We 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 \(H({\text {div}},{\text {curl}})\)-regularity for magnetization and magnetic field. By a subtle approximation procedure, we nevertheless succeed to give a meaning to the Kelvin force \((\mathbf {m}\cdot \nabla )\mathbf {h}\) 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.

我们证明了准静态磁场对流体中稀超顺磁性纳米粒子运动的pde模型的弱解的全局及时存在性。该模型最近在Grün和Weiß中获得（关于磁场诱导的不可压缩流动中的磁性纳米粒子的运输：建模和数值，应用科学中的数学模型和方法，印刷中）。它将颗粒密度和磁化强度的演化方程式与流体动力学和静磁方程式耦合。根据物理论据的建议，我们考虑了磁化方程的无磁通类型边界条件，该边界条件需要\（H（{\ text {div}}，{\ text {curl}}）\）-磁化和磁场的规则性。通过微妙的近似过程，我们仍然成功地给了开尔文力\（（\ mathbf {m} \ cdot \ nabla} \ mathbf {h} \）并建立了在两个意义上的分布解的存在性。空间尺寸。对于三维情况，我们建议系统的两个正则化，每个正则化也保证解的存在。