## Taiwanese Journal of Mathematics

### Global Existence and Exponential Decay of Strong Solutions to the 2D Density-dependent Nematic Liquid Crystal Flows with Vacuum

Yang Liu

#### Abstract

This paper deals with the 2D incompressible nematic liquid crystal flows with density-dependent viscosity in bounded domain. The global well-posedness of strong solutions are established in the vacuum cases, provided the assumption that $\overline{\rho} + \|\nabla d_0\|_{L^2}$ is suitably small with large velocity, which extends the recent work [Discrete Contin. Dyn. Syst. 37 (2017), 4907--4922] and [Methods Appl. Anal. 22 (2015), 201--220] to the case of variable viscosity. Furthermore, the exponential decay of the solution is also obtained.

#### Article information

Source
Taiwanese J. Math., Advance publication (2020), 24 pages.

Dates
First available in Project Euclid: 4 March 2020

https://projecteuclid.org/euclid.twjm/1583290812

Digital Object Identifier
doi:10.11650/tjm/200207

#### Citation

Liu, Yang. Global Existence and Exponential Decay of Strong Solutions to the 2D Density-dependent Nematic Liquid Crystal Flows with Vacuum. Taiwanese J. Math., advance publication, 4 March 2020. doi:10.11650/tjm/200207. https://projecteuclid.org/euclid.twjm/1583290812

#### References

• S. Ding, J. Huang and F. Xia, Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum, Filomat 27 (2013), no. 7, 1247–1257.
• J. L. Ericksen, Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal. 9 (1962), 371–378.
• J. Gao, Q. Tao and Z.-a. Yao, Strong solutions to the density-dependent incompressible nematic liquid crystal flows, J. Differential Equations 260 (2016), no. 4, 3691–3748.
• C. He, J. Li and B. Lü, On the Cauchy problem of 3D nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum, arXiv:1709.05608.
• J. L. Hineman and C. Wang, Well-posedness of nematic liquid crystal flow in $L_{\mathrm{uloc}}^3(\mathbb{R}^3)$, Arch. Ration. Mech. Anal. 210 (2013), no. 1, 177–218.
• D. Hoff, Compressible flow in a half-space with Navier boundary conditions, J. Math. Fluid Mech. 7 (2005), no. 3, 315–338.
• X. Hu and D. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Comm. Math. Phys. 296 (2010), no. 3, 861–880.
• J. Huang, C. Wang and H. Wen, Time decay rate of global strong solutions to nematic liquid crystal flows in $\mathbb{R}_+^3$, J. Differential Equations 267 (2019), no. 3, 1767–1804.
• T. Huang and C. Wang, Blow up criterion for nematic liquid crystal flows, Comm. Partial Differential Equations 37 (2012), no. 5, 875–884.
• X. Huang, J. Li and Z. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math. 65 (2012), no. 4, 549–585.
• X. Huang and Y. Wang, Global strong solution with vacuum to the two dimensional density-dependent Navier-Stokes system, SIAM J. Math. Anal. 46 (2014), no. 3, 1771–1788.
• F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal. 28 (1968), no. 4, 265–283.
• J. Li, Liquid crystal equations with infinite energy local well-posedness and blow up criterion, arXiv:1309.0072.
• ––––, Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions, Nonlinear Anal. 99 (2014), 80–94.
• ––––, Global strong solutions to the inhomogeneous incompressible nematic liquid crystal flow, Methods Appl. Anal. 22 (2015), no. 2, 201–220.
• ––––, Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differential Equations 263 (2017), no. 10, 6512–6536.
• L. Li, Q. Liu and X. Zhong, Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum, Nonlinearity 30 (2017), no. 11, 4062–4088.
• X. Li, Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two, Discrete Contin. Dyn. Syst. 37 (2017), no. 9, 4907–4922.
• X. Li and D. Wang, Global strong solution to the density-dependent incompressible flow of liquid crystals, Trans. Amer. Math. Soc. 367 (2015), no. 4, 2301–2338.
• F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math. 69 (2016), no. 8, 1532–1571.
• P.-L. Lions, Mathematical Topics in Fluid Mechanics I: Incompressible Models, Oxford Lecture Series in Mathematics and its Applications 3, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.
• Q. Liu, S. Liu, W. Tan and X. Zhong, Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows, J. Differential Equations 261 (2016), no. 11, 6521–6569.
• S. Liu and J. Zhang, Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 8, 2631–2648.
• B. Lü and S. Song, On local strong solutions to the three-dimensional nonhomogeneous Navier-Stokes equations with density-dependent viscosity and vacuum, Nonlinear Anal. Real World Appl. 46 (2019), 58–81.
• C. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Ration. Mech. Anal. 200 (2011), no. 1, 1–19.
• H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl. 12 (2011), no. 3, 1510–1531.
• H. Yu and P. Zhang, Global regularity to the 3D incompressible nematic liquid crystal flows with vacuum, Nonlinear Anal. 174 (2018), 209–222.