On the supercritically diffusive magnetogeostrophic equations

Susan Friedlander, Walter Rusin, Vlad Vicol

Research output: Contribution to journalArticle

Abstract

We address the well-posedness theory for the magneto-geostrophic equation, namely an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In the presence of supercritical fractional diffusion given by (Δ) γ with 0<γ<1, we discover that for γ>1/2 the equations are locally well-posed, while for γ<1/2 they are ill-posed, in the sense that there is no Lipschitz solution map. The main reason for the striking loss of regularity when γ goes below 1/2 is that the constitutive law used to obtain the velocity from the active scalar is given by an unbounded Fourier multiplier which is both even and anisotropic. Lastly, we note that the anisotropy of the constitutive law for the velocity may be explored in order to obtain an improvement in the regularity of the solutions when the initial data and the force have thin Fourier support, i.e. they are supported on a plane in frequency space. In particular, for such well-prepared data one may prove the local existence and uniqueness of solutions for all values of γ(0, 1). In fact, these solutions are global in time when γ[1/2, 1).

Original languageEnglish (US)
Pages (from-to)3071-3097
Number of pages27
JournalNonlinearity
Volume25
Issue number11
DOIs
StatePublished - Nov 1 2012

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Constitutive Law
Scalar
scalars
regularity
Regularity
Fractional Diffusion
Fourier multipliers
Divergence-free
Local Existence
multipliers
uniqueness
Existence and Uniqueness of Solutions
Well-posedness
Lipschitz
Anisotropy
divergence
Derivatives
Derivative
anisotropy

ASJC Scopus subject areas

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

Cite this

On the supercritically diffusive magnetogeostrophic equations. / Friedlander, Susan; Rusin, Walter; Vicol, Vlad.

In: Nonlinearity, Vol. 25, No. 11, 01.11.2012, p. 3071-3097.

Research output: Contribution to journalArticle

Friedlander, S, Rusin, W & Vicol, V 2012, 'On the supercritically diffusive magnetogeostrophic equations', Nonlinearity, vol. 25, no. 11, pp. 3071-3097. https://doi.org/10.1088/0951-7715/25/11/3071
Friedlander, Susan ; Rusin, Walter ; Vicol, Vlad. / On the supercritically diffusive magnetogeostrophic equations. In: Nonlinearity. 2012 ; Vol. 25, No. 11. pp. 3071-3097.
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