Blow-up criteria for the 3D liquid crystal flows involving two velocity components

Two regularity criteria of solutions to the liquid crystal flows

“…then the solution (u, d) can be extended smoothly beyond T. Later Zhao et al 13 improved the above regularity criterion (1.2) to the following…”

“…That is, if $$$$ {\int}_0^T{\left\Vert {u}_3\right\Vert}_{L^p}^q+{\left\Vert \nabla d\right\Vert}_{L^p}^q\mathrm{d}t<\infty, \kern0.3em with\kern0.3em \frac{3}{p}+\frac{2}{q}\le \frac{3}{4}+\frac{1}{2p},\kern0.3em \frac{10}{3}\le p, $$$$ then the solution $$$ \left(u,d\right) $$$ can be extended smoothly beyond $$$ T $$$. Later Zhao et al 13 improved the above regularity criterion () to the following $$$$ {\int}_0^T{\left\Vert {u}_h\right\Vert}_{L^p}^q\mathrm{d}t<\infty, \kern0.3em with\kern0.3em \frac{3}{p}+\frac{2}{q}\le \frac{1}{2},\kern0.3em 6\le p\le \infty, $$$$ where $$$ {u}_h=\left({u}_1,{u}_2\right) $$$ is the horizontal component of velocity. And Yuan and Li 10 generalized the regularity cri...…”