# American Institute of Mathematical Sciences

March  2021, 26(3): 1243-1272. doi: 10.3934/dcdsb.2020161

## Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows

 1 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China 2 Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing 100081, China 3 The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong

Received  February 2018 Revised  March 2020 Published  March 2021 Early access  May 2020

In this paper, we consider the initial-boundary value problem to the non-isothermal incompressible liquid crystal system with both variable density and temperature. Global well-posedness of strong solutions is established for initial data being small perturbation around the equilibrium state. As the tools in the proof, we establish the maximal regularities of the linear Stokes equations and parabolic equations with variable coefficients and a rigid lemma for harmonic maps on bounded domains. This paper also generalizes the result in [5] to the inhomogeneous case.

Citation: Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161
##### References:
 [1] H. Abels, Nonstationary Stokes system with variable viscosity in bounded and unbounded domains, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 141-157.  doi: 10.3934/dcdss.2010.3.141. [2] H. Abels and Y. Terasawa, On Stokes operators with variable viscosity in bounded and unbounded domains, Math. Ann., 344 (2009), 381-429.  doi: 10.1007/s00208-008-0311-7. [3] R. A. Adams and J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. [4] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6. [5] D. Bian and Y. Xiao, Global solution to the nematic liquid crystal flows with heat effect, J. Differential Equations, 263 (2017), 5298-5329.  doi: 10.1016/j.jde.2017.06.019. [6] D. Bothe and J. Prüss, $L_p$-theory for a class of Non-Newtonian fluids, SIAM J. Math. Anal., 39 (2007), 379-421.  doi: 10.1137/060663635. [7] R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8 (2006), 333-381.  doi: 10.1007/s00021-004-0147-1. [8] R. Denk, M. Hieber and J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003). doi: 10.1090/memo/0788. [9] W.-Y. Ding and F.-H. Lin, A generalization of Eells-Sampson's theorem, J. Partial Differential Equations, 5 (1992), 13-22. [10] S. J. Ding, C. Y. Wang and H. Y. Wen, Weak solutions to compressible flows of nematic liquid crystals in dimension one, Discrete Contin. Dyn. Syst. B, 15 (2011), 357-371.  doi: 10.3934/dcdsb.2011.15.357. [11] J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), 23-34.  doi: 10.1122/1.548883. [12] J. L. Ericksen, Continuum theory of nematic liquid crystals, Molecular Crystals, 7 (1969), 153-164.  doi: 10.1080/15421406908084869. [13] E. Feireisl, M. Frémond, E. Rocca and G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals, Arch. Rational Mech. Anal., 205 (2012), 651-672.  doi: 10.1007/s00205-012-0517-4. [14] E. Feireisl, E. Rocca and G. Schimperna, On a non-isothermal model for the nematic liquid crystals, Nonlinearity, 24 (2011), 243-257.  doi: 10.1088/0951-7715/24/1/012. [15] G. P. Galdi, An Introduction to the Mathematical Theory of the Nevier-Stokes Equations. Vol. I. Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8. [16] J. C. Gao, Q. Tao and Z.-A. Yao, Long-time behavior of solution for the compressible nematic liquid crystal flows in $\mathbb{R}^3$, J. Differential Equations, 261 (2016), 2334-2383.  doi: 10.1016/j.jde.2016.04.033. [17] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, New York: Oxford University Press, 1993. [18] B. L. Guo, X. Y. Xi and B. Q. Xie, Global well-posedness and decay of smooth solutions to the non-isothermal model for compressible nematic liquid crystals, J. Differential Equations, 262 (2017), 1413-1460.  doi: 10.1016/j.jde.2016.10.015. [19] M. Hieber, M. Nesensohn, J. Prüss and K. Schade, Dynamics of nematic liquid crystal flows: The quasilinear approach, Ann. Ints. H. Poincaré, Analyse Nonlinéaire, 33 (2016), 397-408.  doi: 10.1016/j.anihpc.2014.11.001. [20] M. Hieber and J. Prüss, Dynamics of the Ericksen-Leslie equations with general Leslie stress. I: The incompressible isotropic case, Math. Ann., 369 (2017), 977-996.  doi: 10.1007/s00208-016-1453-7. [21] M. Hieber and J. W. Prüss, Modeling and analysis of the Ericksen-Leslie equations for nematic liquid crystal flows, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2016), 1075–1134. [22] M.-C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36.  doi: 10.1007/s00526-010-0331-5. [23] M.-C. Hong, J. K. Li and Z. P. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in $\mathbb{R}^3$, Comm. Partial Differential Equations, 39 (2014), 1284-1328.  doi: 10.1080/03605302.2013.871026. [24] M.-C. Hong and Z. P. Xin, Global existence of solutions of the nematic liquid crystal flow for the Oseen-Frank model $\mathbb{R}^2$, Adv. Math., 231 (2012), 1364-1400.  doi: 10.1016/j.aim.2012.06.009. [25] X. P. Hu and D. H. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Commun. Math. Phys., 296 (2010), 861-880.  doi: 10.1007/s00220-010-1017-8. [26] X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crytals, SIAM J. Math. Anal., 45 (2013), 2678-2699.  doi: 10.1137/120898814. [27] J. R. Huang, F.-H. Lin and C. Y. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbb{R}^2$, Commun. Math. Phys., 331 (2014), 805-850.  doi: 10.1007/s00220-014-2079-9. [28] F. Jiang, S. Jiang and D. H. Wang, On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain, J. Funct. Anal., 265 (2013), 3369-3397.  doi: 10.1016/j.jfa.2013.07.026. [29] Z. Lei, D. Li and X. Y. Zhang, Remarks of global wellposedness of liquid crystal flows and heat flows of harmonic maps in tow dimensions, Proc. Amer. Math. Soc., 142 (2014), 3801-3810.  doi: 10.1090/S0002-9939-2014-12057-0. [30] F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810. [31] F. Leslie, Theory of Flow Phenomenum in Liquid Crystals, The Theory of Liquid Crystals, 4. London-New York, Academic Press, 1979, 1–81. [32] J. K. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flows in two dimensions, Nonlinear Anal., 99 (2014), 80-94.  doi: 10.1016/j.na.2013.12.023. [33] J. K. Li, E. S. Titi and Z. P. Xin, On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbb{R}^2$, Math. Models Meth. Appl. Sci., 26 (2016), 803-822.  doi: 10.1142/S0218202516500184. [34] J. K. Li and Z. P. Xin, Global existence of weak solutions to the non-isothermal nematic liquid crystals in 2D, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 973-1014.  doi: 10.1016/S0252-9602(16)30054-6. [35] F.-H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814.  doi: 10.1002/cpa.3160420605. [36] F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure. Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503. [37] F.-H. Lin and C. Liu, Partial regularity of the nonlinear dissipative system modeling the flow of nematic liquid crystals, Discrete Comtin. Dyn. Syst., 2 (1996), 1-22.  doi: 10.3934/dcds.1996.2.1. [38] F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336.  doi: 10.1007/s00205-009-0278-x. [39] F.-H. Lin and C. Y. 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. [40] F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532-1571.  doi: 10.1002/cpa.21583. [41] Q. Liu, S. Q. Liu, W. K. Tan and X. Zhong, Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows, J. Differential Equations, 261 (2016), 6521-6569.  doi: 10.1016/j.jde.2016.08.044. [42] P. Oswald and P. Pieranski, Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments, CRC Press, 2005.  doi: 10.1201/9780203023013. [43] V. A. Solonnikov, $L_p$-estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain, J. Math. Sci. (New York), 105 (2001), 2448-2484.  doi: 10.1023/A:1011321430954. [44] A. M. Sonnet and E. G. Virga, Dissipative Ordered Fluids: Theories for Liquid Crystals, Springer, New York, 2012. doi: 10.1007/978-0-387-87815-7. [45] M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Commun. Math. Helv., 60 (1985), 558-581.  doi: 10.1007/BF02567432. [46] C. Y. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Rational Mech. Anal., 200 (2011), 1-19.  doi: 10.1007/s00205-010-0343-5. [47] M. Wang and W. D. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Calc. Var. Partial Differential Equations, 51 (2014), 915-962.  doi: 10.1007/s00526-013-0700-y. [48] X. Xu and Z. F. Zhang, Global regularity and uniqueness of weak solutions for the 2-D liquid crystal flows, J. Differential Equations, 252 (2012), 1169-1181.  doi: 10.1016/j.jde.2011.08.028.

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##### References:
 [1] H. Abels, Nonstationary Stokes system with variable viscosity in bounded and unbounded domains, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 141-157.  doi: 10.3934/dcdss.2010.3.141. [2] H. Abels and Y. Terasawa, On Stokes operators with variable viscosity in bounded and unbounded domains, Math. Ann., 344 (2009), 381-429.  doi: 10.1007/s00208-008-0311-7. [3] R. A. Adams and J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. [4] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6. [5] D. Bian and Y. Xiao, Global solution to the nematic liquid crystal flows with heat effect, J. Differential Equations, 263 (2017), 5298-5329.  doi: 10.1016/j.jde.2017.06.019. [6] D. Bothe and J. Prüss, $L_p$-theory for a class of Non-Newtonian fluids, SIAM J. Math. Anal., 39 (2007), 379-421.  doi: 10.1137/060663635. [7] R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8 (2006), 333-381.  doi: 10.1007/s00021-004-0147-1. [8] R. Denk, M. Hieber and J. Prüss, R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003). doi: 10.1090/memo/0788. [9] W.-Y. Ding and F.-H. Lin, A generalization of Eells-Sampson's theorem, J. Partial Differential Equations, 5 (1992), 13-22. [10] S. J. Ding, C. Y. Wang and H. Y. Wen, Weak solutions to compressible flows of nematic liquid crystals in dimension one, Discrete Contin. Dyn. Syst. B, 15 (2011), 357-371.  doi: 10.3934/dcdsb.2011.15.357. [11] J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheology, 5 (1961), 23-34.  doi: 10.1122/1.548883. [12] J. L. Ericksen, Continuum theory of nematic liquid crystals, Molecular Crystals, 7 (1969), 153-164.  doi: 10.1080/15421406908084869. [13] E. Feireisl, M. Frémond, E. Rocca and G. Schimperna, A new approach to non-isothermal models for nematic liquid crystals, Arch. Rational Mech. Anal., 205 (2012), 651-672.  doi: 10.1007/s00205-012-0517-4. [14] E. Feireisl, E. Rocca and G. Schimperna, On a non-isothermal model for the nematic liquid crystals, Nonlinearity, 24 (2011), 243-257.  doi: 10.1088/0951-7715/24/1/012. [15] G. P. Galdi, An Introduction to the Mathematical Theory of the Nevier-Stokes Equations. Vol. I. Linearized Steady Problems, Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8. [16] J. C. Gao, Q. Tao and Z.-A. Yao, Long-time behavior of solution for the compressible nematic liquid crystal flows in $\mathbb{R}^3$, J. Differential Equations, 261 (2016), 2334-2383.  doi: 10.1016/j.jde.2016.04.033. [17] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, New York: Oxford University Press, 1993. [18] B. L. Guo, X. Y. Xi and B. Q. Xie, Global well-posedness and decay of smooth solutions to the non-isothermal model for compressible nematic liquid crystals, J. Differential Equations, 262 (2017), 1413-1460.  doi: 10.1016/j.jde.2016.10.015. [19] M. Hieber, M. Nesensohn, J. Prüss and K. Schade, Dynamics of nematic liquid crystal flows: The quasilinear approach, Ann. Ints. H. Poincaré, Analyse Nonlinéaire, 33 (2016), 397-408.  doi: 10.1016/j.anihpc.2014.11.001. [20] M. Hieber and J. Prüss, Dynamics of the Ericksen-Leslie equations with general Leslie stress. I: The incompressible isotropic case, Math. Ann., 369 (2017), 977-996.  doi: 10.1007/s00208-016-1453-7. [21] M. Hieber and J. W. Prüss, Modeling and analysis of the Ericksen-Leslie equations for nematic liquid crystal flows, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, (2016), 1075–1134. [22] M.-C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Calc. Var. Partial Differential Equations, 40 (2011), 15-36.  doi: 10.1007/s00526-010-0331-5. [23] M.-C. Hong, J. K. Li and Z. P. Xin, Blow-up criteria of strong solutions to the Ericksen-Leslie system in $\mathbb{R}^3$, Comm. Partial Differential Equations, 39 (2014), 1284-1328.  doi: 10.1080/03605302.2013.871026. [24] M.-C. Hong and Z. P. Xin, Global existence of solutions of the nematic liquid crystal flow for the Oseen-Frank model $\mathbb{R}^2$, Adv. Math., 231 (2012), 1364-1400.  doi: 10.1016/j.aim.2012.06.009. [25] X. P. Hu and D. H. Wang, Global solution to the three-dimensional incompressible flow of liquid crystals, Commun. Math. Phys., 296 (2010), 861-880.  doi: 10.1007/s00220-010-1017-8. [26] X. P. Hu and H. Wu, Global solution to the three-dimensional compressible flow of liquid crytals, SIAM J. Math. Anal., 45 (2013), 2678-2699.  doi: 10.1137/120898814. [27] J. R. Huang, F.-H. Lin and C. Y. Wang, Regularity and existence of global solutions to the Ericksen-Leslie system in $\mathbb{R}^2$, Commun. Math. Phys., 331 (2014), 805-850.  doi: 10.1007/s00220-014-2079-9. [28] F. Jiang, S. Jiang and D. H. Wang, On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain, J. Funct. Anal., 265 (2013), 3369-3397.  doi: 10.1016/j.jfa.2013.07.026. [29] Z. Lei, D. Li and X. Y. Zhang, Remarks of global wellposedness of liquid crystal flows and heat flows of harmonic maps in tow dimensions, Proc. Amer. Math. Soc., 142 (2014), 3801-3810.  doi: 10.1090/S0002-9939-2014-12057-0. [30] F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal., 28 (1968), 265-283.  doi: 10.1007/BF00251810. [31] F. Leslie, Theory of Flow Phenomenum in Liquid Crystals, The Theory of Liquid Crystals, 4. London-New York, Academic Press, 1979, 1–81. [32] J. K. Li, Global strong and weak solutions to inhomogeneous nematic liquid crystal flows in two dimensions, Nonlinear Anal., 99 (2014), 80-94.  doi: 10.1016/j.na.2013.12.023. [33] J. K. Li, E. S. Titi and Z. P. Xin, On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in $\mathbb{R}^2$, Math. Models Meth. Appl. Sci., 26 (2016), 803-822.  doi: 10.1142/S0218202516500184. [34] J. K. Li and Z. P. Xin, Global existence of weak solutions to the non-isothermal nematic liquid crystals in 2D, Acta Math. Sci. Ser. B Engl. Ed., 36 (2016), 973-1014.  doi: 10.1016/S0252-9602(16)30054-6. [35] F.-H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789-814.  doi: 10.1002/cpa.3160420605. [36] F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure. Appl. Math., 48 (1995), 501-537.  doi: 10.1002/cpa.3160480503. [37] F.-H. Lin and C. Liu, Partial regularity of the nonlinear dissipative system modeling the flow of nematic liquid crystals, Discrete Comtin. Dyn. Syst., 2 (1996), 1-22.  doi: 10.3934/dcds.1996.2.1. [38] F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flows in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336.  doi: 10.1007/s00205-009-0278-x. [39] F.-H. Lin and C. Y. 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. [40] F. H. Lin and C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimension three, Comm. Pure Appl. Math., 69 (2016), 1532-1571.  doi: 10.1002/cpa.21583. [41] Q. Liu, S. Q. Liu, W. K. Tan and X. Zhong, Global well-posedness of the 2D nonhomogeneous incompressible nematic liquid crystal flows, J. Differential Equations, 261 (2016), 6521-6569.  doi: 10.1016/j.jde.2016.08.044. [42] P. Oswald and P. Pieranski, Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments, CRC Press, 2005.  doi: 10.1201/9780203023013. [43] V. A. Solonnikov, $L_p$-estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain, J. Math. Sci. (New York), 105 (2001), 2448-2484.  doi: 10.1023/A:1011321430954. [44] A. M. Sonnet and E. G. Virga, Dissipative Ordered Fluids: Theories for Liquid Crystals, Springer, New York, 2012. doi: 10.1007/978-0-387-87815-7. [45] M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Commun. Math. Helv., 60 (1985), 558-581.  doi: 10.1007/BF02567432. [46] C. Y. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data, Arch. Rational Mech. Anal., 200 (2011), 1-19.  doi: 10.1007/s00205-010-0343-5. [47] M. Wang and W. D. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Calc. Var. Partial Differential Equations, 51 (2014), 915-962.  doi: 10.1007/s00526-013-0700-y. [48] X. Xu and Z. F. Zhang, Global regularity and uniqueness of weak solutions for the 2-D liquid crystal flows, J. Differential Equations, 252 (2012), 1169-1181.  doi: 10.1016/j.jde.2011.08.028.
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