Stability and Bifurcation Analysis of Differential Equations and its ApplicationsView this Special Issue
Research Article | Open Access
On Regularity Criteria for the Two-Dimensional Generalized Liquid Crystal Model
We establish the regularity criteria for the two-dimensional generalized liquid crystal model. It turns out that the global existence results satisfy our regularity criteria naturally.
In this paper we consider the following two-dimensional (2D) liquid crystal model: where represents the velocity field, is a vectorial field modeling the orientation of the crystal molecules, and is the scalar pressure, while , are two real parameters. and the operator is defined by , and here denotes the Fourier transform of . We identify the cases or as the 2D generalized liquid crystal model with zero velocity diffusion or zero orientation diffusion, respectively.
We say our system is a generalized form of liquid crystal model. When , , the system is a simplified version of the Ericksen-Leslie system modeling the hydrodynamics of nematic liquid crystals which was developed during the period of 1958 through 1968 [1–3]. We notice that if , , then the system (1)–(4) becomes the Navier-Stokes equations. In this sense, the study of the system (1)–(4) can be valuable and interesting in both mathematical sense and physical sense.
The existence and uniqueness of the weak and smooth solutions for system (1)–(4) are given in [4–6] when . Local existence of classical solutions for the nematic liquid crystal flows was established in .
Now, we mention some known results about regularity theory for the system. In 2010, Zhou and Fan established a regularity criterion for it as with , in . Later, some regularity criteria are proved for the system with zero dissipation in . In , Fan et al. established a global regularity for this system with mixed partial viscosity. Recently, in [11, 12], it is proved that smooth solutions are global in the following three cases: , ; , ; , . Moreover, global strong solution to the density-dependent 2D liquid crystal flows was studied in .
In our Theorems we set , . The first one is for large and .
Theorem 1. Let . Suppose and is a local smooth solution of the 2D generalized liquid crystal model (1)–(4). If , satisfy
then is a regular solution in . Here , .
If in addition, , , satisfies then is a regular solution in , where , .
The following theorems are established for the cases or small.
Remark 4. The results in this paper are motivated by the recent works on 2D incompressible generalized MHD equations (refer  for details). It turns out that our regularity criteria imply previous global existence results naturally. If , or , it is proved in  , , the regularity criteria in Theorem 2 are satisfied naturally. If , , one can prove , , , (refer  for details); the regularity criteria (10) in Theorem 3 are satisfied naturally.
2. Proof of Theorem 1
In this section, we are devoted to prove our main Theorem 1. Under the assumption in Theorem 1, if and , we can deduce and . So we only have to give the regularity criteria to guarantee the estimation for .
Proof. Firstly, we give the following priori estimates.
Multiplying (2) by , integrating over , after integrating by parts, and using the following property, we obtain for any . Applying Gronwall’s inequality, we deduce that Multiplying (1) and (2) by and , respectively, integrating over , and adding the resulting equations together, we obtain where , . Here we have used the following Galiardo-Nirenberg inequality: in the above inequality Applying Gronwall’s inequality and (14), we obtain Now, we are ready to give the estimation for . Multiplying (1) by , applying to (2), and testing it by , then by using (14), (18), we have Here we used (16) and the following Galiardo-Nirenberg inequality: Thanks to Gronwall’s inequality and (5), we deduce If , we have Here we have used In the above estimation we have used (16), (20), and the following Galiardo-Nirenberg inequality: Using the condition (6) we get the estimation (21).
If , we have Here we have used Here and in the following we use the embeddings for and , and the commutator estimate given in  usually. Thanks to condition (7) we have the estimation (21).
If in addition, , , use (16),
where We have Thanks to condition (8) we have the estimation (21). Now we complete the estimation.
Since in 2D, if , then we can deduce that .
Applying to (1) and testing by , applying to (1), and testing by , after suitable integration by parts we obtain If , , we estimate the right hand side of (31) one by one: here we used the following Galiardo-Nirenberg inequalities: Using (14), (16), (20), (21), and the following Galiardo-Nirenberg inequality, we have Finally, putting the above results together, we deduce that By using Gronwall’s inequality and (21), we obtain
The proof of Theorem 1 is finished.
3. Proof of Theorem 2
Now, we give the proof of Theorem 2.
Proof. This section focuses on the case with , or , . Since we cannot estimate (37) as that in Theorem 1, so we should give the estimation of which is defined in Section 2. Using (16), we have
where and . We use the same way to estimate and .
The same as that in Section 2, we have
Finally, putting the above results together, we have Using Gronwall’s inequality and (9) we can deduce (37).
The proof is finished.
4. Proof of Theorem 3
Proof. For , , firstly we give the estimation for :
Then by using Gronwall’s inequality and (10) we obtain (21).
Now, we give the estimation for . We should give the estimation of which is defined in Section 2: The same as that in Section 2, we have Putting the above results together, we deduce Applying Gronwall’s inequality and (10) we obtain (37).
For , , firstly we give the estimation for : and the estimation for : The regularity criteria (11) can guarantee estimations (18) and (21).
Now, we give the estimation of which is defined in Section 2: Using the following Galiardo-Nirenberg inequality, where we can deduce
Finally, putting the above results together, we deduce By using Gronwall’s inequality and (11) we obtain (37). Now we complete our proof.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank Professor Yong Zhou for patient guidance and helpful discussion. This work is partially supported by NSFC (Grant no. 11101376), NSFC (Grant no. 11226176), and ZJNSF (Grant no. LQ13A010008).
- P. G. de Gennes, The Physics of Liquid Crystals, Oxford University Press, Oxford, UK, 1974.
- J. L. Ericksen, “Hydrostatic theory of liquid crystals,” Archive for Rational Mechanics and Analysis, vol. 9, pp. 371–378, 1962.
- F. M. Leslie, “Some constitutive equations for liquid crystals,” Archive for Rational Mechanics and Analysis, vol. 28, no. 4, pp. 265–283, 1968.
- D. Coutand and S. Shkoller, “Well-posedness of the full Ericksen-Leslie model of nematic liquid crystals,” Comptes Rendus de l'Académie des Sciences I: Mathematics, vol. 333, no. 10, pp. 919–924, 2001.
- F. 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. 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.
- H. Sun and C. Liu, “On energetic variational approaches in modeling the nematic liquid crystal flows,” Discrete and Continuous Dynamical Systems A, vol. 23, no. 1-2, pp. 455–475, 2009.
- Y. Zhou and J. Fan, “A regularity criterion for the nematic liquid crystal flows,” Journal of Inequalities and Applications, vol. 2010, Article ID 589697, 9 pages, 2010.
- J. Fan and T. Ozawa, “Regularity criterion for the incompressible viscoelastic fluid system,” Houston Journal of Mathematics, vol. 37, no. 2, pp. 627–636, 2011.
- J. Fan, G. Nakamura, and Y. Zhou, “Global regularity for the 2D liquid crystal model with mixed partial viscosity”.
- Y. Wang, Y. Zhou, A. Alsaedi, T. Hayat, and Z. Jiang, “Global regularity for the incompressible 2D generalized liquid crystal model with fractional diffusions,” Applied Mathematics Letters, vol. 35, pp. 18–23, 2014.
- Y. Jin, Y. Zhou, and M. Zhu, “Global existence of solutions to the 2D incompressible liquid crystal ow with fractional diffusion,” preprint.
- Y. Zhou, J. Fan, and G. Nakamura, “Global strong solution to the density-dependent 2-D liquid crystal flows,” Abstract and Applied Analysis, vol. 2013, Article ID 947291, 5 pages, 2013.
- Z. Jiang and Y. Zhou, “On regularity criteria for the 2D generalized MHD system,” preprint.
- H. Kozono and Y. Taniuchi, “Bilinear estimates in BMO and the Navier-Stokes equations,” Mathematische Zeitschrift, vol. 235, no. 1, pp. 173–194, 2000.
Copyright © 2014 Yanan Wang and Zaihong Jiang. 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.