Homogenization for the Poisson equation in randomly perforated domains under minimal assumptions on the size of the holes

Giunti, Arianna; Höfer, Richard M.; Velázquez, Juan J. L.

doi:10.1080/03605302.2018.1531425

Cited by 31 publications

(81 citation statements)

References 20 publications

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“…These properties are split into two lemmas. The first one is analogous to the corresponding lemma in [10], the other one gives more detailed informations on the geometry of the clusters of H ε and is the result which requires the strengthened version (1.7) of (1.6). In subsection 3.2, we prove the results stated in Section 3.…”

Section: Introductionmentioning

confidence: 92%

“…Assumption (1.6) is minimal in order to have that this quantity is finite in average, but does not exclude that with overwhelming probability some balls generating H ε overlap. For further comments on this, we refer to the introduction in [10]. The main challenge in proving the results of this paper is related to the regions of H ε where there are clustering effects.…”

Section: Introductionmentioning

confidence: 96%

“…This work is an adaptation to the Stokes equations of the homogenization result obtained in [10] for the Poisson equation. In particular, the class of random holes considered in the current paper is included in the class studied in [10]. In the latter, it is assumed that the identically distributed radii ρ i in (1.2) satisfy ρ d−2 < +∞.…”

Section: Introductionmentioning

confidence: 99%

“…In Appendix B, we give some standard estimates for the solutions of the Stokes equations in annuli and exterior domains. In Appendix C, we recall some results concerning the Strong Law of Large Numbers, which have been proved in detail in [10] and which are used also throughout this paper.…”

Section: Introductionmentioning

confidence: 99%

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Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes

Giunti¹,

Höfer²

2019

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We prove the homogenization to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. The fluid satisfies a no-slip boundary condition at the holes. The balls generating the holes have centres distributed according to a Poisson point process and i.i.d. unbounded radii satisfying a suitable moment condition. We stress that our assumption on the distribution of the radii does not exclude that, with overwhelming probability, the holes contain clusters made by many overlapping balls. We show that the formation of these clusters has no effect on the limit Brinkman equations. In contrast with the case of the Poisson equation studied in [A. Giunti, R. Höfer, and J.J.L. Velázquez, Homogenization for the Poisson equation in randomly perforated domains under minimal assumptions on the size of the holes], the incompressibility condition requires a more detailed study of the geometry of the random holes generated by the class of probability measures considered.in a domain D ε , that is obtained by removing from a bounded set D ⊆ R d , d > 2, a random number of small balls having random centres and radii. More precisely, for ε > 0, we define 2)1 similar size. We emphasize, however, that it neither prevents the balls generating H ε from overlapping, nor it implies a uniform upper bound on the number of balls of very different size which combine into a cluster (see Section 6). The main technical effort of this paper goes into developing a strategy to deal with these geometric considerations and succeed in controlling the term in (1.3). We refer to Subsection 2.3 for a more detailed discussion on our strategy. We also mention that, to avoid further technicalities, we only treat the case where the centres of the balls in (1.2) are distributed according to a homogeneous Poisson point process. It is easy to check that our result applies both to the case of periodic centres and to any (short-range) correlated point process for which the results contained in Appendix C hold.After Brinkman proposed the equations (1.3) in [3] for the fluid flow in porous media, an extensive literature has been developed to obtain a rigorous derivation of (1.3) from (1.1) in the case of periodic configuration of holes [2,15,20,16]. We take inspiration in particular from [1], where the method used in [5] for the Poisson equations is adapted to treat the case of the Stokes equations in domains with periodic holes of arbitrary and identical shape. In [1], by a compactness argument, the same techniques used for the Stokes equations also provide the analogous result in the case of the stationary Navier-Stokes equations. The same is true also in our setting (see Remark 2.2 in Section 2). In [6], with methods similar to [1] and [5], the homogenization of stationary Stokes and Navier-Stokes equations has been extended also to the case of spherical holes where different and constant Dirichlet boundary conditions are prescribed at the boundary of each ball. This corre...

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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes

Giunti¹,

Höfer²

2019

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…Moreover, a corresponding framework for quantification was established by Kacimi and Murat [21]. The framework was later extended by Allaire to Stokes and Navier-Stokes problems [1,2], to the obstacle problems by Caffarelli and Mellet [8], and to random settings by Hoàng [18,17]; see also [13] and see [16,15] for randomly perforated domains based on Poisson piont processes.…”

Section: )mentioning

confidence: 99%

A Unified Homogenization Approach for the Dirichlet Problem in Perforated Domains

Jing¹

2020

SIAM J. Math. Anal.

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We revisit the periodic homogenization of Dirichlet problems for the Laplace operator in perforated domains, and establish a unified proof that works for different regimes of hole-cell ratios, that is the ratio between the scaling factor of the holes and that of the periodic cells. The approach is then made quantitative and it yields correctors and error estimates for vanishing holecell ratios. For positive volume fraction of holes, the approach is just the standard oscillating test function method; for vanishing volume fraction of holes, we study asymptotic behaviors of a properly rescaled cell problems and use them to build oscillating test functions. Our method reveals how the different regimes are intrinsically connected through the cell problems and the connection with periodic layer potentials.

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Homogenization of the heat equation in a noncylindrical domain with randomly oscillating boundary

Nandakumaran

Sankar

2022

Math Methods in App Sciences

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Award Number: No.F.4-2/2006 (BSR)/MA/19-20/0060In this article, we study the homogenization of heat equations in a domain with randomly oscillating boundary parts. The random oscillating boundary is time-dependent and confined by a stationary random field. Here, we follow a new homogenization technique that deals with the evolving domains, which covers many applications. We obtain the asymptotic limit as 𝜀 → 0 in the reference configuration, in which the heat equation becomes a parabolic equation with random oscillating coefficients in the reference domain. To the best of our knowledge, this is the first result of the homogenization of problems on the random evolving boundary domain. One of the major contributions is the corrector result which we establish in this article.

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Homogenization for the Poisson equation in randomly perforated domains under minimal assumptions on the size of the holes

Cited by 31 publications

References 20 publications

Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes

Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes

A Unified Homogenization Approach for the Dirichlet Problem in Perforated Domains

Homogenization of the heat equation in a noncylindrical domain with randomly oscillating boundary

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