Equations of magnetodynamics of incompressible thermo-Binghamʹs fluid under the gravity effect

Results of this contribution were obtained in the course of an investigation of a simple model problem of the global geodynamic and geomagnetic fields of the real Earth as well as model problem of astrophysics. The problem leads to an initial boundary value problem for a coupled system of equations of magnetodynamics of incompressible, electrically conducting and thermo viscous Binghamʹs fluid under the gravity effect. The variational formulation of the problem and the existence and uniqueness of the solution will be given. While the existence of the solution is proved for the two- and three-dimensional cases, the uniqueness can be proved for two-dimensional case only.