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

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 $-\mathrm{div}\left[\mathbf{a}\left(x/\epsilon \right)\nabla {\phi }^{\epsilon }(x)\right]=f(x)$, uniform with respect to microstructure scale parameter $\epsilon \ll 1$ in a small interval $(0,{\epsilon }_0)$, where the coefficient $\mathbf{a}$ is only \emph{piecewise} Hölder continuous. Thenceforth, this regularity is used in the derivation of the effective response of the given suspension discussed in \cite{dangHomogenizationNondiluteSuspension2021}.

24 pages, 2 figures, accepted version