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 $\{u^{\nu }{\}}_{{\nu }_{\ge }0}\subset L_t^{\infty }BM{O}_x^{-1}$, 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 $\{$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 MG${}^{\nu }$ equations including the critical equation where $\nu =0$. Furthermore, we obtain the upper semicontinuity of the global attractor as $\nu$ vanishes.

25 pages

References in corpus (2)

• A new Bernstein's Inequality and the 2D Dissipative Quasi-Geostrophic Equation
• Uniformly attracting limit sets for the critically dissipative SQG equation