An existence result for a suspension of rigid magnetizable particles

Abstract

We establish the existence of a weak solution for a strongly coupled, nonlinear Stokes–Maxwell system, originally proposed by Nika and Vernescu (Z Angew Math Phys 71(1):1–19, 2020) in the three-dimensional setting. The model effectively couples the Stokes equation with the quasi-static Maxwell’s equations through the Lorentz force and the Maxwell stress tensor. The proof of existence is premised on: (i) the augmented variational formulation of Maxwell’s equations, (ii) the definition of a new function space for the magnetic induction and the verification of a Poincar’e-type inequality, and (iii) the deployment of the Altman–Shinbrot fixed point theorem when the magnetic Reynolds number, $${\text {R}_{\text {m}}},$$ R m , is small.