Logarithmically improved blow-up criterion for the nematic liquid crystal system with zero viscosity

In this paper, we establish a criterion for the breakdown of local in time classical solutions to the incompressible nematic liquid crystal system with zero viscosity in dimensions three. More precisely, let T∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{*}$$\end{document} be the maximal existence time of the local classical solution, then T∗<+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{*}<+\infty $$\end{document} if and only if ∫0T∗‖∇u‖B˙∞,∞0+‖∇d‖B˙∞,∞021+ln(e+‖∇u‖B˙∞,∞0+‖∇d‖B˙∞,∞0)dt=∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int \limits _{0}^{T_{*}}\frac{\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }}+\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln (e+\Vert \nabla u\Vert _{\dot{B}^{0}_{\infty ,\infty }} +\Vert \nabla d\Vert _{\dot{B}^{0}_{\infty ,\infty }})}}\hbox {d}t=\infty . \end{aligned}$$\end{document}The result can be regarded as a corresponding logarithmical blow-up criterion in Huang and Wang (Commun. Partial Differ. Equ. 37:875–884, 2012) for the nematic liquid crystal system with zero viscosity.