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A Derivation of the Magnetohydrodynamic System from Navier–Stokes–Maxwell Systems

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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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References

  1. Arsénio, D., Saint-Raymond, L.: From the Vlasov-Maxwell-Boltzmann system to incompressible viscous electro-magneto-hydrodynamics

  2. Aubin J.-P.: Un théorème de compacité. C. R. Acad. Sci. Paris 256, 5042–5044 (1963)

    MATH  MathSciNet  Google Scholar 

  3. Biskamp, D.: Nonlinear magnetohydrodynamics. Cambridge Monographs on Plasma Physics, Vol. 1. Cambridge University Press, Cambridge, (1993)

  4. Chemin, J.-Y: Perfect incompressible fluids. Oxford Lecture Series in Mathematics and its Applications, Vol. 14. The Clarendon Press Oxford University Press, New York, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie

  5. Chemin J.-Y., Masmoudi N.: About lifespan of regular solutions of equations related to viscoelastic fluids (English summary). SIAM J. Math. Anal. 33(1), 84–112 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cotsiolis A., Tavoularis N.K.: Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295(1), 225–236 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Davidson P.A.: An introduction to magnetohydrodynamics. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  8. Evard J.-Cl., Jafari F.: A complex Rolle’s theorem. Amer. Math. Monthly 99(9), 858–861 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fujita H., Kato T.: On the Navier–Stokes initial value problem. I. Arch. Rational Mech. Anal. 16, 269–315 (1964)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Germain P., Ibrahim S., Masmoudi N.: Well-posedness of the Navier-Stokes-Maxwell equations. Proc. Roy. Soc. Edinburgh Sect. A 144(1), 71–86 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Giga, Y., Ibrahim, S., Shen, S, Yoneda, T.: Global well-posedness for a two-fluid model. http://arxiv.org/pdf/1411.0917.pdf

  12. Giga Y., Yoshida Z.: On the equations of the two-component theory in magnetohydrodynamics. Comm. Partial Differ. Equ. 9(6), 503–522 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ibrahim S., Keraani S.: Global small solutions for the Navier–Stokes–Maxwell system. SIAM J. Math. Anal. 43(5), 2275–2295 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ibrahim S., Yoneda T.: Local solvability and loss of smoothness of the Navier–Stokes–Maxwell equations with large initial data. J. Math. Anal. Appl. 396(2), 555–561 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jang J., Masmoudi N.: Derivation of Ohm’s law from the kinetic equations. SIAM J. Math. Anal. 44(5), 3649–3669 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lemarié-Rieusset, P.G.: Recent developments in the Navier–Stokes problem. Chapman & Hall/CRC Research Notes in Mathematics, Vol. 431. Chapman & Hall/CRC, Boca Raton, FL, (2002)

  17. Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lions, J.-L.: Équations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111. Springer, Berlin, (1961)

  19. Masmoudi, N.: Global well posedness for the Maxwell–Navier–Stokes system in 2D. J. Math. Pures Appl. (9) 93(6), 559–571 (2010)

  20. Masmoudi N., Nakanishi K.: From the Klein-Gordon-Zakharov system to the nonlinear Schrödinger equation (English summary). J. Hyperbolic Differ. Equ. 2(4), 975–1008 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Murat F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5(3), 489–507 (1978)

    MATH  MathSciNet  Google Scholar 

  22. Murat, F.: Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8(1), 69–102 (1981)

  23. Simon, J.: Compact sets in the space L p(0, T; B). Ann. Mat. Pura Appl. (4) 146, 65–96 (1987)

  24. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4) 110, 353–372 (1976)

  25. Tartar, L.: Compensated compactness and applications to partial differential equations. Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV. Research Notes in Mathematics, Vol. 39. Pitman, Boston, Mass.-London, 136–212, (1979)

  26. Yosida, K.: Functional analysis. Classics in Mathematics. Springer, Berlin, 1995. Reprint of the sixth (1980) edition

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Authors and Affiliations

  1. Univ Paris Diderot, Sorbonne Paris Cité, Sorbonne Universités, UPMC Paris 06, CNRS, UMR 7586, Institut de mathématiques de Jussieu-Paris Rive Gauche, 75013, Paris, France

    Diogo Arsénio

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, Canada

    Slim Ibrahim

  3. Courant Institute of Mathematical Sciences, New York University, New York, USA

    Nader Masmoudi

Authors
  1. Diogo Arsénio
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  2. Slim Ibrahim
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  3. Nader Masmoudi
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Corresponding author

Correspondence to Diogo Arsénio.

Additional information

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

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