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

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 \({\dot{B}}_{\infty ,\infty }^{0}\) ${\dot{B}}_{\infty ,\infty }^0$ with respect to the pressure and the BMO−norm with respect to the gradient of the magnetic field, respectively

\(\underset{0}{\overset{T}{\int }}\left({\Vert \nabla \pi \left(t\right)\Vert }_{{\dot{B}}_{\infty ,\infty }^{0}}^\frac{2}{3}+{\Vert \nabla B\left(t\right)\Vert }_{BMO}^{2}\right)dt<\infty ,\) $\underset{0}{\stackrel{T}{\int }}\left({\Vert \mathrm{\nabla }\pi \left(t\right)\Vert }_{{\dot{B}}_{\infty ,\infty }^0}^{\frac{2}{3}},+,{\Vert \mathrm{\nabla }B\left(t\right)\Vert }_{\mathit{BMO}}^2\right)dt<\infty ,$

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