About this Journal Submit a Manuscript Table of Contents
ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 935045, 10 pages
http://dx.doi.org/10.5402/2012/935045
Research Article

Regularity Criterion for the 3D Nematic Liquid Crystal Flows

1Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
2Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

Received 17 January 2012; Accepted 14 February 2012

Academic Editors: A. Carpio, Y. Liu, and G. A. Seregin

Copyright © 2012 Jishan Fan and Tohru Ozawa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the hydrodynamic theory of liquid crystals. We prove a logarithmically improved regularity criterion for two simplified Ericksen-Leslie systems.

1. Introduction

The hydrodynamic theory of liquid crystals was established by Ericksen and Leslie [14]. However, since the equations are too complicated, we consider the first simplified Ericksen-Leslie system: 𝑢𝑡𝑑+𝑢𝑢+𝜋Δ𝑢=(𝑑𝑑),(1.1)𝑡𝑢+𝑢𝑑=Δ𝑑𝑓(𝑑),(1.2)div𝑢=0,(1.3)(𝑢,𝑑)(𝑥,0)=0,𝑑0(𝑥)in3,(1.4) which include the velocity vector 𝑢=(𝑢1,𝑢2,𝑢3)𝑡, the scalar pressure 𝜋 being and the direction vector 𝑑=(𝑑1,𝑑2,𝑑3)𝑡. 𝑓(𝑑)=1/𝜂(|𝑑|21)𝑑 with 𝜂, a positive constant. (𝑑𝑑)𝑖,𝑗=𝑘𝜕𝑖𝑑𝑘𝜕𝑗𝑑𝑘, and hence (𝑑𝑑)=𝑘Δ𝑑𝑘𝑑𝑘+(1/2)|𝑑|2.

Lin-Liu [5] proved that the system (1.1)–(1.4) has a unique smooth solution globally in 2 space dimensions and locally in 3 dimensions. They also proved the global existence of weak solutions. However, the regularity of solutions to the system is still open. Fan-Guo [6] and Fan-Ozawa [7] showed the following regularity criteria: 𝑢𝐿𝑟0,𝑇;𝐿𝑠32for𝑟+3𝑠=1,3<𝑠,𝑢𝐿𝑟0,𝑇;𝐿𝑠32for𝑟+3𝑠3=2,2<𝑠,𝑢𝐿2̇𝐵0,𝑇;0,,𝜔=curl𝑢𝐿1̇𝐵0,𝑇;0,,(1.5) where ̇𝐵0, denotes the homogeneous Besov space.

The first aim of this paper is to prove a new regularity criterion as follows.

Theorem 1.1. Let 𝑢0𝐻3,𝑑0𝐻4 with div𝑢0=0 in 3. Let (𝑢,𝑑) be a smooth solution to the problem (1.1)–(1.4) on [0,𝑇). If 𝑢 satisfies 𝑇0𝑢(,𝑡)̇𝐵2/(1𝑠)𝑠,1+log𝑒+𝑢(,𝑡)̇𝐵𝑠,𝑑𝑡<(1.6) for some 𝑠 with 0<𝑠<1, then the solution (𝑢,𝑑) can be extended beyond 𝑇>0.

When the penalization parameter 𝜂0, (1.1)–(1.4) reduce to 𝑢𝑡𝑑+𝑢𝑢+𝜋Δ𝑢=(𝑑𝑑),(1.7)div𝑢=0,(1.8)𝑡||||+𝑢𝑑=Δ𝑑+𝑑2||𝑑||𝑢𝑑,=1,(1.9)(𝑢,𝑑)(𝑥,0)=0,𝑑0(𝑥)in3,||𝑑0||=1.(1.10)

When 𝑢=0, then (1.9) is the well-known harmonic heat flow equation onto a sphere.

Fan-Gao-Guo [8] proved the following blow-up criteria:𝑢,𝑑𝐿2̇𝐵0,𝑇;0,,𝜔,Δ𝑑𝐿1̇𝐵0,𝑇;0,.(1.11)

We will prove the folowing theorem

Theorem 1.2. Let 𝑢0,𝑑0𝐻3(3) with div𝑢0=0,|𝑑0|=1 in 3. Let (𝑢,𝑑) be a smooth solution to the problem (1.7)–(1.10) on [0,𝑇). If the following condition is satisfied: 𝑇0𝑢(,𝑡)̇𝐵2/(1𝑠)𝑠,+𝑑(,𝑡)2̇𝐵0,1+log𝑒+𝑢(,𝑡)̇𝐵𝑠,+𝑑(,𝑡)̇𝐵0,𝑑𝑡<,(1.12) for some 𝑠 with 0<𝑠<1, then the solution (𝑢,𝑑) can be extended beyond 𝑇>0.

2. Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1. Since it is well-known that there are 𝑇0>0 and a unique smooth solution (𝑢,𝑑) to the problem (1.1)–(1.4) in (0,𝑇0], we only need to show a priori estimates.

Testing (1.1) by 𝑢 and using (1.3), we see that 12𝑑𝑢𝑑𝑡2||||𝑑𝑥+𝑢2(𝑑𝑥=𝑢)𝑑Δ𝑑𝑑𝑥.(2.1)

Testing (1.2) by Δ𝑑𝑓(𝑑) and using (1.3), we find that 𝑑1𝑑𝑡2||||𝑑2+1||𝑑||4𝜂212||||𝑑𝑥+Δ𝑑𝑓(𝑑)2(𝑑𝑥=𝑢)𝑑Δ𝑑𝑑𝑥.(2.2)

Summing up (2.1) and (2.2), we infer that 12𝑢2+12||||𝑑2+1||𝑑||4𝜂212𝑑𝑥+𝑇0||||𝑢2+||||Δ𝑑𝑓(𝑑)21𝑑𝑥2𝑢20+12||𝑑0||2+1||𝑑4𝜂0||212𝑑𝑥.(2.3)

Testing (1.2) by 𝑑 and using (1.3), we deduce that 12𝑑𝑑𝑑𝑡2||||𝑑𝑥+𝑑21𝑑𝑥+𝜂||𝑑||41𝑑𝑥=𝜂𝑑2𝑑𝑥,(2.4) which yields 𝑑𝐿(0,𝑇;𝐿2)+𝑑𝐿2(0,𝑇;𝐻1)𝐶.(2.5)

Next, we prove the following estimate: 𝑑𝐿(0,𝑇;𝐿)𝑑max0𝐿,1.(2.6)

Without loss of generality, we assume that 1𝑑0𝐿. Multiplying (1.2) by 𝑑, we get 𝜙𝑡||𝑑||+𝑢𝜙Δ𝜙+222𝜙=𝜂𝑑02𝐿||𝑑||12||||2𝑑20(2.7) with 𝜙=|𝑑|2𝑑02𝐿 and 𝜙|𝑡=0=|𝑑0|2𝑑02𝐿0. Then (2.6) follows immediately from 𝜙0 by the maximum principle.

Testing (1.1) by Δ𝑢 and using (1.3), we see that 12𝑑||||𝑑𝑡𝑢2||||𝑑𝑥+Δ𝑢2(𝑑𝑥=𝑢)𝑢Δ𝑢𝑑𝑥𝑖,𝑘Δ𝑑𝑘𝜕𝑖𝑑𝑘𝑢𝑖𝑑𝑥𝑖,𝑘𝜕𝑖𝑑𝑘Δ𝑑𝑘𝑢𝑖𝑑𝑥.(2.8)

Applying Δ to (1.2), testing by Δ𝑑, and using (1.3), we find that 12𝑑||||𝑑𝑡Δ𝑑2||||𝑑𝑥+Δ𝑑2𝑑𝑥=𝑖,𝑘𝜕𝑖𝑑𝑘Δ𝑑𝑘𝑢𝑖𝑑𝑥𝑖,𝑘𝜕𝑖𝜕𝑗𝑑𝑘𝜕𝑗𝑑𝑘𝑢𝑖𝑑𝑥Δ𝑓(𝑑)Δ𝑑𝑑𝑥.(2.9)

Summing up (2.8) and (2.9), we get 12𝑑||||𝑑𝑡𝑢2+||||Δ𝑑2||||𝑑𝑥+Δ𝑢2+||||Δ𝑑2=𝑑𝑥(𝑢)𝑢Δ𝑢𝑑𝑥𝑖,𝑘Δ𝑑𝑘𝜕𝑖𝑑𝑘𝑢𝑖𝑑𝑥𝑖,𝑘𝜕𝑖𝜕𝑗𝑑𝑘𝜕𝑗𝑑𝑘𝑢𝑖𝑑𝑥Δ𝑓(𝑑)Δ𝑑𝑑𝑥=𝐼1+𝐼2+𝐼3+𝐼4.(2.10)

By using (2.6), 𝐼4 is simply bounded as 𝐼4𝐶Δ𝑑2𝐿2.(2.11)

By using the inequalities [9] 𝑢𝑢𝐿2𝐶𝑢̇𝐵𝑠,𝑢̇𝐵𝑠2,2,𝑢̇𝐵𝑠2,2𝐶𝑢𝐿1𝑠2Δ𝑢𝑠𝐿2.(2.12)

𝐼1 can be bounded as follows: 𝐼1𝑢𝑢𝐿2Δ𝑢𝐿2𝐶𝑢̇𝐵𝑠,𝑢̇𝐵1+𝑠2,2Δ𝑢𝐿2𝐶𝑢̇𝐵𝑠,𝑢𝐿1𝑠2Δ𝑢𝐿1+𝑠212Δ𝑢2𝐿2+𝐶𝑢̇𝐵2/(1𝑠)𝑠,𝑢2𝐿2.(2.13)

We bound 𝐼2 and 𝐼3 as follows: 𝐼2,𝐼3𝐶𝑢𝐿2Δ𝑑2𝐿4𝐶𝑢𝐿2Δ𝑑𝐿214Δ𝑑2𝐿2+𝐶𝑢2𝐿2.(2.14)

Here we used the Gagliardo-Nirenberg inequality Δ𝑑2𝐿4𝐶𝑑𝐿Δ𝑑𝐿2.(2.15) Inserting the above estimates into (2.10), we derive 𝑑||||𝑑𝑡𝑢2+||||Δ𝑑2||||𝑑𝑥+Δ𝑢2+||||Δ𝑑2𝑑𝑥𝐶𝑢̇𝐵2/(1𝑠)𝑠,+1𝑢2𝐿2+Δ𝑑2𝐿2𝐶1+𝑢̇𝐵2/(1𝑠)𝑠,1+log𝑒+𝑢̇𝐵𝑠,1+log𝑒+𝑢̇𝐵𝑠,𝑢2𝐿2+Δ𝑑2𝐿2𝐶1+𝑢̇𝐵2/(1𝑠)𝑠,1+log𝑒+𝑢̇𝐵𝑠,(1+log(𝑒+𝑦))𝑢2𝐿2+Δ𝑑2𝐿2.(2.16)

Due to (1.6), one concludes that for any small constant 𝜖>0, there exists 𝑇<𝑇 such that 𝑇𝑇𝑢̇𝐵2/(1𝑠)𝑠,1+log𝑒+𝑢̇𝐵𝑠,𝑑𝑡<𝜖.(2.17)

For any 𝑇<𝑡𝑇, we set 𝑦(𝑡)=sup𝑇𝜏𝑡Λ3(𝑢,𝑑)(,𝜏)𝐿2withΛ=(Δ)1/2.(2.18)

Applying Gronwall’s inequality to (2.16) in the interval [𝑇,𝑡], one has 𝑢(,𝑡)2𝐿2+Δ𝑑(,𝑡)2𝐿2𝐶(1+𝑦)𝐶0𝜖.(2.19)

Now, we derive a bound on 𝑦(𝑡) defined by (2.18). To this end, we will use the following commutator and product estimates due to Kato-Ponce [10]: Λ𝛼(𝑓𝑔)𝑓Λ𝛼𝑔𝐿𝑝𝐶𝑓𝐿𝑝1Λ𝛼1𝑔𝐿𝑞1+Λ𝛼𝑓𝐿𝑝2𝑔𝐿𝑞2,(2.20)Λ𝛼(𝑓𝑔)𝐿𝑝𝐶𝑓𝐿𝑝1Λ𝛼𝑔𝐿𝑞1+Λ𝛼𝑓𝐿𝑝2𝑔𝐿𝑞2,(2.21) with 𝛼>0 and 1/𝑝=1/𝑝1+1/𝑞1=1/𝑝2+1/𝑞2.

Applying Λ3 to (1.1), testing by Λ3𝑢, and using (1.3), (2.20), (2.21) and (2.19), we obtain 12𝑑||Λ𝑑𝑡3𝑢||2||Λ𝑑𝑥+4𝑢||2Λ𝑑𝑥=3(𝑢𝑢)𝑢Λ3𝑢Λ3Λ𝑢𝑑𝑥+3(𝑑𝑑)Λ3𝑢𝑑𝑥𝐶𝑢𝐿3Λ3𝑢2𝐿3+𝐶𝑑𝐿Λ4𝑑𝐿2Λ4𝑢𝐿2𝐶𝑢𝐿3/42Λ3𝑢𝐿1/42𝑢𝐿1/32Λ4𝑢𝐿5/32+1Λ164𝑢2𝐿2+𝐶𝑑2𝐿Λ4𝑑2𝐿214Λ4𝑢2𝐿2+𝐶𝑢𝐿13/22Λ3𝑢𝐿3/22+𝐶Δ𝑑𝐿3/22Λ4𝑑𝐿1/22Δ𝑑𝐿2/32Λ5𝑑𝐿4/3214Λ4𝑢2𝐿2+14Λ5𝑑2𝐿2+𝐶𝑢𝐿13/22Λ3𝑢𝐿3/22+𝐶Δ𝑑𝐿13/22Λ4𝑑𝐿3/22.(2.22) Here we have used the following Gagliardo-Nirenberg inequalities: 𝑢𝐿3𝐶𝑢𝐿3/42Λ3𝑢𝐿1/42,Λ3𝑢𝐿3𝐶𝑢𝐿1/62Λ4𝑢𝐿5/62,𝑑𝐿𝐶Δ𝑑𝐿3/42Λ4𝑑𝐿1/42,Λ4𝑑𝐿2𝐶Δ𝑑𝐿1/32Λ5𝑑𝐿2/32.(2.23)

Taking Λ4 to (1.2), testing by Λ4𝑑, and using (1.3), (2.20), (2.23), and (2.6), we have 12𝑑||Λ𝑑𝑡4𝑑||2||Λ𝑑𝑥+5𝑑||2Λ𝑑𝑥=4(𝑢𝑑)𝑢Λ4𝑑Λ4Λ𝑑𝑑𝑥4𝑓(𝑑)Λ4𝑑𝑑𝑥𝐶𝑢𝐿3Λ4𝑑𝐿6Λ4𝑑𝐿2+𝐶𝑑𝐿Λ4𝑢𝐿2Λ4𝑑𝐿2Λ+𝐶4𝑑2𝐿2𝐶𝑢𝐿3/42Λ3𝑢𝐿1/42Δ𝑑𝐿1/32Λ5𝑑𝐿5/32+14Λ4𝑢2𝐿2+𝐶𝑑2𝐿Λ4𝑑2𝐿2Λ+𝐶4𝑑2𝐿214Λ4𝑢2𝐿2+14Λ5𝑑2𝐿2+𝐶𝑢𝐿9/22Δ𝑑2𝐿2Λ3𝑢𝐿3/22+𝐶Δ𝑑𝐿13/22Λ4𝑑𝐿3/22Λ+𝐶4𝑑2𝐿2.(2.24)

Summing up (2.22) and (2.24) and taking 𝜖 small enough, we arrive at 𝑢𝐿(0,𝑇;𝐻3)+𝑢𝐿2(0,𝑇;𝐻4)𝐶,𝑑𝐿(0,𝑇;𝐻4)+𝑑𝐿2(0,𝑇;𝐻5)𝐶.(2.25)

This completes the proof.

3. Proof of Theorem 1.2

In this section, we will prove Theorem 1.2. Since it is easy to prove that there are 𝑇0>0 and a unique smooth solution (𝑢,𝑝,𝑑) to the problem (1.7)–(1.10) in [0,𝑇0], we only need to prove a priori estimates.

First, as in the previous section, we still have (2.1).

Testing (1.9) by Δ𝑑, using 𝑑Δ𝑑=|𝑑|2 and |𝑑|=1, we see that 12𝑑||||𝑑𝑡𝑑2||||𝑑𝑥+Δ𝑑2=𝑑𝑥(𝑢𝑑)Δ𝑑𝑑𝑥+(𝑑Δ𝑑)2||||𝑑𝑥(𝑢𝑑)Δ𝑑𝑑𝑥+Δ𝑑2𝑑𝑥.(3.1)

Summing up (2.1) and (3.1), we find that 12𝑢2+12||||𝑑2𝑑𝑥+𝑇0||||𝑢21𝑑𝑥𝑑𝑡2𝑢20+||𝑑0||2𝑑𝑥.(3.2)

Similarly to (2.10), we have 12𝑑||||𝑑𝑡𝑢2+||||Δ𝑑2||||𝑑𝑥+Δ𝑢2+||||Δ𝑑2𝑑𝑥=𝐼1+𝐼2+𝐼3+𝐼5.(3.3) Here 𝐼1,𝐼2, and 𝐼3 are the same as that in (2.10) and can be bounded as in the previous section. The corresponding last term 𝐼5 is written and bounded as 𝐼5=𝑘𝜕𝑘||||𝑑2𝑑𝜕𝑘Δ𝑑𝑑𝑥=𝑘𝜕𝑘||||𝑑𝑑2𝜕𝑘Δ𝑑𝑑𝑥𝑘𝑑𝜕𝑘||||𝑑2𝜕𝑘=Δ𝑑𝑑𝑥𝑘𝜕𝑘𝜕𝑘||||𝑑𝑑2Δ𝑑𝑑𝑥𝑘𝑑𝜕𝑘||||𝑑2𝜕𝑘Δ𝑑𝑑𝑥𝐶𝑑2𝐿4Δ𝑑2𝐿4+14Δ𝑑2𝐿2𝐶Δ𝑑𝐿2𝑑̇𝐵0,Δ𝑑𝐿2+14Δ𝑑2𝐿212Δ𝑑2𝐿2+𝐶𝑑2̇𝐵0,Δ𝑑2𝐿2.(3.4) Here we have used the following inequality [11, 12]: Δ𝑑2𝐿4𝐶𝑑̇𝐵0,Δ𝑑𝐿2(3.5) and the Gagliardo-Nirenberg inequality 𝑑2𝐿4𝐶𝑑𝐿Δ𝑑𝐿2.(3.6)

Substituting the above estimates into (3.3), we obtain 𝑑||||𝑑𝑡𝑢2+||||Δ𝑑2𝑑𝑥𝐶𝑢̇𝐵2/(1𝑠)𝑠,+𝑑2̇𝐵0,𝑢2𝐿2+Δ𝑑2𝐿2𝐶𝑢̇𝐵2/(1𝑠)𝑠,+𝑑2̇𝐵0,1+log𝑒+𝑢̇𝐵𝑠,+𝑑̇𝐵0,(1+log(𝑒+𝑦))𝑢2𝐿2+Δ𝑑2𝐿2.(3.7)

Due to (1.12), one concludes that for any small constant 𝜖>0, there exists 𝑇<𝑇 such that 𝑇𝑇𝑢̇𝐵2/(1𝑠)𝑠,+𝑑2̇𝐵0,1+log𝑒+𝑢̇𝐵𝑠,+𝑑̇𝐵0,𝑑𝑡<𝜖.(3.8)

Applying Gronwall’s inequality to (3.7) in the interval [𝑇,𝑡], one has (2.19).

As in the previous section, we still have (2.22).

Similarly to (2.24), we obtain 12𝑑||Λ𝑑𝑡4𝑑||2||Λ𝑑𝑥+5𝑑||2Λ𝑑𝑥=4(𝑢𝑑)𝑢Λ4𝑑Λ4Λ𝑑𝑑𝑥+4𝑑||||𝑑2Λ4𝑑𝑑𝑥=𝐽1+𝐽2.(3.9)𝐽1 is bounded as that in (2.24); 𝐽2𝐶𝑑𝐿Λ4𝑑2𝐿2Λ+𝐶4||||𝑑2𝐿2Λ4𝑑𝐿2,𝐶𝑑2𝐿Λ4𝑑2𝐿2+𝐶𝑑𝐿Λ5𝑑𝐿2Λ4𝑑𝐿2,14Λ5𝑑2𝐿2+𝐶𝑑2𝐿Λ4𝑑2𝐿2,(3.10) then 𝐽2 can be bounded as that in (2.24).

Combining (2.22) and (3.9) and taking 𝜖 small enough, we conclude that 𝑢𝐿(0,𝑇;𝐻3)+𝑢𝐿2(0,𝑇;𝐻4)𝐶,𝑑𝐿(0,𝑇;𝐻3)+𝑑𝐿2(0,𝑇;𝐻4)𝐶.(3.11) This completes the proof.

Acknowledgment

The paper is supported by NSFC (no. 11171154).

References

  1. J. L. Ericksen, “Conservation laws for liquid crystals,” Transactions of The Society of Rheology, vol. 5, pp. 23–34, 1961. View at Publisher · View at Google Scholar
  2. J. Ericksen, “Continuum theory of nematic liquid crystals,” Res Mechanica, vol. 21, pp. 381–392, 1987.
  3. J. L. Ericksen, “Liquid crystals with variable degree of orientation,” Archive for Rational Mechanics and Analysis, vol. 113, no. 2, pp. 97–120, 1990. View at Publisher · View at Google Scholar
  4. F. Leslie, Theory of Flow Phenomenum in Liquid Crystals, vol. 4, Springer, New York, NY, USA, 1979.
  5. F.-H. Lin and C. Liu, “Nonparabolic dissipative systems modeling the flow of liquid crystals,” Communications on Pure and Applied Mathematics, vol. 48, no. 5, pp. 501–537, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J. Fan and B. Guo, “Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in 3,” Science in China A, vol. 51, no. 10, pp. 1787–1797, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. J. Fan and T. Ozawa, “Regularity criteria for a simplified Ericksen-Leslie system modeling the flow of liquid crystals,” Discrete and Continuous Dynamical Systems A, vol. 25, no. 3, pp. 859–867, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. J. Fan, H. Gao, and B. Guo, “Regularity criteria for the Navier-Stokes-Landau-Lifshitz system,” Journal of Mathematical Analysis and Applications, vol. 363, no. 1, pp. 29–37, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. H. Kozono and Y. Shimada, “Bilinear estimates in homogeneous Triebel-Lizorkin spaces and the Navier-Stokes equations,” Mathematische Nachrichten, vol. 276, pp. 63–74, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891–907, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. S. Machihara and T. Ozawa, “Interpolation inequalities in Besov spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 5, pp. 1553–1556, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Y. Meyer, “Oscillating patterns in some nonlinear evolution equations,” in Mathematical Foundation of Turbulent Viscous Flows, vol. 1871 of Lecture Notes in Mathematics, pp. 101–187, Springer, Berlin, Germany, 2006.