On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations

Susan Friedlander, Vlad Vicol

Research output: Contribution to journalArticle

Abstract

We consider an active scalar equation that is motivated by a model for magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is globally well-posed (cf Friedlander and Vicol (2011 Ann. Inst. Henri Poincaré Anal. Non Linéaire 28 283-301)). In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field.

Original languageEnglish (US)
Pages (from-to)3019-3042
Number of pages24
JournalNonlinearity
Volume24
Issue number11
DOIs
StatePublished - Nov 1 2011

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Nonlinear Instability
Sobolev spaces
Well-posedness
Magnetic fields
Sobolev space
Sobolev Spaces
Unstable
Magnetic Field
Scalar
scalars
magnetic fields
Model

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

Cite this

On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations. / Friedlander, Susan; Vicol, Vlad.

In: Nonlinearity, Vol. 24, No. 11, 01.11.2011, p. 3019-3042.

Research output: Contribution to journalArticle

Friedlander, Susan ; Vicol, Vlad. / On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations. In: Nonlinearity. 2011 ; Vol. 24, No. 11. pp. 3019-3042.
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