On the Existence of GlobalWeak Solutions to the 3D Electrically Conductive Rosensweig System and TheirConvergence 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 .