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, Rm,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {R}_{\ ext {m}}},$$\\end{document} is small.
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