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Without accounting for the hydrodynamics part, it has been proved that with appropriated assumptions, the reduced system @ E @t r: Â c C 3. Te /d : 0 is well-posed; see [8]. But the full system is quite difficult to handle. There is also a simpler radiative model called the gray-diffusion approximation which may be derived from the previous one by making some elementary closures. We emphasize it now (disregarding the Compton effect). Te / E /d for the emission/absorption phenomena is simply Er /.

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. 43). B1 ; B2 / C kF k1 /T1 C C6 So we see that kˆ follows. 2 : F k2 D O. / and the remaining part of the proposition Proof of the Lemma 1. x/ the distance from x to the boundary €1 .

On momentum equation [E2T J(ii)], since the current is curl B= 0 ; we display the magnetic pressure tensor r:PB : Then, we get the following system for the evolution of ; U; "0 ; "e ; B 4 As a matter of fact, the electric resistivity is defined as textbooks. B:r/U C curl . 81). Of course, we may state this system in an Euler framework, it suffices to replace equation (i) by @ C r:. U/ D 0 @t D X by and in the other equations to replace Dt equation reads as @ @t . X / C r:. B:r/UCcurl . curlB/ D curl curl @t qe Ne qe Ne curlB 0 B Ã In equation (v), the quadratic term (curl B B), which is related to the Hall effect, may be neglected if the magnetic field is not very strong.

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