Solutions to a class of forced drift-diffusion equations with applications to the magneto-geostrophic equations

Susan FRIEDLANDER, Chun Kit Anthony SUEN

Research output: Contribution to journalArticle

Abstract

We prove the global existence of classical solutions to a class of forced drift-diffusion equations with L² initial data and divergence free drift velocity {u}₀⊂Lᵼ BMOᵪˉ¹, and we obtain strong convergence of solutions as the viscosity ν vanishes. We then apply our results to a family of active scalar equations which includes the three dimensional magneto-geostrophic {MG}₀ equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earth’s fluid core. We prove the existence of a compact global attractor {Aᵛ}₀ in L² (T³) for the MG equations including the critical equation where ν=0 . Furthermore, we obtain the upper semicontinuity of the global attractor as ν vanishes. Copyright © 2018 Springer Nature Switzerland AG.
Original languageEnglish
Article number14
JournalAnnals of PDE
Volume4
Issue number2
Early online dateSep 2018
DOIs
Publication statusPublished - Dec 2018

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Drift-diffusion Equations
Global Attractor
Vanish
Upper Semicontinuity
Convergence of Solutions
Divergence-free
Classical Solution
Strong Convergence
Global Existence
Turbulence
Viscosity
Scalar
Fluid
Three-dimensional
Class

Citation

Friedlander, S., & Suen, A. (2018). Solutions to a class of forced drift-diffusion equations with applications to the magneto-geostrophic equations. Annals of PDE, 4(2). Retrieved from https://doi.org/10.1007/s40818-018-0050-3

Keywords

  • Active scalar equations
  • Vanishing viscosity limit
  • Global attractor