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Annals of PDE

, 4:14 | Cite as

Solutions to a Class of Forced Drift-Diffusion Equations with Applications to the Magneto-Geostrophic Equations

  • Susan Friedlander
  • Anthony SuenEmail author
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Abstract

We prove the global existence of classical solutions to a class of forced drift-diffusion equations with \(L^2\) initial data and divergence free drift velocity \(\{u^\nu \}_{\nu _\ge 0}\subset L^\infty _t BMO^{-1}_x\), and we obtain strong convergence of solutions as the viscosity \(\nu \) vanishes. We then apply our results to a family of active scalar equations which includes the three dimensional magneto-geostrophic \(\{\hbox {MG}^\nu \}_{\nu \ge 0}\) 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 \(\{\mathcal {A}^\nu \}_{\nu \ge 0}\) in \(L^2(\mathbb {T}^3)\) for the \(\hbox {MG}^\nu \) equations including the critical equation where \(\nu =0\). Furthermore, we obtain the upper semicontinuity of the global attractor as \(\nu \) vanishes.

Keywords

Active scalar equations Vanishing viscosity limit Global attractor 

Mathematics Subject Classification

76D03 35Q35 76W05 

Notes

Acknowledgements

We thank Vlad Vicol for his very helpful advice. We also thank the referees for their most valuable comments. AS is partially supported by Hong Kong Early Career Scheme (ECS) Grant Project Number 28300016. SF is partially supported by NSF Grant DMS-1613135.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of Mathematics and Information TechnologyThe Education University of Hong KongNew TerritoriesHong Kong

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