# American Institute of Mathematical Sciences

November  2020, 19(11): 5219-5238. doi: 10.3934/cpaa.2020234

## On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum

 1 College of Mathematics, Changchun Normal University, Changchun 130032, China 2 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author

Received  March 2020 Revised  June 2020 Published  November 2020 Early access  September 2020

Fund Project: Yang Liu is supported by China Postdoctoral Science Foundation (No. 2018M642202) and National Natural Science Foundation of China (No. 11901288). Xin Zhong is supported by National Natural Science Foundation of China (No. 11901474)

This paper deals with the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible nematic liquid crystal flows. The global well-posedness of strong solutions with large velocity is established provided that $\|\rho_0\|_{L^\infty}+\|\nabla d_0\|_{L^3}$ is suitably small. In particular, the initial density may contain vacuum states and even have compact support. Furthermore, the large time behavior of the solution is also obtained.

Citation: Yang Liu, Xin Zhong. On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5219-5238. doi: 10.3934/cpaa.2020234
##### References:
 [1] S. Ding, J. Huang and F. Xia, Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum, Filomat, 27 (2013), 1247-1257.  doi: 10.2298/FIL1307247D.  Google Scholar [2] J. Ericksen, Hydrostatic theory of liquid crystals, Arch. Ration. Mech. Anal., 9 (1962), 371-378.  doi: 10.1007/BF00253358.  Google Scholar [3] M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differ. Equ., 40 (2011), 15-36.  doi: 10.1007/s00526-010-0331-5.  Google Scholar [4] M. Hong and Y. Mei, Well-posedness of the Ericksen-Leslie system with the Oseen-Frank energy in $L^3_uloc(\mathbb{R}^3)$, Calc. Var. Partial Differ. Equ., 58 (2019), . doi: 10.1007/s00526-018-1453-4.  Google Scholar [5] M. Hong and Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\Bbb R^2$, Adv. Math., 231 (2012), 1364-1400.  doi: 10.1016/j.aim.2012.06.009.  Google Scholar [6] F. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.  Google Scholar [7] J. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions, Nonlinear Anal., 99 (2014), 80-94.  doi: 10.1016/j.na.2013.12.023.  Google Scholar [8] J. Li, Global strong solutions to the inhomogeneous incompressible nematic liquid crystal flow, Methods Appl. Anal., 22 (2015), 201-220.  doi: 10.4310/MAA.2015.v22.n2.a4.  Google Scholar [9] J. Li, Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differ. Equ., 263 (2017), 6512-6536.  doi: 10.1016/j.jde.2017.07.021.  Google Scholar [10] X. Li, Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two, Discrete Contin. Dyn. Syst., 37 (2017), 4907-4922.  doi: 10.3934/dcds.2017211.  Google Scholar [11] L. Li, Q. Liu and and X. Zhong, Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum, Nonlinearity, 30 (2017), 4062-4088.  doi: 10.1088/1361-6544/aa8426.  Google Scholar [12] F. Lin, J. Lin and and C. Wang, Liquid crystal flow in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.  doi: 10.1007/s00205-009-0278-x.  Google Scholar [13] F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B, 31 (2010), 921-938.  doi: 10.1007/s11401-010-0612-5.  Google Scholar [14] F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three, Commun. Pure Appl. Math., 69 (2016), 1532-1571.  doi: 10.1002/cpa.21583.  Google Scholar [15] P. L. Lions, Mathematical topics in fluid mechanics, Vol. I: incompressible models, Oxford University Press, Oxford, 1996.   Google Scholar [16] Q. Liu, S. Liu, W. Tan and X. Zhong, Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows, J. Differ. Equ., 261 (2016), 6521-6569.  doi: 10.1016/j.jde.2016.08.044.  Google Scholar [17] Q. Liu, C. Wang, X. Zhang and J. Zhou, On optimal boundary control of Ericksen-Leslie system in dimension two, Calc. Var. Partial Differ. Equ, 59 (2020), 64pp. doi: 10.1007/s00526-019-1676-z.  Google Scholar [18] 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), 2631-2648.  doi: 10.3934/dcdsb.2016065.  Google Scholar [19] B. Lü, X. Shi and and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum, Nonlinearity, 31 (2018), 2617-2632.  doi: 10.1088/1361-6544/aab31f.  Google Scholar [20] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, 13 (1959), 115-162.   Google Scholar [21] R. Temam, Navier-Stokes equations: theory and numerical analysis, Chelsea Publishing, Providence, RI, 2001. Reprint of the 1984 edition. doi: 10.1090/chel/343.  Google Scholar [22] H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl., 12 (2011), 1510-1531.  doi: 10.1016/j.nonrwa.2010.10.010.  Google Scholar [23] H. Yu and P. Zhang, Global regularity to the 3D incompressible nematic liquid crystal flows with vacuum, Nonlinear Anal., 174 (2018), 209-222.  doi: 10.1016/j.na.2018.04.022.  Google Scholar [24] X. Zhong, A note on a global strong solution to the 2D Cauchy problem of density-dependent nematic liquid crystal flows with vacuum, C. R. Math. Acad. Sci. Paris, 356 (2018), 503-508.  doi: 10.1016/j.crma.2018.04.011.  Google Scholar

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##### References:
 [1] S. Ding, J. Huang and F. Xia, Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum, Filomat, 27 (2013), 1247-1257.  doi: 10.2298/FIL1307247D.  Google Scholar [2] J. Ericksen, Hydrostatic theory of liquid crystals, Arch. Ration. Mech. Anal., 9 (1962), 371-378.  doi: 10.1007/BF00253358.  Google Scholar [3] M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differ. Equ., 40 (2011), 15-36.  doi: 10.1007/s00526-010-0331-5.  Google Scholar [4] M. Hong and Y. Mei, Well-posedness of the Ericksen-Leslie system with the Oseen-Frank energy in $L^3_uloc(\mathbb{R}^3)$, Calc. Var. Partial Differ. Equ., 58 (2019), . doi: 10.1007/s00526-018-1453-4.  Google Scholar [5] M. Hong and Z. Xin, Global existence of solutions of the liquid crystal flow for the Oseen-Frank model in $\Bbb R^2$, Adv. Math., 231 (2012), 1364-1400.  doi: 10.1016/j.aim.2012.06.009.  Google Scholar [6] F. Leslie, Some constitutive equations for liquid crystals, Arch. Ration. Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810.  Google Scholar [7] J. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flow in two dimensions, Nonlinear Anal., 99 (2014), 80-94.  doi: 10.1016/j.na.2013.12.023.  Google Scholar [8] J. Li, Global strong solutions to the inhomogeneous incompressible nematic liquid crystal flow, Methods Appl. Anal., 22 (2015), 201-220.  doi: 10.4310/MAA.2015.v22.n2.a4.  Google Scholar [9] J. Li, Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differ. Equ., 263 (2017), 6512-6536.  doi: 10.1016/j.jde.2017.07.021.  Google Scholar [10] X. Li, Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two, Discrete Contin. Dyn. Syst., 37 (2017), 4907-4922.  doi: 10.3934/dcds.2017211.  Google Scholar [11] L. Li, Q. Liu and and X. Zhong, Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum, Nonlinearity, 30 (2017), 4062-4088.  doi: 10.1088/1361-6544/aa8426.  Google Scholar [12] F. Lin, J. Lin and and C. Wang, Liquid crystal flow in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.  doi: 10.1007/s00205-009-0278-x.  Google Scholar [13] F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chin. Ann. Math. Ser. B, 31 (2010), 921-938.  doi: 10.1007/s11401-010-0612-5.  Google Scholar [14] F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three, Commun. Pure Appl. Math., 69 (2016), 1532-1571.  doi: 10.1002/cpa.21583.  Google Scholar [15] P. L. Lions, Mathematical topics in fluid mechanics, Vol. I: incompressible models, Oxford University Press, Oxford, 1996.   Google Scholar [16] Q. Liu, S. Liu, W. Tan and X. Zhong, Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows, J. Differ. Equ., 261 (2016), 6521-6569.  doi: 10.1016/j.jde.2016.08.044.  Google Scholar [17] Q. Liu, C. Wang, X. Zhang and J. Zhou, On optimal boundary control of Ericksen-Leslie system in dimension two, Calc. Var. Partial Differ. Equ, 59 (2020), 64pp. doi: 10.1007/s00526-019-1676-z.  Google Scholar [18] 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), 2631-2648.  doi: 10.3934/dcdsb.2016065.  Google Scholar [19] B. Lü, X. Shi and and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum, Nonlinearity, 31 (2018), 2617-2632.  doi: 10.1088/1361-6544/aab31f.  Google Scholar [20] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, 13 (1959), 115-162.   Google Scholar [21] R. Temam, Navier-Stokes equations: theory and numerical analysis, Chelsea Publishing, Providence, RI, 2001. Reprint of the 1984 edition. doi: 10.1090/chel/343.  Google Scholar [22] H. Wen and S. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. Real World Appl., 12 (2011), 1510-1531.  doi: 10.1016/j.nonrwa.2010.10.010.  Google Scholar [23] H. Yu and P. Zhang, Global regularity to the 3D incompressible nematic liquid crystal flows with vacuum, Nonlinear Anal., 174 (2018), 209-222.  doi: 10.1016/j.na.2018.04.022.  Google Scholar [24] X. Zhong, A note on a global strong solution to the 2D Cauchy problem of density-dependent nematic liquid crystal flows with vacuum, C. R. Math. Acad. Sci. Paris, 356 (2018), 503-508.  doi: 10.1016/j.crma.2018.04.011.  Google Scholar
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