Global gradient estimate for a divergence problem and its application to the homogenization of a magnetic suspension

TL;DR

This paper establishes a global gradient estimate for a divergence problem with piecewise Hölder continuous coefficients, enabling improved homogenization results for magnetic suspensions in viscous flows.

Contribution

It relaxes previous assumptions on coefficients, proving new regularity results that facilitate homogenization analysis of magnetic suspensions.

Findings

Proves a uniform $L^{ abla ext{infinity}}$ bound for solutions with piecewise Hölder coefficients.

Enables derivation of effective macroscopic behavior of magnetic suspensions.

Extends homogenization applicability to more general coefficient structures.

Abstract

This paper generalizes the results obtained by the authors in \cite{dangHomogenizationNondiluteSuspension2021} concerning the homogenization of a non-dilute suspension of magnetic particles in a viscous flow. More specifically, in this paper, a restrictive assumption on the coefficients of the coupled equation, made in \cite{dangHomogenizationNondiluteSuspension2021}, that significantly narrowed the applicability of the homogenization results obtained, is relaxed and a new regularity of the solution of the fine-scale problem is proven. In particular, we obtain a global $L^{\infty}$-bound for the gradient of the solution of the scalar equation $-\mathrm{div} \left[ \mathbf{a} \left( x/\varepsilon \right)\nabla \varphi^{\varepsilon}(x) \right] = f(x)$, uniform with respect to microstructure scale parameter $\varepsilon\ll 1$ in a small interval $(0,\varepsilon_0)$, where the coefficient $\mathbf{a}$ is only \emph{piecewise} H\"{o}lder continuous. Thenceforth, this regularity is used in the derivation of the effective response of the given suspension discussed in \cite{dangHomogenizationNondiluteSuspension2021}.