Initial-boundary value problems for the equations of the global atmospheric electric circuit

Time-dependent and steady-state problems for the equations of the global atmospheric electric circuit are studied in the quasi-stationary approximation. Maxwell's equations in an anisotropic medium are written in terms of the electric potential; several types of boundary conditions (motivated by applications) are considered, including the analogues of the classical Dirichlet, Neumann and mixed conditions, as well as a non-standard condition relating the potential and normal current density at symmetric points on one of the boundary components. In each case the variational formulation of the problem is derived and is shown to be well-posed. Also, certain stabilisation theorems are proved for the solution of time-dependent problem in case the main parameters do not depend on time.