### 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 language | English (US) |
---|---|

Pages (from-to) | 3071-3097 |

Number of pages | 27 |

Journal | Nonlinearity |

Volume | 25 |

Issue number | 11 |

DOIs | |

State | Published - Nov 1 2012 |

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### ASJC Scopus subject areas

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

### Cite this

*Nonlinearity*,

*25*(11), 3071-3097. https://doi.org/10.1088/0951-7715/25/11/3071

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

Research output: Contribution to journal › Article

*Nonlinearity*, vol. 25, no. 11, pp. 3071-3097. https://doi.org/10.1088/0951-7715/25/11/3071

}

TY - JOUR

T1 - On the supercritically diffusive magnetogeostrophic equations

AU - Friedlander, Susan

AU - Rusin, Walter

AU - Vicol, Vlad

PY - 2012/11/1

Y1 - 2012/11/1

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=84868136432&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84868136432&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/25/11/3071

DO - 10.1088/0951-7715/25/11/3071

M3 - Article

VL - 25

SP - 3071

EP - 3097

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 11

ER -