A Single-Component BKM-Type Regularity Criterion for the Inviscid Axially Symmetric Hall-MHD System

摘要

In this paper, we prove a single-component BKM-type regularity criterion for the inviscid axially symmetric Hall-MHD system. More precisely, we show that if the current density is swirl-free, then the L-t(1) L-x(infinity) boundedness of partial vorticity (corresponding to curl of the swirl part of the velocity) implies regularity of the solution. The novelty of our result is that we only impose a critical regularity criterion on one component of the velocity which is new even if we ignore the magnetic field and consider only the 3D axially symmetric Euler equations. Our results can not be easily extended to the classical L-t(1) BMO norm regularity criterion at present, which will be pursued in our further works.