Beale--Kato--Majda-type continuation criteria for Hall- and electron-magnetohydrodynamics

TL;DR

This paper establishes Beale--Kato--Majda-type continuation criteria for electron-MHD and Hall-MHD with resistivity, linking the continuation of solutions to the finiteness of certain integral norms of current and vorticity.

Contribution

It introduces new continuation criteria for electron-MHD and Hall-MHD, extending classical results to these plasma models with resistivity.

Findings

Solutions can be continued if the integral of the current gradient's supremum remains finite.

A similar criterion applies to vorticity, velocity gradient, and current gradient in Hall-MHD.

The criteria generalize Beale--Kato--Majda's result to plasma physics models.

Abstract

We show that regular solutions to electron-MHD with resistivity can be continued as long as the time integral of the supremum of the current gradient remains finite. This dimensionless continuation criterion is analogous to the celebrated result of Beale--Kato--Majda for the incompressible Euler and Navier--Stokes equations. A similar continuation criterion, formulated in terms of the time integral of the supremum of the vorticity, velocity gradient and current gradient, is established for the Hall-MHD with resistivity as well.