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An Osgood type regularity criterion for the liquid crystal flows. (English) Zbl 1301.35118

Summary: In this paper, we prove an Osgood type regularity criterion for the model of liquid crystals, which says that the condition \[ \sup_{2 \leq q< \infty} \int \nolimits_0^T \frac{\| \bar{S}_{q} \nabla {\mathbf u}(t)\|_{L^\infty}}{q \ln q} {\text d} t<\infty \] implies the smoothness of the solution. Here, \({{\bar S_q=\sum\nolimits_{k=-q}^q \dot {\triangle}_k}}\) with \({\dot{\triangle}_k}\) being the frequency localization operator.

MSC:

35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76A15 Liquid crystals
35B65 Smoothness and regularity of solutions to PDEs
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