Bogovskiĭ-Operator in Thin Perforated Domains

• The paper presents a novel construction for the Bogovskiĭ-operator that precisely controls divergence in thin, ε-periodic perforated fluid layers.
• It provides uniform L2 and H1 norm estimates, with scaling laws that manage the ε and εᵅ dependencies critical for homogenization analysis.
• The operator’s mapping properties enable rigorous dimension reduction and facilitate effective macroscopic modeling of incompressible flows.

The Bogovskiĭ-operator in thin perforated domains is a linear mapping constructed to solve divergence equations with Dirichlet-type boundary conditions in thin fluid layers punctuated by ε-periodic micro-inclusions. The operator enables the precise control of divergence and uniform estimates essential for rigorous homogenization of incompressible flow and transport equations, particularly in the dimension reduction limit as the thickness parameter $\varepsilon\to 0$ with thickness scaling $\varepsilon^\alpha$, $\alpha\in(0,1)$, which is large compared to the microstructural porosity. Its construction, estimate, and mapping properties are critical for deriving effective macroscopic models via two-scale convergence and for uniform control of the fluid pressure field in perforated thin geometries (Gahn et al., 4 Dec 2025).

1. Geometric Setting: Thin Perforated Layer

Let $n\geq 2$ and $\Sigma\subset\mathbb{R}^{n-1}$ be a bounded Lipschitz domain. For $\alpha\in(0,1)$ and $\varepsilon>0$ sufficiently small with $\varepsilon^\alpha/\varepsilon\in\mathbb{N}$, the thick layer is defined by

$$\Omega^{(\varepsilon)} := \Sigma\times(-\varepsilon^\alpha, \varepsilon^\alpha)\subset\mathbb{R}^n$$

with top/bottom faces $$S_\varepsilon^\pm := \Sigma\times\{\pm\varepsilon^\alpha\}$$ and lateral boundary $$\partial_D\Omega^{(\varepsilon)} := \partial\Sigma\times(-\varepsilon^\alpha,\varepsilon^\alpha)$$. Inside this layer, an $\varepsilon$-periodic array of solid inclusions is excised:

• Reference (unit) cell: $Y = (0,1)^n$, fluid part $Y_f\subset Y$, solid part $Y_s = Y\setminus\overline{Y_f}$, interface $\Gamma = \partial Y_f\cap\partial Y_s$.
• The collection of indices for full cells is $$K_\varepsilon := \{k\in\mathbb{Z}^n : \varepsilon(Y+k)\subset \Omega^{(\varepsilon)}\}$$.
• The perforated fluid domain is

$$\Omega_\varepsilon := \bigcup_{k\in K_\varepsilon} \varepsilon(Y_f + k)$$

with the oscillating boundary

$$\Gamma_\varepsilon := \bigcup_{k\in K_\varepsilon} \varepsilon(\Gamma + k).$$

Therefore,

$$\Omega_\varepsilon = \Omega^{(\varepsilon)}\setminus \overline{\bigcup_{k\in K_\varepsilon} \varepsilon(Y_s + k)}.$$

This architecture creates a thin, periodically perforated medium with the thickness of order $\varepsilon^\alpha$ and microstructural heterogeneity at scale $\varepsilon$ (Gahn et al., 4 Dec 2025).

2. Functional Framework: Spaces and Constraints

Analysis proceeds in the following spaces:

• $L^2(\Omega_\varepsilon)$: scalar functions;
• $$L^2_0(\Omega_\varepsilon) = \{f\in L^2(\Omega_\varepsilon) : \int_{\Omega_\varepsilon} f\,dx=0\}$$;
• $H^1(\Omega_\varepsilon)$: first-order Sobolev space;
• $H^1_0(\Omega_\varepsilon)$: closure in $H^1$ of $C_c^\infty(\Omega_\varepsilon)$.

For the divergence correction, consider

$$H^1(\Omega_\varepsilon, \partial_D\Omega_\varepsilon\cup\Gamma_\varepsilon) := \{v\in H^1(\Omega_\varepsilon): v|_{\partial_D\Omega_\varepsilon\cup\Gamma_\varepsilon}=0\}.$$

Boundary conditions are:

• No-slip (Dirichlet) along the oscillating boundary $\Gamma_\varepsilon$ and on lateral faces $\partial_D\Omega_\varepsilon$,
• Free or prescribed values allowed on $S_\varepsilon^+$ and $S_\varepsilon^-$.

Given $f\in L^2_0(\Omega_\varepsilon)$, the problem is to construct $$B_\varepsilon(f)\in H^1(\Omega_\varepsilon, \partial_D\Omega_\varepsilon\cup\Gamma_\varepsilon)^n$$ with

$$\operatorname{div} B_\varepsilon(f) = f \quad \text{in } \Omega_\varepsilon,\qquad B_\varepsilon(f) = 0 \text{ on } \partial_D\Omega_\varepsilon\cup\Gamma_\varepsilon.$$

3. Existence, Linearity, and Uniform Estimates

The main result guarantees the existence of a Bogovskiĭ-operator for thin perforated domains: $$B_\varepsilon : L^2_0(\Omega_\varepsilon) \to H^1(\Omega_\varepsilon, \partial_D\Omega_\varepsilon\cup\Gamma_\varepsilon)^n$$ satisfying for all $f\in L^2_0(\Omega_\varepsilon)$:

• $\operatorname{div} B_\varepsilon(f) = f$ in $\Omega_\varepsilon$,
• $B_\varepsilon(f)=0$ on $$\partial_D\Omega_\varepsilon \cup \Gamma_\varepsilon$$,
• The uniform norm estimate: $$\|B_\varepsilon(f)\|_{L^2(\Omega_\varepsilon)}+\varepsilon^\alpha\|\nabla B_\varepsilon(f)\|_{L^2(\Omega_\varepsilon)} \leq C\varepsilon^\alpha \|f\|_{L^2(\Omega_\varepsilon)},$$ where $C>0$ depends only on the geometry of $\Sigma$ and $Y_f$. This can be rewritten as: $$\|B_\varepsilon(f)\|_{H^1(\Omega_\varepsilon)} \leq C\varepsilon^{\alpha-1}\|f\|_{L^2(\Omega_\varepsilon)}$$ indicating that the operator norm grows as $\varepsilon^{\alpha-1}$ as $\varepsilon\to 0$ (Gahn et al., 4 Dec 2025).

4. Explicit Construction: Local-to-Global Procedure

The operator is constructed in two principal steps:

4.1. Local Solution in the Non-Perforated Layer

Decompose $\Omega^{(\varepsilon)}$ into prisms $\varepsilon(Y+k)$ indexed by $$k\in K^0_\varepsilon := \{k\in\mathbb{Z}^{n-1}\times\{0\}: \varepsilon(Y+k)\subset \Omega^{(\varepsilon)}\}$$. For each cell:

• Define $f_k(y) = f(\varepsilon y + \varepsilon k)$ (extended by zero outside $Y_f$).
• Solve:

$$\exists U_k \in H^1(Y, \partial Y \setminus\{\text{top}, \text{bottom}\})^n:\ \operatorname{div}_y U_k = f_k \text{ in } Y,\quad U_k|_{\partial Y\setminus(S^+\cup S^-)}=0,$$

with the standard bound $\|U_k\|_{H^1(Y)}\leq C\|f_k\|_{L^2(Y)}$.

• Define, on $\varepsilon(Y+k)$,

$$\tilde B_\varepsilon f(x) := \varepsilon U_k\left(\frac{x-\varepsilon k}{\varepsilon}\right).$$

Scaling arguments yield: $$\|\tilde B_\varepsilon f\|_{L^2(\varepsilon(Y+k))} \lesssim \varepsilon\|f\|_{L^2(\varepsilon(Y+k))},\qquad \varepsilon^\alpha\|\nabla \tilde B_\varepsilon f\|_{L^2(\varepsilon(Y+k))} \lesssim \varepsilon^\alpha\|f\|_{L^2(\varepsilon(Y+k))}.$$ Summing over all cells establishes the global uniform bound in the whole layer.

4.2. Restriction to the Perforated Geometry

Apply the Allaire restriction operator $R_\varepsilon$: $$R_\varepsilon: H^1(\Omega^{(\varepsilon)})^n \to H^1(\Omega_\varepsilon)^n,$$ mapping vector fields defined in the full layer to those supported on $\Omega_\varepsilon$, preserving boundary conditions and divergence on the fluid domain, with $$\|R_\varepsilon(v)\|_{H^1(\Omega_\varepsilon)}\leq C\|v\|_{H^1(\Omega^{(\varepsilon)})}$$.

The Bogovskiĭ-operator is then: $$B_\varepsilon(f) = R_\varepsilon(\tilde B_\varepsilon(f_{\Omega_\varepsilon}))$$

This operator satisfies the required divergence and boundary conditions, and inherits the uniform norm estimates.

5. Key A Priori and Scaling Estimates

The construction is supported by:

• Poincaré Inequality for Thin Layers: On each $\varepsilon(Y+k)$,

$$\|v\|_{L^2(\Omega^{(\varepsilon)})} \leq C\varepsilon^\alpha\|\nabla v\|_{L^2(\Omega^{(\varepsilon)})}$$

for $v\in H^1(\Omega^{(\varepsilon)})$ vanishing on $\partial_D\Omega^{(\varepsilon)}$.

• Scaling Laws: Rescaling $y\leftrightarrow x/\varepsilon$ (horizontal) and $y_n\leftrightarrow x_n/\varepsilon^\alpha$ (vertical) introduces the $\varepsilon$ and $\varepsilon^\alpha$ weights in the operator norm.
• Effect of Restriction Operator: The mapping $R_\varepsilon$ does not increase the order of $\varepsilon$ in the $H^1$ norm, ensuring uniform asymptotics.

These facts are essential for the norm control of the Bogovskiĭ-operator and ultimately for the uniform a priori bounds used in the two-scale convergence analysis (Gahn et al., 4 Dec 2025).

6. Summary of Analytical Formulas and Estimates

A table summarizing key steps and estimates:

Step	Formula/Operator Definition	Uniform Norm Bound
Cellwise Bogovskiĭ on $Y$	$$U\in H^1(Y,\partial Y\setminus\{\text{top},\text{bottom}\})^n$$: $\operatorname{div}_y U = g$	$\|U\|_{H^1(Y)}\leq C\|g\|_{L^2(Y)}$
Scaling to $\varepsilon$-cell	$$\tilde B_\varepsilon f(x) = \varepsilon U_k((x-\varepsilon k)/\varepsilon)$$	$$\|\tilde B_\varepsilon f\|_{L^2} \lesssim \varepsilon\|f\|_{L^2}$$, $$\varepsilon^\alpha\|\nabla\tilde B_\varepsilon f\|_{L^2} \lesssim \varepsilon^\alpha\|f\|_{L^2}$$
Restriction to $\Omega_\varepsilon$	$$B_\varepsilon(f) := R_\varepsilon(\tilde B_\varepsilon(f_{\Omega_\varepsilon}))$$	$$\|B_\varepsilon(f)\|_{L^2}+\varepsilon^\alpha\|\nabla B_\varepsilon(f)\|_{L^2}\leq C\varepsilon^\alpha\|f\|_{L^2}$$

These formulas encode the operator’s construction, its role in handling divergence with Dirichlet-type boundary in evolving microscopic geometries, and the $\varepsilon$-uniform norm control essential for quantitative homogenization limits.

7. Role in Homogenization and Macroscopic Modeling

The constructed Bogovskiĭ-operator for thin perforated domains enables the rigorous derivation of effective equations for incompressible Stokes flow and transport in the limit $\varepsilon\to 0$. It is instrumental for pressure control in dimension-reduction with $\varepsilon$-dependent geometry and microstructure, forming a technical bedrock for proving strong two-scale convergence and for managing pressure correctors in the limiting Darcy-type law. Its properties, ensuring uniform estimates and mapping divergence data to solenoidal velocities, are indispensable to the homogenization methodology adopted in recent analysis of thin stratified porous media (Gahn et al., 4 Dec 2025).