By Goursat E.

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**Example text**

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 [63] 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 [37] for example. Zel’dovich and Raizer noted a long time ago in [117] that this entropy deposition needs to be made on ion population only. This fact has been justified also in [32] with an asymptotic analysis (in the case where the Debye length goes to zero).