## Abstract and Applied Analysis

### Global Strong Solution to the Density-Dependent 2-D Liquid Crystal Flows

#### Abstract

The initial-boundary value problem for the density-dependent flow of nematic crystals is studied in a 2-D bounded smooth domain. For the initial density away from vacuum, the existence and uniqueness is proved for the global strong solution with the large initial velocity ${u}_{0}$ and small $\nabla {d}_{0}$. We also give a regularity criterion $\nabla d\in {L}^{p}\left(0,T;{L}^{q}\left(\mathrm{\Omega }\right)\right) \left(\left(2/\mathrm{q\right)}+\left(2/\mathrm{p\right)}=1, 2 of the problem with the Dirichlet boundary condition $u=0$, $d={d}_{0}$ on $\partial \mathrm{\Omega }$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 947291, 5 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511778

Digital Object Identifier
doi:10.1155/2013/947291

Mathematical Reviews number (MathSciNet)
MR3035314

Zentralblatt MATH identifier
1328.76008

#### Citation

Zhou, Yong; Fan, Jishan; Nakamura, Gen. Global Strong Solution to the Density-Dependent 2-D Liquid Crystal Flows. Abstr. Appl. Anal. 2013 (2013), Article ID 947291, 5 pages. doi:10.1155/2013/947291. https://projecteuclid.org/euclid.aaa/1393511778

#### References

• S. Chandrasekhar, Liquid Crystalsed, Cambridge University Press, 2nd edition, 1992.
• 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.
• 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.
• R. Danchin, “Density-dependent incompressible fluids in bounded domains,” Journal of Mathematical Fluid Mechanics, vol. 8, no. 3, pp. 333–381, 2006.
• H. Kim, “A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations,” SIAM Journal on Mathematical Analysis, vol. 37, no. 5, pp. 1417–1434, 2006.
• J. Fan and T. Ozawa, “Regularity criteria for the 3D density-dependent Boussinesq equations,” Nonlinearity, vol. 22, no. 3, pp. 553–568, 2009.
• X. Xu and Z. Zhang, “Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows,” Journal of Differential Equations, vol. 252, no. 2, pp. 1169–1181, 2012.
• F. Lin, J. Lin, and C. Wang, “Liquid crystal flows in two dimensions,” Archive for Rational Mechanics and Analysis, vol. 197, no. 1, pp. 297–336, 2010.
• 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.
• 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.
• J. Fan and B. Guo, “Regularity criterion to some liquid crystal models and the Landau-Lifshitz equations in ${\mathcal{R}}^{3}$,” Science in China. Series 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. Series A, vol. 25, no. 3, pp. 859–867, 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.
• H. Wen and S. Ding, “Solutions of incompressible hydrodynamic flow of liquid crystals,” Nonlinear Analysis: Real World Applications, vol. 12, no. 3, pp. 1510–1531, 2011.
• 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.
• J. Li, “Global strong and weak solutions to nematic liquid crystal flow in two dimensions,” http://arxiv.org/abs/1211.0131.