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Instability of the magnetohydrodynamics system at vanishing Reynolds number

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Abstract

The aim of this note is to study the dynamo properties of the magnetohydrodynamics system at vanishing R m . Improving the analysis in Gérard-Varet (SIAM J Math Anal 37(3):815–840, 2006), we shall establish a generic Lyapunov instability result.

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References

  1. Alexakis A.: Searching for the fastest dynamo: Laminar ABC flows. Phys. Rev. E 84(2), 026321 (2011)

    Article  Google Scholar 

  2. Bouya, I.: Instability of the Magnetohydrodynamics System at Small but Finite Reynolds Number. SIAM J. Math. Anal. (2012, accepted)

  3. Fauve S., Petrelis F.: The dynamo effect. Peyresq Lect. Nonlinear Phenom. 2, 1–64 (2003)

    Article  Google Scholar 

  4. Friedlander S., Pavlović N., Shvydkoy R.: Nonlinear instability for the Navier–Stokes equations. Commun. Math. Phys. 264(2), 335–347 (2006)

    Article  MATH  Google Scholar 

  5. Friedlander, S., Strauss, W., Vishik, M.M.: Nonlinear instability in an ideal fluid. In: Annales de l’Institut Henri Poincare/Analyse Non lineaire, volume 14, pp. 187–209. Elsevier, Amsterdam (1997)

  6. Gérard-Varet D.: Oscillating solutions of incompressible magnetohydrodynamics and dynamo effect. SIAM J. Math. Anal. 37(3), 815–840 (2006)

    Article  Google Scholar 

  7. Gilbert A.D.: Dynamo theory. Handb. Math. Fluid Dyn. 2(1), 355–441 (2003)

    Article  Google Scholar 

  8. Guo, Y., Strauss, W.: Nonlinear instability of double-humped equilibria. In: Annales de l’Institut Henri Poincaré. Analyse non linéaire, volume 12, pp. 339–352. Elsevier, Amsterdam (1995)

  9. Larmor, J.: How could a rotating body such as the sun become a magnet. Rep. Brit. Assoc. Adv. Sci. 159, 412 (1919)

    Google Scholar 

  10. Otani N.: A fast kinematic dynamo in two-dimensional time-dependent flows. J. Fluid Mech. 253, 327–340 (1993)

    Article  MATH  Google Scholar 

  11. Parker E.N.: Hydromagnetic dynamo models. Astrophys. J. 122, 293–314 (1955)

    Article  MathSciNet  Google Scholar 

  12. Soward A.: On the role of stagnation points and periodic particle paths in a two-dimensional pulsed flow fast dynamo model. Physica D: Nonlinear Phenom. 76(1–3), 181–201 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Soward A.M.: Fast dynamo action in a steady flow. J. Fluid Mech. 180, 267–295 (1987)

    Article  MATH  Google Scholar 

  14. Stieglitz R., Müller U.: Experimental demonstration of a homogeneous two-scale dynamo. Phys. Fluids 13, 561 (2001)

    Article  Google Scholar 

  15. Temam R.: Navier–Stokes Equations: Theory and Numerical Analysis. American Mathematical Society, Providence (2001)

    Google Scholar 

  16. Vishik M.M.: Magnetic field generation by the motion of a highly conducting fluid. Geophys. Astrophys. Fluid Dyn. 48(1), 151–167 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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  1. UMR 7586 Institut de mathématiques de Jussieu—Analyse fonctionnelle, Paris, France

    Ismaël Bouya

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  1. Ismaël Bouya
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Correspondence to Ismaël Bouya.

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Bouya, I. Instability of the magnetohydrodynamics system at vanishing Reynolds number. Z. Angew. Math. Phys. 64, 1689–1698 (2013). https://doi.org/10.1007/s00033-013-0309-1

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  • DOI: https://doi.org/10.1007/s00033-013-0309-1

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