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)

$${\partial }_t(\rho u)+\text{div}(\rho u\otimes u)+\nabla p(\rho )-\mu \Delta u-(\lambda +\mu \nabla \text{div}u=-\Delta d\cdot \nabla d,$$

(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) := aρ^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:

$$\mu >0,3\lambda +2\mu \ge 0.$$

(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 ∈ W^1,q∩ H¹ ∩ L¹ for some q ∈ (3, 6] and ρ₀ ≥ 0 in ℝ³, ∇u₀ ∈ H¹, ∇d₀ ∈ H² and |d₀| = 1 in ℝ³. If, in additions, the following compatibility condition

$$-\mu \Delta u_0-\left(\lambda +\mu \right)\nabla \text{div}{u}_0-\nabla p\left({\rho }_0\right)-\Delta d_0\cdot d_0=\sqrt{{\rho }_0}gforsomeg\in L^2\left({ℝ}^3\right)$$

(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]:

$${∥u∥}_{Lp}\le C{∥u∥}_{BMO}^{1-\frac{q}{p}}{∥u∥}_{L^q}^{\frac{q}{p}}\left(1<q<p<\infty \right).$$

(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

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

(2.2)

Using (1.8), we see that

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

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-2∇d (r ≥ 2), using (1.8), we infer that

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

which yields

$$\underset{0\le t<T}{\text{sup}}{∥\nabla d∥}_{L^r}+\underset{0}{\overset{T}{\int }}\int {\left|\nabla d\right|}^{r-2}{\left|{\nabla }^{2}d\right|}^{2}dxdt\le C.$$

(2.4)

Let

$$ḟ:=f_t+u\cdot \nabla f$$

denotes the material derivative of f.

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

$$\begin{array}{l}\frac{1}{2}\frac{d}{dt}\int \mu {\left|\nabla u\right|}^2+\left(\lambda +\mu \right){\left|\text{div}u\right|}^{2}dx+\int {\rho \left|\dot{u}\right|}^{2}dx\\ =\int <\left(u\cdot \nabla \right)u,-\mu \Delta u-\left(\lambda +\mu \right)\nabla \text{div}u>dx\\ -\int u\cdot \nabla u\cdot \nabla p\left(\rho \right)dx-\int u_t\cdot \nabla p\left(\rho \right)dx\\ -\int u\cdot \nabla u\cdot <\Delta d,\nabla d>dx-\int u_t<\Delta d,\nabla d>dx\\ =:I_1+I_2+I_3+I_4+I_5.\end{array}$$

(2.5)

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

$$\begin{array}{ll}\hfill I_1& =\mu \int \mathcal{D}\left(u\right):\text{curl}u\otimes \text{curl}u-\frac{1}{2}\text{div}u{\left(\text{curl}u\right)}^{2}dx\\ -\left(2\mu +\lambda \right)\int \left(\nabla u{:}^t\nabla u\right)\text{div}u-\frac{1}{2}{\left(\text{div}u\right)}^{3}dx\\ \le C{∥\mathcal{D}\left(u\right)∥}_{L^{\infty }}{∥\nabla u∥}_{L^2}^2,\\ \hfill I_2& =\int p\left(\rho \right)\left(\nabla u{:}^t\nabla u-{\left(\text{div}u\right)}^2\right)dx-\int u\text{div}u\cdot \nabla p\left(\rho \right)dx\\ \le C{∥\nabla u∥}_{L^2}^2+C\int \left|\nabla \rho ∥u∥\text{div}u\right|dx\\ \le C{∥\nabla u∥}_{L^2}^2+C{∥u∥}_{L^6}{∥\nabla \rho ∥}_{L^2}{∥\text{div}u∥}_{L^3}\\ \le C{∥\nabla u∥}_{L^2}^2+C{∥\nabla u∥}_{L^2}{∥\nabla \rho ∥}_{L^2}{∥\mathcal{D}\left(u\right)∥}_{L^{\infty }}^{\frac{1}{3}}{∥\mathcal{D}\left(u\right)∥}_{L^2}^{\frac{2}{3}}\\ \le C{∥\nabla u∥}_{L^2}^2+C{∥\nabla u∥}_{L^2}^2{∥\nabla \rho ∥}_{L^2}^2+C{∥\mathcal{D}\left(u\right)∥}_{L^{\infty }}^{\frac{2}{3}}{∥\mathcal{D}\left(u\right)∥}_{L^2}^{\frac{4}{3}}\\ \le C{∥\nabla u∥}_{L^2}^2+C{∥\nabla u∥}_{L^2}^2{∥\nabla \rho ∥}_{L^2}^2+C\left(1+{∥\mathcal{D}\left(u\right)∥}_{L^{\infty }}\right){∥\nabla u∥}_{L^2}^2,\\ \hfill I_3& =\frac{d}{dt}\int p\left(\rho \right)\text{div}udx-\int \text{div}u{\partial }_{t}p\left(\rho \right)dx\\ \le \frac{d}{dt}\int p\left(\rho \right)\text{div}udx+C+C\left(1+{∥\mathcal{D}\left(u\right)∥}_{L^{\infty }}\right){∥\nabla u∥}_{L^2}^2\\ +C{∥\nabla u∥}_{L^2}^2{∥\nabla \rho ∥}_{L^2}^2,\\ \hfill I_4& \le {∥u∥}_{L^6}{∥\nabla u∥}_{L^6}{∥\Delta d∥}_{L^2}{∥\nabla d∥}_{L^6}\\ \le C{∥\nabla u∥}_{L^2}{∥\Delta u∥}_{L^2}{∥\Delta d∥}_{L^2}\left(by\left(2.4\right)forr=6\right)\\ \le \epsilon {∥\Delta u∥}_{L^2}^2+C{∥\nabla u∥}_{L^2}^2{∥\Delta d∥}_{L^2}^2,\end{array}$$

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]:

$$\begin{array}{ll}\hfill I_5& =\frac{d}{dt}\int M\left(d\right):\nabla udx-\int {\partial }_{t}M\left(d\right):\nabla udx\\ \le \frac{d}{dt}\int M\left(d\right):\nabla udx+C\int \left|\nabla d_t∥\nabla d∥\nabla u\right|dx\\ \le \frac{d}{dt}\int M\left(d\right):\nabla udx+C{∥\nabla d∥}_{L^6}{∥\nabla u∥}_{L^3}{∥\nabla d_t∥}_{L^2}\\ \le \frac{d}{dt}\int M\left(d\right):\nabla udx+C{∥\nabla u∥}_{L^3}{∥\nabla d_t∥}_{L^2}\left(by\left(2.4\right)forr=6\right)\\ \le \frac{d}{dt}\int M\left(d\right):\nabla udx+\epsilon {∥\nabla d_t∥}_{L^2}^2+\epsilon {∥\Delta u∥}_{L^2}^2+C{∥\nabla u∥}_{L^2}^2.\end{array}$$

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

$$\begin{array}{l}\frac{1}{2}\frac{d}{dt}\int \mu +{\left|\nabla u\right|}^2\left(\lambda +\mu \right){\left|\text{div}u\right|}^{2}dx+\int \rho {\left|\dot{u}\right|}^{2}dx\\ \le C{∥\nabla u∥}_{L^2}^2{∥\nabla \rho ∥}_{L^2}^2+C\left(1+{∥\mathcal{D}\left(u\right)∥}_{L^{\infty }}\right){∥\nabla u∥}_{L^2}^2\\ +C+\frac{d}{dt}\int p\left(\rho \right)\text{div}u+M\left(d\right):\nabla udx\\ +C{∥\nabla u∥}_{L^2}^2{∥\Delta d∥}_{L^2}^2+2\epsilon \left({∥\nabla d_t∥}_{L^2}^2+{∥\Delta u∥}_{L^2}^2\right)\end{array}$$

(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

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

(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

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

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

$$\begin{array}{ll}\hfill {∥\Delta u∥}_{L^2}& \le C{∥\rho \dot{u}∥}_{L^2}+C{∥\nabla p\left(\rho \right)∥}_{L^2}+C{∥\Delta d\cdot \nabla d∥}_{L^2}\\ \le C{∥\rho \dot{u}∥}_{L^2}+C{∥\nabla \rho ∥}_{L^2}+C{∥\nabla d∥}_{L^6}{∥\Delta d∥}_{L^3}\\ \le C{∥\rho \dot{u}∥}_{L^2}+C{∥\nabla \rho ∥}_{L^2}+C{∥\Delta d∥}_{L^3}\\ \le C{∥\rho \dot{u}∥}_{L^2}+C{∥\nabla \rho ∥}_{L^2}+C{∥\Delta d∥}_{L^2}+C{∥\nabla \Delta d∥}_{L^2}\\ \le C{∥\rho \dot{u}∥}_{L^2}+C{∥\nabla \rho ∥}_{L^2}+C{∥\Delta d∥}_{L^2}+C{∥\nabla d_t∥}_{L^2}\\ +C{∥\nabla u∥}_{L^2}^2+C{∥\nabla u∥}_{L^2}+\frac{1}{2}{∥\Delta u∥}_{L^2}+C,\end{array}$$

which implies

$${∥\Delta u∥}_{L^2}\le C{∥\rho \dot{u}∥}_{L^2}+C{∥\nabla \rho ∥}_{L^2}+C{∥\Delta d∥}_{L^2}+C{∥\nabla d_t∥}_{L^2}+C{∥\nabla u∥}_{L^2}^2+C.$$

(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.