On the Field-Induced Transport of Magnetic Nanoparticles in Incompressible Flow: Existence of Global Solutions

Abstract

AbstractWe prove global-in-time existence of weak solutions to a pde-model for the motion of dilute superparamagnetic nanoparticles in fluids influenced by quasi-stationary magnetic fields. This model has recently been derived in Grün and Weiß(On the field-induced transport of magnetic nanoparticles in incompressible flow: modeling and numerics, Mathematical Models and Methods in the Applied Sciences, in press). It couples evolution equations for particle density and magnetization to the hydrodynamic and magnetostatic equations. Suggested by physical arguments, we consider no-flux-type boundary conditions for the magnetization equation which entails $$H({\text {div}},{\text {curl}})$$ H ( div , curl ) -regularity for magnetization and magnetic field. By a subtle approximation procedure, we nevertheless succeed to give a meaning to the Kelvin force $$(\mathbf {m}\cdot \nabla )\mathbf {h}$$ ( m · ∇ ) h and to establish existence of solutions in the sense of distributions in two space dimensions. For the three-dimensional case, we suggest two regularizations of the system which each guarantee existence of solutions, too.