to journal
publication status
published
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in
Archive for Rational Mechanics and Analysis
volume
248
issue
3
article number
50
publisher
Springer
external identifiers
  • scopus:85194076215
ISSN
0003-9527
DOI
10.1007/s00205-024-01993-x
language
English
LU publication?
yes
id
ac9f6bbb-479a-485f-a2aa-b17c0ee84740
date added to LUP
2024-05-31 09:20:26
date last changed
2025-04-04 14:08:02
@article{ac9f6bbb-479a-485f-a2aa-b17c0ee84740,
  abstract     = {{<p>We study the homogenization of the Dirichlet problem for the Stokes equations in R<sup>3</sup> perforated by m spherical particles. We assume the positions and velocities of the particles to be identically and independently distributed random variables. In the critical regime, when the radii of the particles are of order m<sup>-1</sup>, the homogenization limit u is given as the solution to the Brinkman equations. We provide optimal rates for the convergence u<sub>m</sub>→u in L<sup>2</sup>, namely m<sup>-β</sup> for all β&lt;1/2. Moreover, we consider the fluctuations. In the central limit scaling, we show that these converge to a Gaussian field, locally in L<sup>2</sup>(R<sup>3</sup>), with an explicit covariance. Our analysis is based on explicit approximations for the solutions u<sub>m</sub> in terms of u as well as the particle positions and their velocities. These are shown to be accurate in H˙<sup>1</sup>(R<sup>3</sup>) to order m<sup>-β</sup> for all β&lt;1. Our results also apply to the analogous problem regarding the homogenization of the Poisson equations.</p>}},
  author       = {{Höfer, Richard M. and Jansen, Jonas}},
  issn         = {{0003-9527}},
  language     = {{eng}},
  number       = {{3}},
  publisher    = {{Springer}},
  series       = {{Archive for Rational Mechanics and Analysis}},
  title        = {{Convergence Rates and Fluctuations for the Stokes–Brinkman Equations as Homogenization Limit in Perforated Domains}},
  url          = {{http://dx.doi.org/10.1007/s00205-024-01993-x}},
  doi          = {{10.1007/s00205-024-01993-x}},
  volume       = {{248}},
  year         = {{2024}},
}