On the non-diffusive Magneto-Geostrophic equation

TL;DR

This paper investigates the well-posedness of the non-diffusive magneto-geostrophic equation, demonstrating stability and existence results for specific steady states and perturbations, despite known ill-posedness in Sobolev spaces.

Contribution

It provides the first example of nonlinear stability for a steady state and establishes local and global existence results under certain conditions for the non-diffusive magneto-geostrophic equation.

Findings

Existence of a nonlinearly stable steady state.

Local well-posedness in Sobolev spaces for frequency-localized perturbations.

Global existence under smallness conditions on the perturbation's Sobolev norm.

Abstract

Motivated by an equation arising in magnetohydrodynamics, we address the well-posedness theroy for the non-diffusive magneto-geostrophic equation. Namely, an active scalar equation in which the divergence-free drift velocity is one derivative more singular that the active scalar. In \cite{Friedlander-Vicol_3}, the authors prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces, but locally well posed in spaces of analytic functions. Here, we give an example of a steady state that is nonlinearly stable for periodic perturbations with initial data localized in frequency straight lines crossing the origin. For such well-prepared data, the local existence and uniqueness of solutions can be obtained in Sobolev spaces and the global existence holds under a size condition over the $H^{5/2^{+}}(\mathbb{T}^3)$ norm of the perturbation.