On the Non-diffusive Magneto-Geostrophic Equation

Motivated by an equation arising in magnetohydrodynamics, we address the well-posedness theroy for non-diffusive magneto-geostrophic equation. Namely, active scalar which divergence-free drift velocity is one derivative more singular than scalar. In Friedlander and Vicol (Nonlinearity 24(11)::3019–3042, 2011), authors prove that ill-posed sense of Hadamard Sobolev spaces, but locally well posed spaces analytic functions. Here, give example a steady state nonlinearly stable periodic perturbations with initial data localized frequency straight lines crossing origin. For such well-prepared data, local existence uniqueness solutions can be obtained global holds under size condition over $$H^{5/2^{+}}({{\mathbb {T}}}^3)$$ norm perturbation.