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Logarithmically improved blow-up criterion for the nematic liquid crystal system with zero viscosity. (English) Zbl 1326.35296

Summary: 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 \(T_*\) be the maximal existence time of the local classical solution, then \(T_*<+\infty\) if and only if \[ \int \limits _{0}^{T_*}\frac{\| \nabla u\| _{\dot{B}^{0}_{\infty ,\infty }}+\| \nabla d\| _{\dot{B}^{0}_{\infty ,\infty }}^{2}}{\sqrt{1+\ln (e+\| \nabla u\| _{\dot{B}^{0}_{\infty ,\infty }} +\| \nabla d\| _{\dot{B}^{0}_{\infty ,\infty }})}}\mathrm{d}t=\infty. \] The result can be regarded as a corresponding logarithmical blow-up criterion in [T. Huang and C. Wang, Commun. Partial Differ. Equations 37, No. 4–6, 875–884 (2012; Zbl 1247.35103)] for the nematic liquid crystal system with zero viscosity.

MSC:

35Q35 PDEs in connection with fluid mechanics
76A15 Liquid crystals
35B44 Blow-up in context of PDEs

Citations:

Zbl 1247.35103
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References:

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