On the Existence of Global Weak Solutions to the 3D Electrically Conductive Rosensweig System and Their Convergence Towards Quasi-Equilibrium

Abstract

In this article, we study an electrically conductive Rosensweig model for ferrofluids, whose Bloch–Torrey regularization was studied by Hamdache and Hamroun (Appl Math Optim 81(2):479–509, 2020). We mainly prove the global existence of weak solutions to the non-regularized model under a certain smallness condition on the electric conductivity. Hence, our result not only solves a problem that was left open by Hamdache and Hamroun, but it can also serve as a confirmation that ferrofluids are naturally poor conductors of electric current. The proof, which is interesting in itself, is quite involved and relies on the Helmohltz–Leray decomposition of the magnetic fields and the use of renormalized solutions for the magnetization. We also give a rigorous and detailed description of the convergence of the global weak solutions towards the quasi-equilibrium in the relaxation time limit regime τ→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au \\rightarrow 0$$\\end{document}.