Analysis of a coupled system of equations of a global geodynamic model of the earth

The investigated global geodyanmic model of the Earth is described by an initial boundary value problem for a coupled system of equations of motion in Bingham visco-plastic rheology, equation of Stefan-like problem (describing melting, recrystallization and solidification processes in the Earth's interior), equations of concentration of solid and liquid phases (namely in the mushy zones), equations of magnetodynamics of incompressible, electrically conducting visco-plastic fluid and equation of a time-dependent potential gravity field. The variational formulation of the problem and the existence and uniqueness of the solution will be given. The variational formulation of the problem leads to analyses of the system of variational equalities and inequalities. The proof of the existence theorem proceeds by regularization of the non-linear degenerated terms and of the non-differentiable functional j(·). For construction of sequences of regularized solutions the Galerkin approximation technique will be used. For proving the solution monotonicity arguments will also be used.