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 Dang et al. (SIAM J. Appl. Math. 81(6):2547–2568, 2021) 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 Dang et al. (SIAM J. Appl. Math. 81(6):2547–2568, 2021), 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∞-bound for the gradient of the solution of the scalar equation − div [a(x∕ ε) ∇ φε(x) ] = f(x), uniform with respect to microstructure scale parameter ε ≪ 1 in a small interval (0, ε0), where the coefficient a is only piecewise Hölder continuous. Thenceforth, this regularity is used in the derivation of the effective response of the given suspension discussed in Dang et al. (SIAM J. Appl. Math. 81(6):2547–2568, 2021).