- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
International Journal of Analysis
Volume 2013 (2013), Article ID 718173, 5 pages
Regularity Criteria for a Coupled Navier-Stokes and Q-Tensor System
1Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
2Department 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.
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.
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.  and Kozono and Shimada  proved the well-known regularity criteria where denotes the homogeneous Besov spaces .
Very recently, Paicu and Zarnescu  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.
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  that the system (1)–(4) has a Lyapunov functional: which satisfies from which we easily obtain  When , (13) give  thus (5) and (8) hold true; this proves the existence of global-in-time strong solutions when . In , 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  much simpler. However, in , 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 ). 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
Let us observe that for , a traceless, symmetric, matrix, 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 : and the Gagliardo-Nirenberg inequality
This completes the proof.
(2) Let (6) hold true
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
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.
This completes the proof.
This paper is supported by NSFC (no. 11171154). The authors are indebted to the referee for nice suggestions which improved the paper.
- 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.