International Journal of AnalysisVolume 2013 (2013), Article ID 718173, 5 pageshttp://dx.doi.org/10.1155/2013/718173

Research Article

Regularity Criteria for a Coupled Navier-Stokes and Q-Tensor System

Jishan Fan¹ and Tohru Ozawa²

¹Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China²Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

Received 14 September 2012; Accepted 7 April 2013

Academic Editor: Jens Lorenz

Copyright © 2013 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 a system describing the evolution of a nematic liquid crystal flow. The system couples a forced Navier-Stokes system describing the flow with a parabolic-type system describing the evolution of the nematic crystal director fields (Q-tensors). We prove some regularity criteria for the local strong solutions. However, we do not provide estimates on the rates of increase of high norms.

1. Introduction

We consider the following coupled Navier-Stokes and -tensor system [1–4]: Here the unknowns , and denote the velocity field of the fluid, the pressure, and the order parameter, respectively. A -tensor is a symmetric and traceless -matrix, and are physical constants, is the space dimension, , and thus .

When , (2) and (3) are the well-known Navier-Stokes system, for which Kozono et al. [5] and Kozono and Shimada [6] proved the well-known regularity criteria where denotes the homogeneous Besov spaces [7].

Very recently, Paicu and Zarnescu [8] proved the existence of global-in-time weak solutions in 3-dimensional space and of smooth solutions in 2-dimensional space. The aim of this paper is to study the regularity criteria.

If one formally takes , with a constant, then the equations reduce to the generally accepted equations of Leslie [9], which have been studied in [10–15]. We will prove the following.

Theorem 1. Let in with . Let be a unique strong solution in with . (i)If and satisfies one of the conditions (5), (6), or (7) and satisfies for some finite , then the solution can be extended beyond .(ii)If and satisfies for some finite , then the solution can be extended beyond .

Remark 2. By the well-known inequality , the condition (9) can be replaced by

It has been proved in [8] that the system (1)–(4) has a Lyapunov functional: which satisfies from which we easily obtain [8] When , (13) give [8] thus (5) and (8) hold true; this proves the existence of global-in-time strong solutions when . In [8], this result was proven by complicated Littlewood-Paley theory, Bony’s paraproduct decomposition, and the logarithmic Sobolev inequality. The purpose of this paper is to make the argument in [8] much simpler. However, in [8], they obtained in addition the rate of increase of high norms.

Our proof uses an energy method and relies on a simple estimate of and the following cancellation property:

Lemma 3 (see [8]). Let be symmetric matrix-valued functions and let be smooth and decaying and sufficiently fast at infinity (so that one can integrate by parts without boundary terms). Then

2. Proof of Theorem 1

This section is devoted to the proof of Theorem 1. Since it is easy to prove that there are and a unique strong solution to the problem (1)–(4) in , we only need to prove a priori estimates.

First, we prove the following key estimate: To prove (16), we multiply (1) by and take the trace to obtain

Let us observe that for , a traceless, symmetric, matrix, we have

Integrating over , integrating by parts, and using (3), (18), and the assumption , we obtain which gives with independent of .

Thanks to a simple lemma in [16, Page 102], we take in (20), this proves (16).

Taking to (1), testing scalarly by , and using (3); we find that Here “testing scalarly by ” means multiplying with respect to the Frobenius inner product of matrices, and integrating over .

Testing (2) by and using (3), we infer that

Summing (21) and (22) and using the cancellation , due to Lemma 3, we have

(i) Let (8) hold true

Using the integration by parts, can be bounded as Here and satisfy the relation (8), and we have used the Gagliardo-Nirenberg inequality

Similarly, we get

By using (16), is simply bounded as Here we treat the term by the Gagliardo-Nirenberg inequality

Inserting the above estimates into (23), we derive

Now we estimate as follows.

(1) Let (5) hold true

We will use the following inequality [6]: and the Gagliardo-Nirenberg inequality

Using (31) and (32), we bound as follows: Substituting the above estimates into (30), we reach

This completes the proof.

(2) Let (6) hold true

Using the following elegant inequality [17, 18]: we bound as follows:

Substituting the above estimates into (30), we have (34).

This completes the proof.

(3) Let (7) hold true

Let be the Littlewood-Paley dyadic decomposition of unity that satisfies , and for any , where is the Fourier transform and is the ball with radius centered at the origin.

We decompose as follows: where is a positive integer to be chosen later. Plugging this decomposition into , we derive

Recalling Bernstein’s inequality, with being a positive constant independent of and , we apply Hölder’s inequality to deduce that

Now we choose so that and to conclude that Substituting the above estimates into (30), we arrive at (34).

This completes the proof of part (i).

(ii) Let (9) hold true

We still have (23).

is simply bounded as Here we have used the Gagliardo-Nirenberg inequality

is simply bounded as Here we have used (43) and the Gagliardo-Nirenberg inequality Similarly, , and can be bounded as follows:

is bounded as above.

Inserting the above estimates into (23), using the Gronwall inequality, we arrive at (34).

This completes the proof.

Acknowledgments

This paper is supported by NSFC (no. 11171154). The authors are indebted to the referee for nice suggestions which improved the paper.

References

A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, vol. 36 of Oxford Engineering Science Series, Oxford University Press, Oxford, UK, 1994.

C. Denniston, E. Orlandini, and J. M. Yeomans, “Lattice Boltzmann simulations of liquid crystal hydrodynamics,” Physical Review E, vol. 63, no. 5, pp. 567021–5670210, 2001.

P. G. De Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford, UK, 1974.

G. Tóth, C. Denniston, and J. M. Yeomans, “Hydrodynamics of domain growth in nematic liquid crystals,” Physical Review E, vol. 67, no. 5, Article ID 051705, 11 pages, 2003.

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.

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.

H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, Switzerland, 1983.

M. Paicu and A. Zarnescu, “Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system,” Archive for Rational Mechanics and Analysis, vol. 203, pp. 45–67, 2012.

F. M. Leslie, “Some constitutive equations for liquid crystals,” Archive for Rational Mechanics and Analysis, vol. 28, no. 4, pp. 265–283, 1968.

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.

F. H. Lin and C. Liu, “Partial regularity of the dynamic system modeling the flow of liquid crystals,” Discrete and Continuous Dynamical Systems, vol. 2, no. 1, pp. 1–22, 1996.

F. H. Lin and C. Liu, “Existence of solutions for the Ericksen-Leslie system,” Archive for Rational Mechanics and Analysis, vol. 154, no. 2, pp. 135–156, 2000.

Z. Lei, N. Masmoudi, and Y. Zhou, “Remarks on the blowup criteria for Oldroyd models,” Journal of Differential Equations, vol. 248, no. 2, pp. 328–341, 2010.

J. Fan and B. Guo, “Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in ℝ³,” Science in China A, vol. 51, no. 10, pp. 1787–1797, 2008.

J. Fan and T. Ozawa, “Regularity criteria for a simplified Ericksen-Leslie system modeling the flow of liquid crystals,” Discrete and Continuous Dynamical Systems, vol. 25, no. 3, pp. 859–867, 2009.

M. H. Giga, Y. Giga, and J. Saal, Nonlinear Partial Differential Equations, Birkhäauser, 2010.

S. Machihara and T. Ozawa, “Interpolation inequalities in Besov spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 5, pp. 1553–1556, 2003.

Y. Meyer, “Oscillating patterns in some nonlinear evolution equations,” Lecture Notes in Mathematics, vol. 1871, pp. 101–187, 2005.