[Paper Review] Periodic homogenization of Green's functions for Stokes systems

Shu Gu, Jinping Zhuge|arXiv (Cornell University)|Oct 15, 2017

Advanced Mathematical Modeling in Engineering3 citations

TL;DR

This paper establishes uniform regularity estimates and asymptotic expansions for Green's functions and their derivatives in Stokes systems with periodically oscillating coefficients. By introducing adjustable uniform estimates—Lipschitz for velocity and oscillation-based for pressure—it achieves asymptotic expansions with a small error loss, offering new quantitative homogenization results applicable to both Stokes flow and linearized elasticity in heterogeneous media.

ABSTRACT

This paper is devoted to establishing the uniform estimates and asymptotic behaviors of the Green's functions $(G_\varepsilon,Π_\varepsilon)$ (and fundamental solutions $(Γ_\varepsilon, Q_\varepsilon)$) for the Stokes system with periodically oscillating coefficients (including a system of linear incompressible elasticity). Particular emphasis will be placed on the new oscillation estimates for the pressure component $Π_\varepsilon$. Also, for the first time we prove the extit{adjustable} uniform estimates (i.e., Lipschitz estimate for velocity and oscillation estimate for pressure) by making full use of the Green's functions. Via these estimates, we establish the asymptotic expansions of $G_\varepsilon, abla G_\varepsilon, Π_\varepsilon$ and more, with a tiny loss on the errors. Some estimates obtained in this paper are new even for Stokes system with constant coefficients, and possess potential applications in homogenization of Stokes or elasticity system.

Motivation & Objective

• To establish uniform regularity estimates for Green's functions and pressure components in Stokes systems with rapidly oscillating periodic coefficients.
• To derive asymptotic expansions of Green's functions and their derivatives with controlled error terms, particularly for the pressure component.
• To extend these results to the fundamental solutions (Γε, Qε) and apply them to homogenization of linear incompressible elasticity.
• To provide new quantitative estimates even in the constant-coefficient case, enhancing applicability to multiscale PDE problems.
• To bridge the gap in uniform regularity theory for pressure in periodic homogenization, where oscillation estimates were previously lacking.

Proposed method

• Derives adjustable uniform estimates by leveraging the structure of Green's functions, distinguishing between Lipschitz control for velocity and oscillation-based control for pressure.
• Employs a mollification technique on data (f, g, h, F) to approximate Hölder continuous and Sobolev-class data with smooth ones, enabling error quantification.
• Applies energy estimates and compactness arguments to compare solutions of the oscillatory system (ε > 0) with the homogenized limit (ε = 0), using the difference in solutions to bound error terms.
• Introduces a corrector term πε∇u₀ to capture the oscillatory behavior of the pressure, improving the convergence rate in L² norms.
• Uses the method of oscillating test functions and the theory of periodic correctors to control the error in the asymptotic expansion up to a small loss in the exponent.
• Establishes convergence rates of order ε^γ with γ arbitrarily close to η/d, where η is the Hölder exponent of the data, by optimizing the mollification parameter r = ε^{2σ/d}.

Experimental results

Research questions

• RQ1 How can uniform regularity estimates be established for the pressure component Πε in periodic homogenization of the Stokes system?
• RQ2 What is the optimal asymptotic expansion of the Green’s function Gε and its derivatives, including ∇xGε and ∇x∇yGε, with controlled error terms?
• RQ3 Can adjustable uniform estimates—Lipschitz for velocity and oscillation-based for pressure—be derived via Green’s functions to improve convergence rates?
• RQ4 What is the convergence rate of the pressure component pε to its homogenized limit, including the corrector term πε∇u₀?
• RQ5 To what extent do the derived estimates extend to the fundamental solutions (Γε, Qε), and are they new even in the constant-coefficient case?

Key findings

• The paper proves the first adjustable uniform estimates for the pressure component Πε in periodic homogenization, combining Lipschitz control for velocity and oscillation-based estimates for pressure.
• Asymptotic expansions of Gε, ∇xGε, and Πε are established with an error loss of order ε^γ, where γ can be made arbitrarily close to η/d, with η the Hölder exponent of the data.
• The convergence rate for the pressure is improved by introducing the corrector term πε∇u₀, leading to the estimate ‖pε − p₀ − πε∇u₀‖_{L²₀} ≤ Cε^γ × (data norms).
• The method yields new estimates even in the constant-coefficient case, demonstrating broader applicability to homogenization of Stokes and elasticity systems.
• The analysis confirms that the mollification parameter r = ε^{2σ/d} optimizes the error balance between data approximation and homogenization error, yielding γ = 2ση/d.
• The results are extended to the fundamental solutions (Γε, Qε), showing that the same asymptotic structure and error control hold in the singular limit.