A logarithmically improved regularity criterion for the $3$D MHD system involving the velocity field in homogeneous Besov spaces

Zujin Zhang¹

¹ School of Mathematics and Computer Sciences Gannan Normal University Ganzhou 341000, Jiangxi, P.R. China

Annales Polonici Mathematici, Tome 118 (2016) no. 1, pp. 51-57.

Résumé

We consider a regularity criterion for the $3$D MHD equations. It is proved that if \[ \int _0^T\frac {\|\boldsymbol {u}(\tau )\| _{\dot B^r_{\infty ,\infty }}^{2/(1+r)}}{1+\ln(e+\| \boldsymbol {u}(\tau )\| _{\dot B^r_{\infty ,\infty }})}\,d \tau \lt \infty \] for some $0 \lt r \lt 1$, then the solution is actually smooth on $(0,T)$.

Détail

DOI : 10.4064/ap3952-9-2016

Keywords: consider regularity criterion mhd equations proved int frac boldsymbol tau dot infty infty boldsymbol tau dot infty infty tau infty solution actually smooth