A variety of approaches have been developed in the literature towards modelling of cathode spots in vacuum arcs. These include space-resolved descriptions based on numerical solution of 1D or 2D partial differential equations, both stationary and non-stationary. However, the simplest model, namely, a stationary spot on an infinite planar cathode, has been studied mostly in the 0D approximation, where the spot is described only on the integral level. Some authors even presumed that due to thermal runaway steady-state solutions describing a stationary spot on an infinite planar vacuum arc cathode do not exist or are unstable. The lack of a mathematically accurate solution of the most basic problem of the theory of plasma-cathode interaction in vacuum arc discharges is unfortunate and detrimental to the theory.
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