On the dynamics of ferrofluids: Global weak solutions to the Rosensweig system and rigorous convergence to equilibrium

TL;DR

This paper proves the global existence of weak solutions for the Rosensweig ferrofluid model and rigorously demonstrates convergence to equilibrium using advanced mathematical techniques.

Contribution

It establishes the first rigorous proof of global weak solutions and convergence to equilibrium for the ferrofluid dynamics model by Rosensweig.

Findings

Existence of global weak solutions is proven.

Convergence to equilibrium is rigorously demonstrated.

The relative entropy method is effectively applied.

Abstract

This article establishes the global existence of weak solutions to a model proposed by Rosensweig (Rosensweig, Ferrohydrodynamics (1985)) for the dynamics of ferrofluids. The system is expressed by the conservation of linear momentum, the incompressibility condition, the conservation of angular momentum, and the evolution of the magnetization. The existence proof is inspired by the DiPerna-Lions theory of renormalized solutions. In addition, the rigorous relaxation limit of the equations of ferrohydrodynamics towards the quasi-equilibrium is investigated. The proof relies on the relative entropy method, which involves constructing a suitable functional, analyzing its time evolution and obtaining convergence results for the sequence of approximating solutions.