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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])

t ρ + div ( ρ u ) = 0 ,
(1.1)
t ( ρ u ) + div ( ρ u u ) + p ( ρ ) μ Δ u ( λ + μ div u = Δ d d ,
(1.2)
t d + u d = Δ d + d 2 d ,
(1.3)
( ρ , u , d ) ( x , 0 ) = ( ρ 0 , u 0 , d 0 ) ( x ) , d 0 = 1 , x 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:

μ > 0 , 3 λ + 2 μ 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

- μ Δ u 0 - ( λ + μ ) div u 0 - p ( ρ 0 ) - Δ d 0 d 0 = ρ 0 g f o r s o m e g L 2 ( 3 )
(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:

0 T D ( u ) L + d L 2 d t < ,
(1.6)

where D ( u ) := 1 2 u + t u .

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

0 T D ( u ) L + d B M O 2 d t < ,
(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 L p C u B M O 1 - q p u L q q p ( 1 < q < p < ) .
(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

d L 2 ( 0 , T ; L ) .
(2.1)

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

ρ L ( 0 , T ; L ) C , ρ u 2 + d 2 d x + 0 T u 2 + Δ d + d 2 d 2 d x d t C .
(2.2)

Using (1.8), we see that

0 T d 4 d x d t C 0 T d B M O 2 d L 2 2 d t max t d L 2 2 0 T d B M O 2 d t C ,

from which and (2.2), we get

0 T Δ d 2 d x d t 2 0 T Δ d + d 2 d 2 d x d t + 2 0 T d 4 d x d t C .
(2.3)

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

d d t d r d x + r d r - 2 2 d 2 + ( r - 2 ) d r - 2 d 2 d x = r d 2 d d r - 2 d d x - r ( u d ) d r - 2 d d x = r d r + 2 d x - r d r - 2 i u j < j d , i , d > d x - u d r d x = r d r + 2 d x - r d r - 2 D ( u ) : d d d x + ( div u ) d r d x C d B M O 2 d L r r + C D ( u ) L d L r r ,

which yields

sup 0 t < T d L r + 0 T d r - 2 2 d 2 d x d t C .
(2.4)

Let

: = f t + u f

denotes the material derivative of f.

Testing (1.2) by u ˙ , we derive

1 2 d d t μ u 2 + ( λ + μ ) div u 2 d x + ρ u ˙ 2 d x = < ( u ) u , - μ Δ u - ( λ + μ ) div u > d x - u u p ( ρ ) d x - u t p ( ρ ) d x - u u < Δ d , d > d x - u t < Δ d , d > d x = : I 1 + I 2 + I 3 + I 4 + I 5 .
(2.5)

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

I 1 = μ D ( u ) : curl u curl u - 1 2 div u ( curl u ) 2 d x - ( 2 μ + λ ) ( u : t u ) div u - 1 2 ( div u ) 3 d x C D ( u ) L u L 2 2 , I 2 = p ( ρ ) ( u : t u - ( div u ) 2 ) d x - u div u p ( ρ ) d x C u L 2 2 + C ρ u div u d x C u L 2 2 + C u L 6 ρ L 2 div u L 3 C u L 2 2 + C u L 2 ρ L 2 D ( u ) L 1 3 D ( u ) L 2 2 3 C u L 2 2 + C u L 2 2 ρ L 2 2 + C D ( u ) L 2 3 D ( u ) L 2 4 3 C u L 2 2 + C u L 2 2 ρ L 2 2 + C 1 + D ( u ) L u L 2 2 , I 3 = d d t p ( ρ ) div u d x - div u t p ( ρ ) d x d d t p ( ρ ) div u d x + C + C 1 + D ( u ) L u L 2 2 + C u L 2 2 ρ L 2 2 , I 4 u L 6 u L 6 Δ d L 2 d L 6 C u L 2 Δ u L 2 Δ d L 2 ( b y ( 2 . 4 ) f o r r = 6 ) ε Δ u L 2 2 + C u L 2 2 Δ d L 2 2 ,

for any ϵ > 0.

We denote M ( d ) : = d d - 1 2 d 2 I 3 , I5 is estimated as follows, which is slightly different from that in [4]:

I 5 = d d t M ( d ) : u d x - t M ( d ) : u d x d d t M ( d ) : u d x + C d t d u d x d d t M ( d ) : u d x + C d L 6 u L 3 d t L 2 d d t M ( d ) : u d x + C u L 3 d t L 2 ( b y ( 2 . 4 ) f o r r = 6 ) d d t M ( d ) : u d x + ε d t L 2 2 + ε Δ u L 2 2 + C u L 2 2 .

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

1 2 d d t μ + u 2 ( λ + μ ) div u 2 d x + ρ u ˙ 2 d x C u L 2 2 ρ L 2 2 + C 1 + D ( u ) L u L 2 2 + C + d d t p ( ρ ) div u + M ( d ) : u d x + C u L 2 2 Δ d L 2 2 + 2 ε d t L 2 2 + Δ u L 2 2
(2.6)

for any 0 < ϵ < 1.

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

d d t ρ L 2 2 C 1 + D ( u ) L ρ L 2 2 + ε Δ u L 2 2 .
(2.7)

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

1 2 d d t Δ d 2 d x + d t 2 d x = u d - d 2 d Δ d t d x = - u d - d 2 d d t d x u L 3 d L 6 + u L 6 Δ d L 3 + d L 6 3 + d L 6 Δ d L 3 d t L 2 C u L 3 + u L 2 Δ d L 3 + 1 + Δ d L 3 d t L 2 C u L 2 1 / 2 Δ u L 2 1 / 2 + u L 2 d L 6 1 / 2 Δ d L 2 1 / 2 + 1 + d L 6 1 / 2 Δ d L 2 1 / 2 d t L 2 C u L 2 1 / 2 Δ u L 2 1 / 2 + u L 2 Δ d L 2 1 / 2 + 1 + Δ d L 2 1 / 2 d t L 2 ε d t L 2 2 + ε Δ u L 2 2 + C u L 2 2 + C u L 2 4 + C + ε Δ d L 2 2 .
(2.8)

Here we have used the Gagliardo-Nirenberg inequality

Δ d L 3 2 C d L 6 Δ d L 3 .
(2.9)

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

Δ d L 2 d t + ( u d ) - d 2 d L 2 C d t L 2 + C u L 6 Δ d L 3 + C u L 3 d L 6 + C d L 6 3 + C d L 6 Δ d L 3 C d t L 2 + C u L 2 Δ d L 3 1 / 2 + C u L 2 1 / 2 Δ u L 2 1 / 2 + C + C Δ d L 2 1 / 2 ,

whence

Δ d L 2 C d t L 2 + C u L 2 2 + C u L 2 1 / 2 Δ u L 2 1 / 2 + C .
(2.10)

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

Δ u L 2 C ρ u ˙ L 2 + C p ( ρ ) L 2 + C Δ d d L 2 C ρ u ˙ L 2 + C ρ L 2 + C d L 6 Δ d L 3 C ρ u ˙ L 2 + C ρ L 2 + C Δ d L 3 C ρ u ˙ L 2 + C ρ L 2 + C Δ d L 2 + C Δ d L 2 C ρ u ˙ L 2 + C ρ L 2 + C Δ d L 2 + C d t L 2 + C u L 2 2 + C u L 2 + 1 2 Δ u L 2 + C ,

which implies

Δ u L 2 C ρ u ˙ L 2 + C ρ L 2 + C Δ d L 2 + C d t L 2 + C 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

d L 2 ( 0 , T ; H 2 ) ,

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

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Acknowledgements

The authors would like to thank the referees for careful reading and helpful comments. This article is supported by NSFC (No. 11171154).

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Correspondence to Xiaochun Chen.

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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

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Keywords

  • blow-up criterion
  • compressible
  • nematic
  • liquid crystals