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The limit states of magnetic relaxation

Published online by Cambridge University Press:  21 May 2007

MANUEL NÚÑEZ
MANUEL NÚÑEZ*
Affiliation:
Departamento de Análisis Matemático, Universidad de Valladolid, 47005 Valladolid, Spain e-mail: mnjmhd@am.uva.es
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Abstract

It is generally believed that a viscous, non-resistive plasma will eventually decay to a magnetostatic state, probably possessing contact discontinuities. We prove that even in the presence of a decaying forcing, the kinetic energy of the system tends to zero, which justifies the belief that the limit state is static. Regarding the magnetic field, the fact that the magnetic energy remains bounded proves the existence of weak sequential limits of the field as the time goes to infinity, but this does not imply that the magnetic field tends to a single state: we present an example where there is no limit, even in a weak sense. One additional condition upon the velocity, however, is enough to guarantee existence of a single limit magnetic configuration.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 580 , 10 June 2007 , pp. 251 - 260
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Agapito, R. & Schonbek, M. E. 2006 Non-uniform decay of MHD equations with and without magnetic diffusion. arXiv:math AP/0605638.Google Scholar
Deny, J. & Lions, J. L. 1954 Les espaces du type de Beppo-Levi. Ann. Inst. Fourier 5, 305–370.CrossRefGoogle Scholar
Kukavica, I. 2001 Space-time decay for solutions of the Navier-Stokes equations. Indiana Univ. Math. J. 50, 205–222.CrossRefGoogle Scholar
Lions, P. L. 1996 Mathematical Topics in Fluid Mechanics 1: Incompressible Models. Oxford University Press.Google Scholar
Low, B. C. & Wolfson, R. 1988 Spontaneous formation of electric current sheets and the origin of solar flares. Astrophys. J. 324, 574–581.CrossRefGoogle Scholar
Moffatt, H. K. 1985 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topolog: Fundamentals. J. Fluid Mech. 159, 359–378.CrossRefGoogle Scholar
Nishiyama, T. 2002 Magnetohydrodynamic approach to solvability of the three-dimensional stationary Euler equations. Glasgow Math. J. 44, 411–418.CrossRefGoogle Scholar
Nishiyama, T. 2003 Constructions of three-dimensional stationary Euler flows from pseudo-advected vorticity equations. Proc. R. Soc Lond. A 459, 2392–2398.CrossRefGoogle Scholar
Parker, E. N. 1994 Spontaneous Current Sheets in Magnetic Fields. Oxford University Press.CrossRefGoogle Scholar
Temam, R. 1980 Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer.Google Scholar