An Improved Blow-Up Criterion for the Magnetohydrodynamics with the Hall and Ion-Slip Effects

Abstract

In this work, we consider the magnetohydrodynamics system with the Hall and ion-slip effects in R³. The main result is a sufficient condition for regularity on a time interval [0,T] expressed in terms ∞,∞ of the norm of the homogeneous Besov space ${\dot{B}_{\infty,\infty}^0}$ with respect to the pressure and the BMO-norm with respect to the gradient of the magnetic field, respectively

${\int_{0}^{T} ({\| \Delta \pi(t)\|}^{2/3}_{\dot{B}_{\infty,\infty}^0} + {\| \Delta B (t)\|}^{2}_{BMO} ) dt<\infty}$,

which can be regarded as improvement of the result in [3].