Existence, uniqueness and regularity results for the viscous magneto-geostrophic equation

We study the three dimensional active scalar equation called the magneto-geostropic equation which was proposed by Moffatt and Loper as a model for the geodynamo processes in the Earth's fluid core. When the viscosity of the fluid is positive, the constitutive law that relates the drift velocity $u(x,t)$ and the scalar temperature $\theta(x,t)$ produces two orders of smoothing. We study the implications of this property. For example, we prove that in the case of the non-diffusive ($\varepsilon_\kappa=0$) active scalar equation, initial data $\theta_0\in L^3$ implies the existence of unique, global weak solutions. If $\theta_0\in W^{s,3}$ with $s>0$, then the solution $\theta(x,t)\in W^{s,3}$ for all time. In the case of positive diffusivity ($\varepsilon_\kappa>0$), even for singular initial data $\theta_0\in L^3$, the global solution is instantaneously $C^\infty$-smoothed and satisfies the drift-diffusion equation classically for all $t>0$. We demonstrate, via a particular example, that the viscous magneto-geostrophic equation permits exponentially growing "dynamo type" instabilities.