On the supercritically diffusive magneto-geostrophic equations

TL;DR

This paper investigates the well-posedness of the supercritically diffusive magneto-geostrophic equations, revealing a critical threshold at =1/2 for local well-posedness and exploring anisotropic effects on regularity.

Contribution

It establishes the well-posedness threshold for the equations depending on , and shows how anisotropic Fourier support can improve regularity results for all  in (0,1).

Findings

For >1/2, the equations are locally well-posed.

For <1/2, the equations are ill-posed with no Lipschitz solution map.

Anisotropic Fourier support allows for improved regularity and well-posedness for all  in (0,1).

Abstract

We address the well-posedness theory for the magento-geostrophic equation, namely an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In the presence of supercritical fractional diffusion given by (-\Delta)^\gamma, where 0<\gamma<1, we discover that for \gamma>1/2 the equations are locally well-posed, while for \gamma<1/2 they are ill-posed, in the sense that there is no Lipschitz solution map. The main reason for the striking loss of regularity when \gamma goes below 1/2 is that the constitutive law used to obtain the velocity from the active scalar is given by an unbounded Fourier multiplier which is both even and anisotropic. Lastly, we note that the anisotropy of the constitutive law for the velocity may be explored in order to obtain an improvement in the regularity of the solutions when the initial data and the force have thin Fourier support, i.e. they are supported on a plane in frequency space. In particular, for such well-prepared data one may prove the local existence and uniqueness of solutions for all values of \gamma \in (0,1).