Regularity criteria for liquid crystal system involving one derivative component of pressure

Abstract

This paper is devoted to investigating regularity criteria for the nematic liquid crystal system in terms of one derivative component of the pressure and gradient of the orientation field. More precisely, we mainly proved that the strong solution $(u,d)$ can be extended beyond T, provided that one derivative component of the pressure ${\mathrm{\partial }}_{x_3}P$ and gradient of the orientation field satisfy $\begin{array}{r}{\mathrm{\partial }}_{x_3}P\in L^s(0,T;L^q({\mathbb{R}}^3\left)\right),\frac{2}{s}+\frac{3}{q}<\frac{29}{10},\frac{30}{23}<q\le \frac{10}{3},\\ {\mathrm{\partial }}_{x_3}d\in L^{\gamma }(0,T;L^{\text{β}}({\mathbb{R}}^3\left)\right),\frac{2}{\gamma }+\frac{3}{\text{β}}<\frac{19}{20},\frac{60}{13}<\text{β}\le \frac{20}{3}\end{array}$ and ${\mathrm{\nabla }}_{h}d\in L^{{\gamma }_1}(0,T;L^{{\text{β}}_1}({\mathbb{R}}^3\left)\right),\frac{2}{{\gamma }_1}+\frac{3}{{\text{β}}_1}\le \frac{3}{4}+\frac{1}{2{\text{β}}_1},\frac{60}{13}<{\text{β}}_1\le \frac{20}{3}.$

COMMUNICATED BY:

• M. Mei

Keywords:

• Regularity criteria
• nematic liquid crystal
• pressure
• Navier–Stokes equations

AMS CLASSIFICATIONS:

• 35B65
• 35Q35
• 76A15

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partly supported by the NSFC (National Natural Sciences Foundation of China) (No: 11571057).