An improved blow-up criterion for the magnetohydrodynamics with the Hall and ion-slip effects

S. Gala ; M. A. Ragusa

Contemporary Mathematics. Fundamental Directions, Dedicated to 70th anniversary of the President of the RUDN University V. M. Filippov, Tome 67 (2021) no. 3, pp. 526-534

Résumé

In this work, we consider the magnetohydrodynamics system with the Hall and ion-slip effects in $\mathbb{R}^{3}$. 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 \begin{equation*} \int_{0}^{T}\left( \left\Vert \nabla \pi (t)\right\Vert _{\dot{B}_{\infty ,\infty }^{0}}^{\frac{2}{3}}+\left\Vert \nabla B(t)\right\Vert _{BMO}^{2}\right) dt\infty , \end{equation*} which can be regarded as improvement of the result in [3].