A unified variational formulation for the parabolic-elliptic eddy current equations (Q2903530)

The linear eddy current problem with a variable conductivity term \(\sigma\) (Ohm's law) is considered in \(\mathbb{R}^{3}\). The resulting system for the electric field is of parabolic type where \(\sigma\) is positive and (quasi-static) elliptic where \(\sigma\) vanishes. A variational formulation covering this parabolic-elliptic situation is established. Particular difficulties arise from the large null space of the spatial operator and the fact that we have purely continuous spectrum. The first incurs some non-uniqueness issues (choice of gauge field), the latter requires weighted \(L^{2}\)-type spaces for establishing well-posedness. Particular attention is given to the aspect of the limiting behavior as the conductivity tends to \(0\).