# American Institute of Mathematical Sciences

October  2016, 21(8): 2631-2648. doi: 10.3934/dcdsb.2016065

## Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density

 1 Institute of Applied Physics and Computational Mathematics, Beijing 100088, China 2 School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received  January 2013 Revised  May 2016 Published  September 2016

In this paper, we first establish the global well-posedness of strong solutions of the simplified Ericksen-Leslie model for nonhomogeneous incompressible nematic liquid crystal flows in dimensions two, if the initial data satisfies some smallness condition. It is worth pointing out that the initial density is allowed to contain vacuum states and the initial velocity can be arbitrarily large. Next, we present a Serrin's type criterion, depending only on $\nabla d$, for the breakdown of local strong solutions. As a byproduct, the global strong solutions with large initial data are obtained, provided the macroscopic molecular orientation of the liquid crystal materials satisfies a natural geometric angle condition (cf. [19]).
Citation: Shengquan Liu, Jianwen Zhang. Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2631-2648. doi: 10.3934/dcdsb.2016065
##### References:
 [1] R. A. Adams, Sobolev Space, Academic Press, New York, 1975. [2] H. Brezis and S. Wainger, A note on the limiting cases of Sobolev embedding and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154. [3] K. Chang, Heat flow and boudnary value problem for harmonic maps, Annales de l'Institut Henri Poincaré (C) Analyse non éarire, 6 (1989), 363-395. [4] K. Chang, W. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differ. Geom., 36 (1992), 507-515. [5] Y. Chen and W. Ding, Blow-up and global existence for heat flows of harmonic maps, Invent. Math., 99 (1990), 567-578. doi: 10.1007/BF01234431. [6] Y. Chen and M. Struwe, Existence and partial regularity for heat flow for harmonic maps, Math. Z., 201 (1989), 83-103. doi: 10.1007/BF01161997. [7] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201. doi: 10.1081/PDE-120021191. [8] M. M. Dai, J. Qing and M. Schonbek, Regularity of solutions to the liquid crystals systems in $R^2$ and $R^3$, Nonlinearity, 25 (2012), 513-532. doi: 10.1088/0951-7715/25/2/513. [9] J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34. doi: 10.1122/1.548883. [10] J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. [11] G. P. Galdi, An introduction to the Mathematical Theory of Navier-Stokes Equations, Vol. I: Linearized Steady Problems, Springer Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8. [12] P. G. de Gennes, The Physics of Liquid Crystals, $2^{nd}$ edition, Oxford University Press, Oxford, 1995. [13] 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. [14] T. Huang and C. Y. Wang, Blow up criterion for nematic liquid crystal flows, Commun. Partial Differntial Equations, 37 (2012), 875-884. doi: 10.1080/03605302.2012.659366. [15] X. D. Huang and Y. Wang, Global strong solution with vacuum to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527. doi: 10.1016/j.jde.2012.08.029. [16] F. Jiang and Z. Tan, Global weak soutions to the flow of liquid crystals system, Math. Methods Appl. Sci., 32 (2009), 2243-2266. doi: 10.1002/mma.1132. [17] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids, $2^{nd}$ edition, Gordon and Breach, New York, 1969. [18] O. A. Ladyzhenskaya and T. N. Shilkin, On coercive estimates for solutions of linear systems of hydrodynamic type, J. Math. Sci., 123 (2004), 4580-4596. doi: 10.1023/B:JOTH.0000041476.69749.31. [19] Z. Lei, D. Li and X. Y. Zhang, Remarks of global well posedness of liquid crystal flows and heat flows of harmonic maps in two dimensions, Proc. Amer. Math. Soc., 142 (2014), 3801-3810. doi: 10.1090/S0002-9939-2014-12057-0. [20] F. M. Leslie, Some contitutive equations for liquid crystals, Arch Ration. Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810. [21] F. M. Leslie, Theory of flow phenomena in liquid crystals, Advances in Liquid Crystals, 4 (1979), 1-81. doi: 10.1016/B978-0-12-025004-2.50008-9. [22] X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals, J. Differential Equations, 252 (2012), 745-767. doi: 10.1016/j.jde.2011.08.045. [23] 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. [24] F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flow in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. [25] 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. [26] F. H. Lin and C. Liu, Partial regularity of the nonlinear dissipative system modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 2 (1996), 1-22. [27] P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1, Incompressible Models, Oxford Science Publication, Oxford, 1996. [28] X. Liu and Z. Zhang, Existence of the flow of liquid crystal system, Chinese Ann. Math., 30 (2009), 1-20. [29] H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251-278. doi: 10.1007/s002090100332. [30] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7. [31] T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012. [32] M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Comment. Math. Helv., 60 (1985), 558-581. doi: 10.1007/BF02567432. [33] M. Wang and W. D. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Cal. Var. Partial Differ. Equ., 51 (2014), 915-962. doi: 10.1007/s00526-013-0700-y. [34] W. Wang, P. W. Zhang and Z. F. Zhang, Well-Posedness of the Ericksen-Leslie System, Arch. Rational Mech. Anal., 210 (2013), 837-855. doi: 10.1007/s00205-013-0659-z. [35] C. Y. Wang, Well-posedeness 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. [36] H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. RWA, 12 (2011), 1510-1531. doi: 10.1016/j.nonrwa.2010.10.010. [37] X. Xu and Z. F. Zhang, Global regularity and uniqueness of weak solution 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] R. A. Adams, Sobolev Space, Academic Press, New York, 1975. [2] H. Brezis and S. Wainger, A note on the limiting cases of Sobolev embedding and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789. doi: 10.1080/03605308008820154. [3] K. Chang, Heat flow and boudnary value problem for harmonic maps, Annales de l'Institut Henri Poincaré (C) Analyse non éarire, 6 (1989), 363-395. [4] K. Chang, W. Ding and R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differ. Geom., 36 (1992), 507-515. [5] Y. Chen and W. Ding, Blow-up and global existence for heat flows of harmonic maps, Invent. Math., 99 (1990), 567-578. doi: 10.1007/BF01234431. [6] Y. Chen and M. Struwe, Existence and partial regularity for heat flow for harmonic maps, Math. Z., 201 (1989), 83-103. doi: 10.1007/BF01161997. [7] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201. doi: 10.1081/PDE-120021191. [8] M. M. Dai, J. Qing and M. Schonbek, Regularity of solutions to the liquid crystals systems in $R^2$ and $R^3$, Nonlinearity, 25 (2012), 513-532. doi: 10.1088/0951-7715/25/2/513. [9] J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34. doi: 10.1122/1.548883. [10] J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. [11] G. P. Galdi, An introduction to the Mathematical Theory of Navier-Stokes Equations, Vol. I: Linearized Steady Problems, Springer Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8. [12] P. G. de Gennes, The Physics of Liquid Crystals, $2^{nd}$ edition, Oxford University Press, Oxford, 1995. [13] 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. [14] T. Huang and C. Y. Wang, Blow up criterion for nematic liquid crystal flows, Commun. Partial Differntial Equations, 37 (2012), 875-884. doi: 10.1080/03605302.2012.659366. [15] X. D. Huang and Y. Wang, Global strong solution with vacuum to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527. doi: 10.1016/j.jde.2012.08.029. [16] F. Jiang and Z. Tan, Global weak soutions to the flow of liquid crystals system, Math. Methods Appl. Sci., 32 (2009), 2243-2266. doi: 10.1002/mma.1132. [17] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids, $2^{nd}$ edition, Gordon and Breach, New York, 1969. [18] O. A. Ladyzhenskaya and T. N. Shilkin, On coercive estimates for solutions of linear systems of hydrodynamic type, J. Math. Sci., 123 (2004), 4580-4596. doi: 10.1023/B:JOTH.0000041476.69749.31. [19] Z. Lei, D. Li and X. Y. Zhang, Remarks of global well posedness of liquid crystal flows and heat flows of harmonic maps in two dimensions, Proc. Amer. Math. Soc., 142 (2014), 3801-3810. doi: 10.1090/S0002-9939-2014-12057-0. [20] F. M. Leslie, Some contitutive equations for liquid crystals, Arch Ration. Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810. [21] F. M. Leslie, Theory of flow phenomena in liquid crystals, Advances in Liquid Crystals, 4 (1979), 1-81. doi: 10.1016/B978-0-12-025004-2.50008-9. [22] X. L. Li and D. H. Wang, Global solution to the incompressible flow of liquid crystals, J. Differential Equations, 252 (2012), 745-767. doi: 10.1016/j.jde.2011.08.045. [23] 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. [24] F. H. Lin, J. Y. Lin and C. Y. Wang, Liquid crystal flow in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336. doi: 10.1007/s00205-009-0278-x. [25] 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. [26] F. H. Lin and C. Liu, Partial regularity of the nonlinear dissipative system modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst., 2 (1996), 1-22. [27] P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1, Incompressible Models, Oxford Science Publication, Oxford, 1996. [28] X. Liu and Z. Zhang, Existence of the flow of liquid crystal system, Chinese Ann. Math., 30 (2009), 1-20. [29] H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251-278. doi: 10.1007/s002090100332. [30] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7. [31] T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012. [32] M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Comment. Math. Helv., 60 (1985), 558-581. doi: 10.1007/BF02567432. [33] M. Wang and W. D. Wang, Global existence of weak solution for the 2-D Ericksen-Leslie system, Cal. Var. Partial Differ. Equ., 51 (2014), 915-962. doi: 10.1007/s00526-013-0700-y. [34] W. Wang, P. W. Zhang and Z. F. Zhang, Well-Posedness of the Ericksen-Leslie System, Arch. Rational Mech. Anal., 210 (2013), 837-855. doi: 10.1007/s00205-013-0659-z. [35] C. Y. Wang, Well-posedeness 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. [36] H. Y. Wen and S. J. Ding, Solutions of incompressible hydrodynamic flow of liquid crystals, Nonlinear Anal. RWA, 12 (2011), 1510-1531. doi: 10.1016/j.nonrwa.2010.10.010. [37] X. Xu and Z. F. Zhang, Global regularity and uniqueness of weak solution 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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