A logarithmically improved regularity criterion for the $3$D MHD system involving the velocity field in homogeneous Besov spaces
Volume 118 / 2016
Annales Polonici Mathematici 118 (2016), 51-57
MSC: Primary 35B65; Secondary 35Q35, 76D03.
DOI: 10.4064/ap3952-9-2016
Published online: 21 October 2016
Abstract
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 $(0,T)$.
