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Corrector estimates and homogenization errors of unsteady flow ruled by Darcy's law

Li Wang, Qiang Xu, Zhifei Zhang

TL;DR

This work analyses unsteady Stokes flow on perforated domains under memory-bearing Darcy-type limits. It derives sharp energy-norm homogenization errors for velocity and pressure by explicitly constructing boundary-layer correctors using Bogovskii's operator and by introducing a radial cut-off to handle boundary incompatibilities between initial and boundary data. The authors establish robust regularity for correctors, develop two-scale-like correctors (including a flux corrector) and prove well-posedness of the homogenized Darcy-with-memory model, achieving an $O(ε^{1/2})$ convergence rate. The approach blends interior regularity, semigroup decay, and careful boundary-layer analysis, potentially extendable to other evolutionary PDEs in homogenization settings. Overall, the paper provides a comprehensive framework for quantitative homogenization with memory effects in porous media and advances the treatment of incompressibility-induced boundary layers.

Abstract

Focusing on Darcy's law incorporating memory effects, this paper studies non-stationary Stokes equations on perforated domains. We establish a sharp homogenization error for both velocity and pressure in terms of the energy norm. The main challenge lies in gauging the boundary layers induced by the incompressibility condition. To address this, we construct boundary-layer correctors using Bogovskii's operator. Also, the present work provides detailed regularity estimates for these correctors, where a significant difficulty arises from the incompatibility between initial and boundary values. The methodologies developed herein hold great potential for tackling the same issue in other evolutionary models beyond a homogenization setting.

Corrector estimates and homogenization errors of unsteady flow ruled by Darcy's law

TL;DR

This work analyses unsteady Stokes flow on perforated domains under memory-bearing Darcy-type limits. It derives sharp energy-norm homogenization errors for velocity and pressure by explicitly constructing boundary-layer correctors using Bogovskii's operator and by introducing a radial cut-off to handle boundary incompatibilities between initial and boundary data. The authors establish robust regularity for correctors, develop two-scale-like correctors (including a flux corrector) and prove well-posedness of the homogenized Darcy-with-memory model, achieving an convergence rate. The approach blends interior regularity, semigroup decay, and careful boundary-layer analysis, potentially extendable to other evolutionary PDEs in homogenization settings. Overall, the paper provides a comprehensive framework for quantitative homogenization with memory effects in porous media and advances the treatment of incompressibility-induced boundary layers.

Abstract

Focusing on Darcy's law incorporating memory effects, this paper studies non-stationary Stokes equations on perforated domains. We establish a sharp homogenization error for both velocity and pressure in terms of the energy norm. The main challenge lies in gauging the boundary layers induced by the incompressibility condition. To address this, we construct boundary-layer correctors using Bogovskii's operator. Also, the present work provides detailed regularity estimates for these correctors, where a significant difficulty arises from the incompatibility between initial and boundary values. The methodologies developed herein hold great potential for tackling the same issue in other evolutionary models beyond a homogenization setting.
Paper Structure (26 sections, 24 theorems, 240 equations, 3 figures)

This paper contains 26 sections, 24 theorems, 240 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Relation to other works
  4. Structure of the paper
  5. Notations
  6. Sketch of the proof
  7. On the error term (main ideas)
  8. Outline the proof of Theorem $\ref{['thm1']}$
  9. Correctors
  10. Semigroup estimate I
  11. Semigroup estimate II
  12. Proof of Proposition $\ref{['prop2.1']}$
  13. Boundary-layer correctors
  14. Radial cut-off functions
  15. Proof of Proposition $\ref{['lemma3.6-1']}$
  16. ...and 11 more sections

Key Result

Theorem 1

Let $0<T<\infty$, and $d\geq 2$. There exists an extension $(\tilde{u}_\varepsilon,\tilde{p}_\varepsilon)$ of the solution $(u_\varepsilon,p_\varepsilon)$ to pde1.1, which weakly converges in $L^2(0,T;L^2(\Omega)^d)\times L^2(0,T;L^2(\Omega)/\mathbb{R})$ to the unique solution $(u_0,p_0)$ of the hom where $\vec{n}$ is the unit outward normal vector of $\partial\Omega$. The quantity $A=(A_{ij})$ is

Figures (3)

  • Figure 1: This figure provides a heuristic illustration of the decomposition of region $Y_f$ in two dimensional case. Furthermore, it is important to note that region $Y$ is a periodic domain.
  • Figure 2: The proof structure of Proposition \ref{['prop2.1']}
  • Figure 3: A typical example for $d=2$

Theorems & Definitions (46)

  • Theorem 1: homogenization theorem Allaire92Allaire-Mikelic97Lions81Mikelic94Sandrakov97
  • Theorem 2: error estimates
  • Corollary 3
  • Remark 5
  • Remark 6
  • Proposition 7: corrector $\&$ flux corrector
  • Remark 8
  • Proposition 9: corrector of Bogovskii's operator
  • Remark 10
  • lemma 11: semigroup estimate I
  • ...and 36 more