Why is there no generally accepted theory of the plasma-sheath transition and the Bohm criterion?

Despite the efforts of able researchers over decades and despite the existence of results of a mathematical nature, including recently obtained pure-mathematics results, there is still no general agreement in gas discharge science regarding the theoretical description of the plasma-sheath transition and the Bohm criterion. The question then arises: what should the mathematical results be to have a better chance of being accepted in gas discharge science? This paper formulates the criteria for “good mathematics,” which, in the authors' opinion, are the most important for the theory of the plasma-sheath transition. Existing theoretical results on the plasma-sheath transition, particularly those concerning the Bohm criterion, are reviewed in this context. The necessity of presenting the essence of a mathematical treatment and its results in a form understandable to physicists who have little interest in mathematical details is emphasized, and it is shown that the fluid and kinetic versions of the classical collisionless Bohm criterion can indeed be presented in this way. In contrast, the Bohm criterion for collisional sheaths derived from the assumption of a maximum of the Sagdeev potential at the sheath edge, which has become popular in the modern literature, cannot be presented in this way, and it is difficult to see how such criterion could be useful.