Note on global regular solution to the 3D liquid crystal equations

Two regularity criteria of solutions to the liquid crystal flows

“…Hence, in what follows, we shall focus on a priori estimates for a local smooth solution. For the standard $$$ {L}^2 $$$ estimate of $$$ u $$$ and $$$ \nabla d $$$, by a direct estimate as we did in Yuan and Li, 10, p.2‐3 one has $$$$ {\displaystyle \begin{array}{cc}\hfill & \left({\left\Vert u\right\Vert}_{L^2}^2+{\left\Vert \nabla d\right\Vert}_{L^2}^2\right)+2{\int}_0^T\left({\left\Vert \nabla u\right\Vert}_{L^2}^2+{\left\Vert \Delta d\right\Vert}_{L^2}^2+{\left\Vert d|\nabla d|\right\Vert}_{L^2}^2+\frac{1}{2}{\left\Vert \nabla ...$$…”

“…It is obvious that system () reduces to the well‐known Navier–Stokes equations when the orientation field $$$ d\equiv 1 $$$. Therefore, in view of the results of regularity criterion for the 3D Navier–Stokes equations, some Prodi–Serrin type regularity criteria of the nematic liquid crystal flows have been obtained based on the velocity field $$$ u $$$, the gradient of the velocity field $$$ \nabla u $$$, and their components; see earlier studies 3–12 and the references therein.…”

“…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) $$$ is the horizontal component of velocity. And Yuan and Li 10 generalized the regularity criterion () to $$$$ {\int}_0^T{\left\Vert \nabla {u}_h\right\Vert}_{L^p}^q\mathrm{d}t<\infty, \kern0.3em with\kern0.3em \frac{3}{p}+\frac{2}{q}\le \frac{3}{2},\kern0.3em 2\le p\le \infty . $$$$ …”

Two Logarithmically Improved Regularity Criteria for the 3D Nematic Liquid Crystal Flows

“…are approximate, it is obvious that the condition (10) is weaker than the condition (8) and (9) in some sense. And it can be seen that if the condition (10)…”

“…Hence much eforts have been paid to study the regularity criteria to extend local solutions. For the regularity criteria readers may refer to [5][6][7][8][9][10][11][12][13] and references therein.…”