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On the localization of the magnetic and the velocity fields in the equations of magnetohydrodynamics

Published online by Cambridge University Press:  21 May 2007

Lorenzo Brandolese
Affiliation:
Université de Lyon and Université Lyon 1, UMR 5208 du CNRS, Institut Camille Jordan, Université Claude Bernard Lyon 1, 21 avenue Claude Bernard, 69622 Villeurbanne Cedex, France (brandolese@math.univ-lyon1.fr)
François Vigneron
Affiliation:
Centre de Mathématiques L. Schwartz, UMR 7640 du CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France (francois.vigneron@normalesup.org)
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Abstract

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We study the behaviour at infinity, with respect to the spatial variable, of solutions to the magnetohydrodynamics equations in $\mathbb{R}^d$. We prove that if the initial magnetic field decays sufficiently fast, then the plasma flow behaves as a solution of the free non-stationary Navier–Stokes equations when $|x|\to\infty$, and that the magnetic field will govern the decay of the plasma, if it is poorly localized at the beginning of the evolution. Our main tools are new boundedness criteria for convolution operators in weighted spaces.

Type
Research Article
Copyright
2007 Royal Society of Edinburgh