# A note on regularity criterion for 3D compressible nematic liquid crystal flows

## Abstract

In this article, we prove a regularity criterion for the local strong solutions to a simplified hydrodynamic flow modeling the compressible, nematic liquid crystal materials.

Mathematics Subject Classifications (2010): 35M10; 76N10; 82D30.

## 1 Introduction

In this article, we consider the following simplified version of Ericksen-Leslie system modeling the hydrodynamic flow of compressible, nematic liquid crystals (see: [1, 2])

${\partial }_{t}\rho +\text{div}\left(\rho u\right)=0,$
(1.1)
${\partial }_{t}\left(\rho u\right)+\text{div}\left(\rho u\otimes u\right)+\nabla p\left(\rho \right)-\mu \Delta u-\left(\lambda +\mu \nabla \text{div}u=-\Delta d\cdot \nabla d,$
(1.2)
${\partial }_{t}d+u\cdot \nabla d=\Delta d+{\left|\nabla d\right|}^{2}d,$
(1.3)
$\left(\rho ,u,d\right)\left(x,0\right)=\left({\rho }_{0},{u}_{0},{d}_{0}\right)\left(x\right),\left|{d}_{0}\right|=1,x\in {ℝ}^{3}.$
(1.4)

Here ρ is the density, u is the fluid velocity and d represents the macroscopic average of the nematic liquid crystal orientation field, p(ρ) := γis the pressure with positive constants a > 0 and γ ≥ 1. μ and λ are the shear viscosity and the bulk viscosity coefficients of the fluid respectively, which are assumed to satisfy the following physical condition:

$\mu >0,\phantom{\rule{1em}{0ex}}3\lambda +2\mu \ge 0.$

(1.1) and (1.2) is the well-known compressible Navier-Stokes system with the external force -Δd·d. (1.3) is the well-known heat flow of harmonic map when u = 0.

Very recently, Ericksen [3] proved the following local-in-time well-posedness:

Proposition 1.1. Let ρ0 W1,qH1L1 for some q (3, 6] and ρ0 ≥ 0 in 3, u0 H1, d0 H2 and |d0| = 1 in 3. If, in additions, the following compatibility condition

$-\mu \Delta {u}_{0}-\left(\lambda +\mu \right)\nabla \text{div}{u}_{0}-\nabla p\left({\rho }_{0}\right)-\Delta {d}_{0}\cdot {d}_{0}=\sqrt{{\rho }_{0}}g\phantom{\rule{1em}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}some\phantom{\rule{2.77695pt}{0ex}}g\in {L}^{2}\left({ℝ}^{3}\right)$
(1.5)

holds, then there exist T0 > 0 and a unique strong solution (ρ, u, d) to the problem (1.1)-(1.4).

Based on the above Proposition 1.1, Huang et al. [4] proved the following regularity criterion:

$\underset{0}{\overset{T}{\int }}{∥\mathcal{D}\left(u\right)∥}_{{L}^{\infty }}+{∥\nabla d∥}_{{L}^{\infty }}^{2}dt<\infty ,$
(1.6)

where $\mathcal{D}\left(u\right):=\frac{1}{2}\left(\nabla u{+}^{t}\nabla u\right)$.

The aim of this note is to refine (1.6) as follows.

Theorem 1.2. Let the assumptions in Proposition 1.1 holds true. If

$\underset{0}{\overset{T}{\int }}{∥\mathcal{D}\left(u\right)∥}_{{L}^{\infty }}+{∥\nabla d∥}_{BMO}^{2}dt<\infty ,$
(1.7)

then the solution (ρ,u,d) can be extended beyond T > 0.

Here BMO denotes the space of functions of bounded mean oscillations.

In this note, we will use the following inequality [5]:

${∥u∥}_{Lp}\le C{∥u∥}_{BMO}^{1-\frac{q}{p}}{∥u∥}_{{L}^{q}}^{\frac{q}{p}}\phantom{\rule{2.77695pt}{0ex}}\left(1
(1.8)

For the standard nematic liquid crystal flows, we refer to recent studies in [6, 7].

## 2 Proof of Theorem 1.2

Since (ρ,u,d) is the local strong solution, we only need to prove

$\nabla d\in {L}^{2}\left(0,T;{L}^{\infty }\right).$
(2.1)

By the same calculations as that in [4], it is easy to show that

$\begin{array}{l}{∥\rho ∥}_{{L}^{\infty }\left(0,T;{L}^{\infty }\right)}\le C,\phantom{\rule{2em}{0ex}}\\ \int \rho {\left|u\right|}^{2}+{\left|\nabla d\right|}^{2}dx+\underset{0}{\overset{T}{\int }}\int {\left|\nabla u\right|}^{2}+{\left|\Delta d+{\left|\nabla d\right|}^{2}d\right|}^{2}dxdt\le C.\phantom{\rule{2em}{0ex}}\end{array}$
(2.2)

Using (1.8), we see that

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\underset{0}{\overset{T}{\int }}\int {\left|\nabla d\right|}^{4}dxdt\le C\underset{0}{\overset{T}{\int }}{∥\nabla d∥}_{BMO}^{2}{∥\nabla d∥}_{{L}^{2}}^{2}dt\phantom{\rule{2em}{0ex}}\\ \le \underset{t}{\text{max}}{∥\nabla d∥}_{{L}^{2}}^{2}\underset{0}{\overset{T}{\int }}{∥\nabla d∥}_{BMO}^{2}dt\le C,\phantom{\rule{2em}{0ex}}\end{array}$

from which and (2.2), we get

$\underset{0}{\overset{T}{\int }}\int {\left|\Delta d\right|}^{2}dxdt\le 2\underset{0}{\overset{T}{\int }}\int {\left|\Delta d+{\left|\nabla d\right|}^{2}d\right|}^{2}dxdt+2\underset{0}{\overset{T}{\int }}\int {\left|\nabla d\right|}^{4}dxdt\le C.$
(2.3)

Applying to (1.3), testing by r|d|r-2d (r ≥ 2), using (1.8), we infer that

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\frac{d}{dt}\int {\left|\nabla d\right|}^{r}dx+r\int {\left|\nabla d\right|}^{r-2}{\left|{\nabla }^{2}d\right|}^{2}+\left(r-2\right){\left|\nabla d\right|}^{r-2}{\left|\nabla \left|\nabla d\right|\right|}^{2}dx\phantom{\rule{2em}{0ex}}\\ =r\int \nabla \left({\left|\nabla d\right|}^{2}d\right){\left|\nabla d\right|}^{r-2}\nabla ddx-r\int \nabla \left(u\cdot \nabla d\right){\left|\nabla d\right|}^{r-2}\nabla ddx\phantom{\rule{2em}{0ex}}\\ =r\int {\left|\nabla d\right|}^{r+2}dx-r\int {\left|\nabla d\right|}^{r-2}{\nabla }_{i}{u}^{j}<{\nabla }_{j}d,{\nabla }_{i},d>dx-\int u\cdot \nabla {\left|\nabla d\right|}^{r}dx\phantom{\rule{2em}{0ex}}\\ =r\int {\left|\nabla d\right|}^{r+2}dx-r\int {\left|\nabla d\right|}^{r-2}\mathcal{D}\left(u\right):\nabla d\otimes \nabla ddx+\int \left(\text{div}\phantom{\rule{2.77695pt}{0ex}}u\right){\left|\nabla d\right|}^{r}dx\phantom{\rule{2em}{0ex}}\\ \le C{∥\nabla d∥}_{BMO}^{2}{∥\nabla d∥}_{{L}^{r}}^{r}+C{∥\mathcal{D}\left(u\right)∥}_{{L}^{\infty }}{∥\nabla d∥}_{{L}^{r}}^{r},\phantom{\rule{2em}{0ex}}\end{array}$

which yields

$\underset{0\le t
(2.4)

Let

$ḟ:={f}_{t}+u\cdot \nabla f$

denotes the material derivative of f.

Testing (1.2) by $\stackrel{˙}{u}$, we derive

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\frac{1}{2}\frac{d}{dt}\int \mu {\left|\nabla u\right|}^{2}+\left(\lambda +\mu \right){\left|\text{div}\phantom{\rule{2.77695pt}{0ex}}u\right|}^{2}dx+\int {\rho \left|\stackrel{˙}{u}\right|}^{2}dx\phantom{\rule{2em}{0ex}}\\ =\int <\left(u\cdot \nabla \right)u,-\mu \Delta u-\left(\lambda +\mu \right)\nabla \text{div}\phantom{\rule{2.77695pt}{0ex}}u>dx\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\int u\cdot \nabla u\cdot \nabla p\left(\rho \right)dx-\int {u}_{t}\cdot \nabla p\left(\rho \right)dx\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\int u\cdot \nabla u\cdot <\Delta d,\nabla d>dx-\int {u}_{t}<\Delta d,\nabla d>dx\phantom{\rule{2em}{0ex}}\\ =:{I}_{1}+{I}_{2}+{I}_{3}+{I}_{4}+{I}_{5}.\phantom{\rule{2em}{0ex}}\end{array}$
(2.5)

By the same calculations as that in [4], we have

$\begin{array}{ll}\hfill {I}_{1}& =\mu \int \mathcal{D}\left(u\right):\text{curl}\phantom{\rule{2.77695pt}{0ex}}u\otimes \text{curl}\phantom{\rule{2.77695pt}{0ex}}u-\frac{1}{2}\text{div}\phantom{\rule{2.77695pt}{0ex}}u{\left(\text{curl}\phantom{\rule{2.77695pt}{0ex}}u\right)}^{2}dx\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\left(2\mu +\lambda \right)\int \left(\nabla u{:}^{t}\nabla u\right)\text{div}\phantom{\rule{2.77695pt}{0ex}}u-\frac{1}{2}{\left(\text{div}\phantom{\rule{2.77695pt}{0ex}}u\right)}^{3}dx\phantom{\rule{2em}{0ex}}\\ \le C{∥\mathcal{D}\left(u\right)∥}_{{L}^{\infty }}{∥\nabla u∥}_{{L}^{2}}^{2},\phantom{\rule{2em}{0ex}}\\ \hfill {I}_{2}& =\int p\left(\rho \right)\left(\nabla u{:}^{t}\nabla u-{\left(\text{div}\phantom{\rule{2.77695pt}{0ex}}u\right)}^{2}\right)dx-\int u\phantom{\rule{2.77695pt}{0ex}}\text{div}\phantom{\rule{2.77695pt}{0ex}}u\cdot \nabla p\left(\rho \right)dx\phantom{\rule{2em}{0ex}}\\ \le C{∥\nabla u∥}_{{L}^{2}}^{2}+C\int \left|\nabla \rho ∥u∥\text{div}\phantom{\rule{2.77695pt}{0ex}}u\right|dx\phantom{\rule{2em}{0ex}}\\ \le C{∥\nabla u∥}_{{L}^{2}}^{2}+C{∥u∥}_{{L}^{6}}{∥\nabla \rho ∥}_{{L}^{2}}{∥\text{div}\phantom{\rule{2.77695pt}{0ex}}u∥}_{{L}^{3}}\phantom{\rule{2em}{0ex}}\\ \le C{∥\nabla u∥}_{{L}^{2}}^{2}+C{∥\nabla u∥}_{{L}^{2}}{∥\nabla \rho ∥}_{{L}^{2}}{∥\mathcal{D}\left(u\right)∥}_{{L}^{\infty }}^{\frac{1}{3}}{∥\mathcal{D}\left(u\right)∥}_{{L}^{2}}^{\frac{2}{3}}\phantom{\rule{2em}{0ex}}\\ \le C{∥\nabla u∥}_{{L}^{2}}^{2}+C{∥\nabla u∥}_{{L}^{2}}^{2}{∥\nabla \rho ∥}_{{L}^{2}}^{2}+C{∥\mathcal{D}\left(u\right)∥}_{{L}^{\infty }}^{\frac{2}{3}}{∥\mathcal{D}\left(u\right)∥}_{{L}^{2}}^{\frac{4}{3}}\phantom{\rule{2em}{0ex}}\\ \le C{∥\nabla u∥}_{{L}^{2}}^{2}+C{∥\nabla u∥}_{{L}^{2}}^{2}{∥\nabla \rho ∥}_{{L}^{2}}^{2}+C\left(1+{∥\mathcal{D}\left(u\right)∥}_{{L}^{\infty }}\right){∥\nabla u∥}_{{L}^{2}}^{2},\phantom{\rule{2em}{0ex}}\\ \hfill {I}_{3}& =\frac{d}{dt}\int p\left(\rho \right)\text{div}\phantom{\rule{2.77695pt}{0ex}}udx-\int \text{div}\phantom{\rule{2.77695pt}{0ex}}u{\partial }_{t}p\left(\rho \right)dx\phantom{\rule{2em}{0ex}}\\ \le \frac{d}{dt}\int p\left(\rho \right)\text{div}\phantom{\rule{2.77695pt}{0ex}}udx+C+C\left(1+{∥\mathcal{D}\left(u\right)∥}_{{L}^{\infty }}\right){∥\nabla u∥}_{{L}^{2}}^{2}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+C{∥\nabla u∥}_{{L}^{2}}^{2}{∥\nabla \rho ∥}_{{L}^{2}}^{2},\phantom{\rule{2em}{0ex}}\\ \hfill {I}_{4}& \le {∥u∥}_{{L}^{6}}{∥\nabla u∥}_{{L}^{6}}{∥\Delta d∥}_{{L}^{2}}{∥\nabla d∥}_{{L}^{6}}\phantom{\rule{2em}{0ex}}\\ \le C{∥\nabla u∥}_{{L}^{2}}{∥\Delta u∥}_{{L}^{2}}{∥\Delta d∥}_{{L}^{2}}\left(by\phantom{\rule{2.77695pt}{0ex}}\left(2.4\right)\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}r=6\right)\phantom{\rule{2em}{0ex}}\\ \le \epsilon {∥\Delta u∥}_{{L}^{2}}^{2}+C{∥\nabla u∥}_{{L}^{2}}^{2}{∥\Delta d∥}_{{L}^{2}}^{2},\phantom{\rule{2em}{0ex}}\end{array}$

for any ϵ > 0.

We denote $M\left(d\right):=\nabla d\otimes \nabla d-\frac{1}{2}{\left|\nabla d\right|}^{2}{I}_{3}$, I5 is estimated as follows, which is slightly different from that in [4]:

$\begin{array}{ll}\hfill {I}_{5}& =\frac{d}{dt}\int M\left(d\right):\nabla udx-\int {\partial }_{t}M\left(d\right):\nabla udx\phantom{\rule{2em}{0ex}}\\ \le \frac{d}{dt}\int M\left(d\right):\phantom{\rule{2.77695pt}{0ex}}\nabla udx+C\int \left|\nabla {d}_{t}∥\nabla d∥\nabla u\right|dx\phantom{\rule{2em}{0ex}}\\ \le \frac{d}{dt}\int M\left(d\right):\phantom{\rule{2.77695pt}{0ex}}\nabla udx+C{∥\nabla d∥}_{{L}^{6}}{∥\nabla u∥}_{{L}^{3}}{∥\nabla {d}_{t}∥}_{{L}^{2}}\phantom{\rule{2em}{0ex}}\\ \le \frac{d}{dt}\int M\left(d\right):\nabla udx+C{∥\nabla u∥}_{{L}^{3}}{∥\nabla {d}_{t}∥}_{{L}^{2}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(by\phantom{\rule{2.77695pt}{0ex}}\left(2.4\right)\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}r=6\right)\phantom{\rule{2em}{0ex}}\\ \le \frac{d}{dt}\int M\left(d\right):\nabla udx+\epsilon {∥\nabla {d}_{t}∥}_{{L}^{2}}^{2}+\epsilon {∥\Delta u∥}_{{L}^{2}}^{2}+C{∥\nabla u∥}_{{L}^{2}}^{2}.\phantom{\rule{2em}{0ex}}\end{array}$

Substituting the above estimates into (2.5), we deduce that

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\frac{1}{2}\frac{d}{dt}\int \mu +{\left|\nabla u\right|}^{2}\left(\lambda +\mu \right){\left|\text{div}\phantom{\rule{2.77695pt}{0ex}}u\right|}^{2}dx+\int \rho {\left|\stackrel{˙}{u}\right|}^{2}dx\phantom{\rule{2em}{0ex}}\\ \le C{∥\nabla u∥}_{{L}^{2}}^{2}{∥\nabla \rho ∥}_{{L}^{2}}^{2}+C\left(1+{∥\mathcal{D}\left(u\right)∥}_{{L}^{\infty }}\right){∥\nabla u∥}_{{L}^{2}}^{2}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+C+\frac{d}{dt}\int p\left(\rho \right)\text{div}\phantom{\rule{2.77695pt}{0ex}}u+M\left(d\right):\nabla udx\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+C{∥\nabla u∥}_{{L}^{2}}^{2}{∥\Delta d∥}_{{L}^{2}}^{2}+2\epsilon \left({∥\nabla {d}_{t}∥}_{{L}^{2}}^{2}+{∥\Delta u∥}_{{L}^{2}}^{2}\right)\phantom{\rule{2em}{0ex}}\end{array}$
(2.6)

for any 0 < ϵ < 1.

By the same calculations as that in [4], we write

$\frac{d}{dt}{∥\nabla \rho ∥}_{{L}^{2}}^{2}\le C\left(1+{∥\mathcal{D}\left(u\right)∥}_{{L}^{\infty }}\right){∥\nabla \rho ∥}_{{L}^{2}}^{2}+\epsilon {∥\Delta u∥}_{{L}^{2}}^{2}.$
(2.7)

Testing (1.3) by Δd t , using (2.4), we obtain

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\frac{1}{2}\frac{d}{dt}\int {\left|\Delta d\right|}^{2}dx+\int {\left|\nabla {d}_{t}\right|}^{2}dx=\int \left(u\cdot \nabla d-{\left|\nabla d\right|}^{2}d\right)\Delta {d}_{t}dx\phantom{\rule{2em}{0ex}}\\ =-\int \nabla \left(u\cdot \nabla d-{\left|\nabla d\right|}^{2}d\right)\cdot \nabla {d}_{t}dx\phantom{\rule{2em}{0ex}}\\ \le \left({∥\nabla u∥}_{{L}^{3}}{∥\nabla d∥}_{{L}^{6}}+{∥u∥}_{{L}^{6}}{∥\Delta d∥}_{{L}^{3}}+{∥\nabla d∥}_{{L}^{6}}^{3}+{∥\nabla d∥}_{{L}^{6}}{∥\Delta d∥}_{{L}^{3}}\right){∥\nabla {d}_{t}∥}_{{L}^{2}}\phantom{\rule{2em}{0ex}}\\ \le C\left({∥\nabla u∥}_{{L}^{3}}+{∥\nabla u∥}_{{L}^{2}}{∥\Delta d∥}_{{L}^{3}}+1+{∥\Delta d∥}_{{L}^{3}}\right){∥\nabla {d}_{t}∥}_{{L}^{2}}\phantom{\rule{2em}{0ex}}\\ \le C\left({∥\nabla u∥}_{{L}^{2}}^{1/2}{∥\Delta u∥}_{{L}^{2}}^{1/2}+{∥\nabla u∥}_{{L}^{2}}{∥\nabla d∥}_{{L}^{6}}^{1/2}{∥\nabla \Delta d∥}_{{L}^{2}}^{1/2}+1+{∥\nabla d∥}_{{L}^{6}}^{1/2}{∥\nabla \Delta d∥}_{{L}^{2}}^{1/2}{∥\nabla {d}_{t}∥}_{{L}^{2}}\right)\phantom{\rule{2em}{0ex}}\\ \le C\left({∥\nabla u∥}_{{L}^{2}}^{1/2}{∥\Delta u∥}_{{L}^{2}}^{1/2}+{∥\nabla u∥}_{{L}^{2}}{∥\nabla \Delta d∥}_{{L}^{2}}^{1/2}+1+{∥\nabla \Delta d∥}_{{L}^{2}}^{1/2}{∥\nabla {d}_{t}∥}_{{L}^{2}}\right)\phantom{\rule{2em}{0ex}}\\ \le \epsilon {∥\nabla {d}_{t}∥}_{{L}^{2}}^{2}+\epsilon {∥\Delta u∥}_{{L}^{2}}^{2}+C{∥\nabla u∥}_{{L}^{2}}^{2}+C{∥\nabla u∥}_{{L}^{2}}^{4}+C+\epsilon {∥\nabla \Delta d∥}_{{L}^{2}}^{2}.\phantom{\rule{2em}{0ex}}\end{array}$
(2.8)

Here we have used the Gagliardo-Nirenberg inequality

${∥\Delta d∥}_{{L}^{3}}^{2}\le C{∥\nabla d∥}_{{L}^{6}}{∥\nabla \Delta d∥}_{{L}^{3}}.$
(2.9)

Using (1.3), (2.4), and (2.9), we have

$\begin{array}{ll}\hfill {∥\nabla \Delta d∥}_{{L}^{2}}& \le {∥\nabla {d}_{t}+\nabla \left(u\cdot \nabla d\right)-\nabla \left({\left|\nabla d\right|}^{2}d\right)∥}_{{L}^{2}}\phantom{\rule{2em}{0ex}}\\ \le C{∥\nabla {d}_{t}∥}_{{L}^{2}}+C{∥u∥}_{{L}^{6}}{∥\Delta d∥}_{{L}^{3}}+C{∥\nabla u∥}_{{L}^{3}}{∥\nabla d∥}_{{L}^{6}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+C{∥\nabla d∥}_{{L}^{6}}^{3}+C{∥\nabla d∥}_{{L}^{6}}{∥\Delta d∥}_{{L}^{3}}\phantom{\rule{2em}{0ex}}\\ \le C{∥\nabla {d}_{t}∥}_{{L}^{2}}+C{∥\nabla u∥}_{{L}^{2}}{∥\nabla \Delta d∥}_{{L}^{3}}^{1/2}+C{∥\nabla u∥}_{{L}^{2}}^{1/2}{∥\Delta u∥}_{{L}^{2}}^{1/2}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+C+C{∥\nabla \Delta d∥}_{{L}^{2}}^{1/2},\phantom{\rule{2em}{0ex}}\end{array}$

whence

${∥\nabla \Delta d∥}_{{L}^{2}}\le C{∥\nabla {d}_{t}∥}_{{L}^{2}}+C{∥\nabla u∥}_{{L}^{2}}^{2}+C{∥\nabla u∥}_{{L}^{2}}^{1/2}{∥\Delta u∥}_{{L}^{2}}^{1/2}+C.$
(2.10)

On the other hand, it follows from (1.2), (2.4), and (2.10) that

$\begin{array}{ll}\hfill {∥\Delta u∥}_{{L}^{2}}& \le C{∥\rho \stackrel{˙}{u}∥}_{{L}^{2}}+C{∥\nabla p\left(\rho \right)∥}_{{L}^{2}}+C{∥\Delta d\cdot \nabla d∥}_{{L}^{2}}\phantom{\rule{2em}{0ex}}\\ \le C{∥\rho \stackrel{˙}{u}∥}_{{L}^{2}}+C{∥\nabla \rho ∥}_{{L}^{2}}+C{∥\nabla d∥}_{{L}^{6}}{∥\Delta d∥}_{{L}^{3}}\phantom{\rule{2em}{0ex}}\\ \le C{∥\rho \stackrel{˙}{u}∥}_{{L}^{2}}+C{∥\nabla \rho ∥}_{{L}^{2}}+C{∥\Delta d∥}_{{L}^{3}}\phantom{\rule{2em}{0ex}}\\ \le C{∥\rho \stackrel{˙}{u}∥}_{{L}^{2}}+C{∥\nabla \rho ∥}_{{L}^{2}}+C{∥\Delta d∥}_{{L}^{2}}+C{∥\nabla \Delta d∥}_{{L}^{2}}\phantom{\rule{2em}{0ex}}\\ \le C{∥\rho \stackrel{˙}{u}∥}_{{L}^{2}}+C{∥\nabla \rho ∥}_{{L}^{2}}+C{∥\Delta d∥}_{{L}^{2}}+C{∥\nabla {d}_{t}∥}_{{L}^{2}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+C{∥\nabla u∥}_{{L}^{2}}^{2}+C{∥\nabla u∥}_{{L}^{2}}+\frac{1}{2}{∥\Delta u∥}_{{L}^{2}}+C,\phantom{\rule{2em}{0ex}}\end{array}$

which implies

${∥\Delta u∥}_{{L}^{2}}\le C{∥\rho \stackrel{˙}{u}∥}_{{L}^{2}}+C{∥\nabla \rho ∥}_{{L}^{2}}+C{∥\Delta d∥}_{{L}^{2}}+C{∥\nabla {d}_{t}∥}_{{L}^{2}}+C{∥\nabla u∥}_{{L}^{2}}^{2}+C.$
(2.11)

Combining (2.6), (2.7), (2.8), (2.10), and (2.11), taking ϵ small enough, using the Gron-wall inequality, we arrive at

$\nabla d\in {L}^{2}\left(0,T;{H}^{2}\right),$

whence (2.1) holds true.

This completes the proof.

## Authors' information

Both X. Chen and J. Fan are professors. J. Fan has published more than 90 scientific papers on nonlinear partial differential equations.

## References

1. Ericksen J: Liquid crystals with variable degree of orientation. Arch Rat Mech Anal 1991, 113: 97–120. 10.1007/BF00380413

2. Leslie F: Theory of flow phenomena in liquid crystals. Advances in Liquid Crystals 1979, 4: 1–81.

3. Huang T, Wang CY, Wen HY: Strong solutions of the compressible nematic liquid crystal flow. Preprint; 2011.

4. Huang T, Wang CY, Wen HY: Blow up criterion for compressible nematic liquid crystal flows in dimension three. 2011. arXiv:1104.5685v1[math.AP]

5. Fefferman C, Stein EM: Hpspaces of several variables. Acta Math 1972, 129: 137–193. 10.1007/BF02392215

6. Fa J, Ozawa T: Regularity criteria for a simplified Ericksen-Leslie system modeling the flow of liquid crystals. Discrete Contin Dyn Syst 2009, 25: 859–867.

7. Zhou Y, Fan J: A regularity criterion for the nematic liquid crystal flows. J Inequal Appl 2010, 2010: 9. Article ID 589697

## Author information

Authors

### Corresponding author

Correspondence to Xiaochun Chen.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

XC wrote the manuscript and did partial computation. JF proposed the problem and did the main estimates. All authors read and approved the final manuscript.

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Chen, X., Fan, J. A note on regularity criterion for 3D compressible nematic liquid crystal flows. J Inequal Appl 2012, 59 (2012). https://doi.org/10.1186/1029-242X-2012-59