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

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 {uᵛ}ᵥ_≥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 {MGᵛ}ᵥ_≥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}ᵥ_≥₀ in L² (T³) for the MGᵛ 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.