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

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.