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ISRN Mathematical Analysis
Volume 2012 (2012), Article ID 248473, 7 pages
http://dx.doi.org/10.5402/2012/248473
Research Article

Continuation Criterion for the 2D Liquid Crystal Flows

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

Received 16 December 2011; Accepted 17 January 2012

Academic Editors: A. Montes-Rodriguez and T.-P. Tsai

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 consider the 2D liquid crystal systems, which consists of Navier-Stokes system coupled with wave maps or biharmonic wave maps, respectively. By logarithmic Sobolev inequalities, we obtain a blow-up criterion 𝑑,𝜕𝑡𝑑𝐿1̇𝐵(0,𝑇;0,(2)) for the case with wave maps, and we prove the existence of a global-in-time strong solutions for the case with biharmonic wave maps.

1. Introduction

First, we consider the following simplified liquid crystal flows in two space dimensions [1]:𝜕𝑡𝑢+𝑢𝑢+𝜋Δ𝑢=𝑘𝜕𝑡𝑑𝑘𝑑𝑘,𝜕(1.1)div𝑢=0,(1.2)2𝑡||||𝑑+𝑢𝑑Δ𝑑=𝑑𝑑2||𝜕𝑡𝑑||2,||𝑑||=1,(1.3)𝑢,𝑑,𝜕𝑡𝑑𝑢(𝑥,0)=0,𝑑0,𝑑1(𝑥),𝑥2,||𝑑0||=1,𝑑0𝑑1=0,(1.4) where 𝑢 is the velocity, 𝜋 is the pressure, and 𝑑 represents the macroscopic average of the liquid crystal orientation field with values in the unit circle.

The first two equations (1.1) and (1.2) are the well-known Navier-Stokes system with the Lorentz force 𝑘𝜕𝑡𝑑𝑘𝑑𝑘. The last equation (1.3) is the well-known wave maps when 𝑢=0.

It is a simple matter to show that the system (1.1)–(1.4) has a unique local-in-time smooth solution when 𝑢0,𝑑0,𝑑1𝐻1+𝑠(2) with 𝑠>0,div𝑢0=0,|𝑑0|=1,𝑑0𝑑1=0 in 2. The aim of this paper is to study the regularity criterion of smooth solutions to the problem (1.1)–(1.4). We will prove the following.

Theorem 1.1. Let 𝑢0,𝑑0,𝑑1𝐻1+𝑠(2) with 𝑠>0,div𝑢0=0,|𝑑0|=1,𝑑0𝑑1=0 in 2 and let (𝑢,𝑑) be a smooth solution of (1.1)–(1.4) on some interval [0,𝑇] with 0<𝑇<. Assume that 𝑑,𝜕𝑡𝑑𝐿1̇𝐵0,𝑇;0,2.(1.5) Then the solution (𝑢,𝑑) can be extended beyond 𝑇>0.

̇𝐵0, is the homogeneous Besov space. We have 𝐿̇𝐵𝐵𝑀𝑂0,; see Triebel [2].

In the proof of Theorem 1.1, we will use the logarithmic Sobolev inequalities [36]:𝑢𝐿𝐶𝑢𝐻1log1/2𝑒+𝑢𝐻1+𝑠,(1.6)𝑑𝐿𝐶1+𝑑̇𝐵0,log𝑒+𝑑𝐻1+𝑠𝜕,(1.7)𝑡𝑑𝐿𝜕𝐶1+𝑡𝑑̇𝐵0,𝜕log𝑒+𝑡𝑑𝐻1+𝑠,(1.8) for 𝑠>0, and the Gagliardo-Nirenberg inequalities:𝑤𝐿4𝐶𝑤𝛼𝐿2Λ1+𝑠𝑤𝐿1𝛼2,Λ𝑠𝑤𝐿4𝐶𝑤𝐿1𝛼2Λ1+𝑠𝑤𝛼𝐿2,(1.9) with Λ=(Δ)1/2,𝛼=1(1/2)(1/(1+𝑠)), and 𝑠>0, and the product estimate due to Kato-Ponce [7]:Λ𝑠(𝑓𝑔)𝐿𝑝𝐶𝑓𝐿𝑝1Λ𝑠𝑔𝐿𝑞1+𝑔𝐿𝑝2Λ𝑠𝑓𝐿𝑞2,(1.10) with 𝑠>0 and 1/𝑝=1/𝑝1+1/𝑞1=1/𝑝2+1/𝑞2.

Motivated by the problem (1.1)–(1.4), we consider the following liquid crystal flows:𝜕𝑡𝑢+𝑢𝑢+𝜋Δ𝑢=𝑘𝜕𝑡𝑑𝑘𝑑𝑘,𝜕(1.11)div𝑢=0,(1.12)2𝑡𝑑+𝑢𝑑+(Δ)2||𝑑||||𝜕𝑑=𝜆𝑑,=1,(1.13)𝜆=𝑡𝑑||2+||||Δ𝑑2||||+Δ𝑑2+2𝑘𝜕𝑘𝑑Δ𝜕𝑘𝑑,(1.14)𝑢,𝑑,𝜕𝑡𝑑𝑢(𝑥,0)=0,𝑑0,𝑑1(𝑥,0),𝑥2,||𝑑0||=1.(1.15) The last two equations (1.13) and (1.14) are the biharmonic wave maps. It is also a simple matter to show that the problem (1.11)–(1.15) has at least one local-in-time strong solution. The aim of this paper is to prove the global-in-time regularity. We obtain the following.

Theorem 1.2. Let 𝑢0𝐻2,(𝑑0,𝑑1)𝐻3×𝐻2 with div𝑢0=0,|𝑑0|=1,𝑑0𝑑1=0 in 2. Then there exists at least a global-in-time smooth solution: 𝑢,𝑑,𝜕𝑡𝑑𝐿0,𝑇;𝐻2×𝐿0,𝑇;𝐻3×𝐿0,𝑇;𝐻2(1.16) for any 𝑇>0.

Remark 1.3. We are unable to prove the uniqueness of strong solutions in Theorem 1.2.

2. Proof of Theorem 1.1

We only need to prove a priori estimates.

Testing (1.1) by 𝑢, using (1.2), we see that12𝑑𝑢𝑑𝑡2||||𝑑𝑥+𝑢2(𝑑𝑥=𝑢)𝑑𝜕𝑡𝑑𝑑𝑥.(2.1)

Testing (1.3) by 𝜕𝑡𝑑, using |𝑑|=1 and 𝑑𝜕𝑡𝑑=0, we find that12𝑑||||𝑑𝑡𝑑2+||𝜕𝑡𝑑||2(𝑑𝑥=𝑢)𝑑𝜕𝑡𝑑𝑑𝑥.(2.2)

Summing up (2.1) and (2.2), we get12𝑑𝑢𝑑𝑡2+||||𝑑2+||𝜕𝑡𝑑||2||||𝑑𝑥+𝑢2𝑑𝑥=0,(2.3) from which we get𝑢2+||||𝑑2+||𝜕𝑡𝑑||2𝑑𝑥+𝑇0||||𝑢2𝑑𝑥𝑑𝑡𝐶.(2.4)

Applying Λ1+𝑠 to (1.1), testing by Λ1+𝑠𝑢, using (1.2) and (1.10), we derive12𝑑||Λ𝑑𝑡1+𝑠𝑢||2||Λ𝑑𝑥+2+𝑠𝑢||2Λ𝑑𝑥=1+𝑠div(𝑢𝑢)Λ1+𝑠Λ𝑢𝑑𝑥+1+𝑠𝜕𝑡𝑑𝑑Λ1+𝑠𝑢𝑑𝑥𝐶𝑢𝐿Λ2+𝑠𝑢𝐿2Λ1+𝑠𝑢𝐿2𝜕+𝐶𝑡𝑑𝐿Λ2+𝑠𝑑𝐿2+𝑑𝐿Λ1+𝑠𝜕𝑡𝑑𝐿2Λ1+𝑠𝑢𝐿212Λ2+𝑠𝑢2𝐿2+𝐶𝑢2𝐿Λ1+𝑠𝑢2𝐿2𝜕+𝐶𝑡𝑑,𝑑𝐿𝑦2+Λ1+𝑠𝑢2𝐿2,(2.5) where𝑦2Λ=1+𝑠𝜕𝑡𝑑2𝐿2+Λ2+𝑠𝑑2𝐿2.(2.6)

Taking Λ1+𝑠 to (1.3), testing by Λ1+𝑠𝜕𝑡𝑑, we have12𝑑𝑦𝑑𝑡2=Λ1+𝑠𝑑||||𝑑2||𝜕𝑡𝑑||2Λ1+𝑠𝜕𝑡Λ𝑑𝑑𝑥1+𝑠(𝑢𝑑)Λ1+𝑠𝜕𝑡𝑑𝑑𝑥=𝐼1+𝐼2.(2.7)

By using (1.10), (2.4), and (1.9), 𝐼1 can be bounded as follows:𝐼1𝐶𝑑𝐿Λ1+𝑠||||𝑑2||𝜕𝑡𝑑||2𝐿2+Λ1+𝑠𝑑𝐿4𝑑𝐿𝑑𝐿4+𝜕𝑡𝑑𝐿𝜕𝑡𝑑𝐿4Λ1+𝑠𝜕𝑡𝑑𝐿2𝐶𝑑𝐿Λ2+𝑠𝑑𝐿2+𝜕𝑡𝑑𝐿Λ1+𝑠𝜕𝑡𝑑𝐿2+𝑐𝑦𝛼𝑑𝐿𝑦1𝛼+𝜕𝑡𝑑𝐿𝑦1𝛼Λ1+𝑠𝜕𝑡𝑑𝐿2𝜕𝐶𝑡𝑑,𝑑𝐿𝑦2.(2.8)

By using (1.10), 𝐼2 can be bounded as𝐼2𝐶𝑢𝐿Λ2+𝑠𝑑𝐿2+𝑑𝐿Λ1+𝑠𝑢𝐿2Λ1+𝑠𝜕𝑡𝑑𝐿2𝐶𝑢𝐿𝑦2+𝐶𝑑𝐿𝑦2+Λ1+𝑠𝑢2𝐿2.(2.9)

Combining (2.5), (2.7), (2.8), and (2.9) and using (1.6), (1.7), (1.8), and the Gronwall lemma, we arrive at𝑢𝐿(0,𝑇;𝐻1+𝑠)+𝑢𝐿2(0,𝑇;𝐻2+𝑠)𝐶,𝑑,𝜕𝑡𝑑𝐿(0,𝑇;𝐻1+𝑠)𝐶.(2.10)

This completes the proof.

3. Proof of Theorem 1.2

For simplicity, we only present a priori estimates.

First, we still have (2.1).

Testing (1.13) by 𝜕𝑡𝑑, using 𝑑𝜕𝑡𝑑=0, we have12𝑑||||𝑑𝑡Δ𝑑2+||𝜕𝑡𝑑||2(𝑑𝑥=𝑢)𝑑𝜕𝑡𝑑𝑑𝑥.(3.1)

Summing up (2.1) and (3.1), we get𝑢2+||||Δ𝑑2+||𝜕𝑡𝑑||2𝑑𝑥+𝑇0||||𝑢21𝑑𝑥𝑑𝑡𝐶,2𝑑||||𝑑𝑡𝑑2𝑑𝑥=Δ𝑑𝜕𝑡𝑑𝑑𝑥Δ𝑑𝐿2𝜕𝑡𝑑𝐿2𝐶,(3.2)

which yields||||𝑑2𝑑𝑥𝐶.(3.3)

Applying Δ to (1.11), testing by Δ𝑢, using (1.2) and (1.10), we deduce that12𝑑||||𝑑𝑡Δ𝑢2||||𝑑𝑥+Δ𝑢2Δ𝜕𝑑𝑥=Δdiv(𝑢𝑢)Δ𝑢𝑑𝑥+𝑡𝑑,𝑑Δ𝑢𝑑𝑥𝐶𝑢𝐿Δ𝑢𝐿2Δ𝑢𝐿2𝜕+𝐶𝑡𝑑𝐿Δ𝑑𝐿2+𝑑𝐿Δ𝜕𝑡𝑑𝐿2Δ𝑢𝐿212Δ𝑢2𝐿2+𝐶𝑢2𝐿Δ𝑢2𝐿2𝜕+𝐶𝑡𝑑𝐿1/22Δ𝜕𝑡𝑑𝐿1/22Δ𝑑𝐿1/22Δ2𝑑𝐿1/22+𝑑𝐿Δ𝜕𝑡𝑑𝐿2Δ𝑢𝐿212Δ𝑢2𝐿2+𝐶𝑢2𝐿Δ𝑢2𝐿2+𝐶𝑦2+𝐶Δ𝑢2𝐿2+𝐶𝑑𝐿𝑦2+Δ𝑢2𝐿2,(3.4) where𝑦2=Δ𝜕𝑡𝑑2𝐿2+Δ2𝑑2𝐿2.(3.5)

Applying Δ to (1.13), we haveΔ𝜕2𝑡𝑑+Δ3𝑑=(𝜆Δ𝑑+2𝜆𝑑+𝑑Δ𝜆)Δ(𝑢𝑑).(3.6)

Since0=Δ𝑑𝜕𝑡𝑑=𝑑Δ𝜕𝑡𝑑+𝜕𝑡𝑑Δ𝑑+2𝑘𝜕𝑘𝑑𝜕𝑘𝜕𝑡𝑑,(3.7) we easily see that𝑑Δ𝜕𝑡𝑑=𝜕𝑡𝑑Δ𝑑+𝜕𝑡||||𝑑2.(3.8)

Testing (3.6) by Δ𝜕𝑡𝑑, using (3.8), we obtain12𝑑𝑦𝑑𝑡2=𝜆Δ𝑑Δ𝜕𝑡𝑑+𝜆2𝑑Δ𝜕𝑡𝜕𝑑+𝑡𝑑Δ𝑑+2𝑑𝜕𝑡𝑑𝑑𝑥Δ(𝑢𝑑)Δ𝜕𝑡𝑑𝑑𝑥=𝐽1+J2.(3.9)

By the same calculations as those in [8], we have𝐽1𝐶1+𝑑𝐿𝑦2𝐶1+𝑑𝐻1𝑦log(𝑒+𝑦)2.(3.10)

By using (1.10), 𝐽2 can be bounded as𝐽2𝐶Δ𝑢𝐿2𝑑𝐿+𝑢𝐿Δ𝑑𝐿2Δ𝜕𝑡𝑑𝐿2𝐶𝑑𝐿𝑦2+Δ𝑢2𝐿2+𝐶𝑢𝐿1/22Δ𝑢𝐿1/22Δ𝑑𝐿1/22Δ2𝑑𝐿1/22Δ𝜕𝑡𝑑𝐿2𝐶𝑑𝐿𝑦2+Δ𝑢2𝐿2+𝐶𝑦2+𝐶Δ𝑢2𝐿2.(3.11)

Combining (3.4), (3.9), (3.10), and (3.11) and using (1.6) and the Gronwall lemma, we conclude that𝑢𝐿(0,𝑇;𝐻2)+𝑢𝐿2(0,𝑇;𝐻3)𝐶,𝑑𝐿(0,𝑇;𝐻3)+𝜕𝑡𝑑𝐿(0,𝑇;𝐻2)𝐶.(3.12)

This completes the proof.

Acknowledgment

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

References

  1. F. Lin and C. Liu, “Static and dynamic theories of liquid crystals,” Journal of Partial Differential Equations, vol. 14, no. 4, pp. 289–330, 2001.
  2. H. Triebel, Theory of Function Spaces. II, vol. 84, Birkhäuser, Basel, Switzerland, 1992. View at Publisher · View at Google Scholar
  3. H. Brézis and T. Gallouet, “Nonlinear Schrödinger evolution equations,” Nonlinear Analysis, vol. 4, no. 4, pp. 677–681, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. H. Brézis and S. Wainger, “A note on limiting cases of Sobolev embeddings and convolution inequalities,” Communications in Partial Differential Equations, vol. 5, no. 7, pp. 773–789, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. T. Ozawa, “On critical cases of Sobolev's inequalities,” Journal of Functional Analysis, vol. 127, no. 2, pp. 259–269, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. H. Kozono, T. Ogawa, and Y. Taniuchi, “The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations,” Mathematische Zeitschrift, vol. 242, no. 2, pp. 251–278, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. 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
  8. J. Fan and T. Ozawa, “On regularity criterion for the 2D wave maps and the 4D biharmonic wave maps,” in Current Advances in Nonlinear Analysis and Related Topics, vol. 32 of GAKUTO International Series Mathematical Sciences and Applications, pp. 69–83, Gakkōtosho, Tokyo, Japan, 2010. View at Zentralblatt MATH