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.