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By Scarborough J.B.

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Ne Te /: Then, the result follows. Proof of Proposition 3. e e ' /. '/ . ZN0 =Nref / ! 0 in L2 ; ZN0 =Nref ! 45) holds. Remark 4. O/; we may show that ˆ ! O/ weakly, but one can only prove that rˆ ! ZN0 / in distribution meaning. Proof of Proposition 4. We will use the following lemmas, the proof of which is given below. 34 2 Quasi-Neutrality and Magneto-Hydrodynamics Lemma 1. 62) ‚ C ‚ D 0; @ supplemented with the boundary conditionW ‚ j€1 D 1; ‚ j€2 D 0; @n ‚ j@On€1 [€2 D 0: Then, we have ˇ Z ˇ Z ˇ @‚ ˇ 1 1 ˇ ˇ (iii) ‚ ÄC ; (ii) kr‚ k Ä C0 ; (i) ˇ @n ˇ Ä C1 : O @O Lemma 2.

U B/ D curl rPe curl B B curl . 24) It is called the diffusion magnetic equation. On the right-hand side, the last term is the usual resistive diffusion operator; moreover, the quadratic term, curl. N1e curl B B/ is called the Hall’s effect term (it is taken into account only if the electron density is small enough and if the magnetic field is strong enough). Lastly, we have the Gauss relation r:B D 0I of course, if this relation holds at initial time, it holds at any time. (a) The Ion Euler–Poisson Model Without Resistivity.

Now, to state a rigorous mathematical result, we need to make some technical assumptions. 41) Moreover, for the sake of simplicity, we impose @ N0 D 0; @n on @O: Proposition 3. 41) hold. ˆ / ! O/, rˆ ! ZN0 / 1 A sequence un converges weakly to u; if R un v ! O/ weakly: uv for any v in L2 . 26 2 Quasi-Neutrality and Magneto-Hydrodynamics The proof is given below. Notice that there are also results with weaker assumptions on the regularity of N0 (see the proof). This proposition shows first that the quasi-neutrality holds Ne ' ZN0 ; Pe ' ZN0 Te : Moreover, this implies that the electric field E satisfies qe E D Te rˆ rTe !

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Differential equations and applications by Scarborough J.B.

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