Formation of small scales in nonlinear PDEs

Relaxation to the quasi-static equilibrium in the Rosensweig model for ferrofluids

Franziska Weber

Carnegie Mellon University


We show global existence of weak solutions to a model proposed by Rosensweig for the dynamics of ferrofluids. The system consists of the equations for the conservation of mass, linear momentum, and angular momentum, and the evolution of the magentization coupled to the magnetostatics equations. The existence proof is inspired by the DiPerna-Lions theory of renormalized solutions. Then we show rigorous relaxation to the quasi-static equilibrium where the magnetization is parallel to the magnetic field using the relative entropy method.