Abstract
We provide a full and rigorous derivation of the standard viscous magnetohydrodynamic system (MHD) as the asymptotic limit of Navier–Stokes–Maxwell systems when the speed of light is infinitely large. We work in the physical setting provided by the natural energy bounds and therefore mainly consider Leray solutions of fluid dynamical systems. Our methods are based on a direct analysis of frequencies and we are able to establish the weak stability of a crucial nonlinear term (the Lorentz force), neither assuming any strong compactness of the components nor applying standard compensated compactness methods (which actually fail in this case).
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Communicated by C. Le Bris
Slim Ibrahimwas partially supported by the NSERC grant 371637-2014. Slim Ibrahim wants also to thank the Division de Mathématiques Appliquées à l’École Normale Supérieure de Paris for their great hospitality and support during his visit when the first part of this paper was completed. That visit was part of a PIMS-CNRS project. NM was partially supported by NSF-DMS grant 1211806.
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Arsénio, D., Ibrahim, S. & Masmoudi, N. A Derivation of the Magnetohydrodynamic System from Navier–Stokes–Maxwell Systems. Arch Rational Mech Anal 216, 767–812 (2015). https://doi.org/10.1007/s00205-014-0819-9
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DOI: https://doi.org/10.1007/s00205-014-0819-9