We study a nonlinear coupling system of partial differential equations describing the dynamic of a magnetic fluid with internal rotations. The present mathematical model generalizes those discussed previously in the literature since actually the fluid is electrically conducting inducing additional nonlinearities in the problem and the dynamics of the magnetic field is described by the quasi-static Maxwell equations instead of the usual magnetostatic ones. We prove existence of weak solutions with finite energy first for the unsteady problem then for the steady one.

中文翻译：

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导电磁流体不稳定和稳定模型的弱解

我们研究了偏微分方程的非线性耦合系统，该系统描述了具有内部旋转的磁性流体的动力学。本数学模型概括了先前在文献中讨论的那些数学模型，因为实际上流体正在导电中引发问题中的其他非线性，并且磁场的动力学由准静态Maxwell方程而不是通常的静磁方程描述。我们首先证明对于不稳定的问题，然后对于稳定的问题，具有有限能量的弱解的存在。