By Goursat E.
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Additional resources for A course in mathematical analysis. - part.2 Differential equations
23). ZN0 Ne /U: rPe ZN0 C. J B/ J:rPe D J:E; so we have the previous balance Ne qe Ne by using equation (ii) multiplied by U and remembering that r:PB D J B. 20)]. moreover ZN0 Remark 1. , to let the ratio of electron mass to ion mass me =m0 converge to zero in the ion–electron Euler system); see  for a proof in a monodimensional case. ZN0 Ne /U may be neglected with respect to J=qe : This system looks quite complicated. As a matter of fact, it is an open problem to show its well-posedness from a mathematical point of view.
52) Thus, the basic principles of quasi-neutral approximation may be summarized as follows. 52) for the momentum equation. E C U B/ comes into sight; it is a natural quantity in all the MHD models. Moreover, it is worth noticing that the electron velocity never appears explicitly in these models (it is always evaluated thanks to U and J) and that, generally, one has to account for the evolution of the electron temperature. 3 Proofs of the Propositions of Sects. 2 Proof of Proposition 1. Let us set D D which is fixed.
U /: Summary. 66), it is not the case. 66). In its reduced form without the thermal conduction term qth,e , it has been analyzed recently from a mathematical point of view; it is related to the definition of the nonconservative products like Pe r:U when U is discontinuous. It may be shown that one has to add some information on the entropy deposition, see  for example. Zel’dovich and Raizer noted a long time ago in  that this entropy deposition needs to be made on ion population only. This fact has been justified also in  with an asymptotic analysis (in the case where the Debye length goes to zero).