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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

Shengquan Liu 1, and  Jianwen Zhang 2,

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]).
Keywords: blowup criterion., nonhomogeneous incompressible flows, Liquid crystals, vacuum, global strong solutions.
Mathematics Subject Classification: Primary: 35B45, 76A15; Secondary: 76D03, 76D0.
Citation: Shengquan Liu, Jianwen Zhang. Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34. doi: 10.1122/1.548883.  Google Scholar

[10]

J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378.  Google Scholar

[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.  Google Scholar

[12]

P. G. de Gennes, The Physics of Liquid Crystals, $2^{nd}$ edition, Oxford University Press, Oxford, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids, $2^{nd}$ edition, Gordon and Breach, New York, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

F. M. Leslie, Some contitutive equations for liquid crystals, Arch Ration. Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1, Incompressible Models, Oxford Science Publication, Oxford, 1996.  Google Scholar

[28]

X. Liu and Z. Zhang, Existence of the flow of liquid crystal system, Chinese Ann. Math., 30 (2009), 1-20.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012.  Google Scholar

[32]

M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Comment. Math. Helv., 60 (1985), 558-581. doi: 10.1007/BF02567432.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Space, Academic Press, New York, 1975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5 (1961), 22-34. doi: 10.1122/1.548883.  Google Scholar

[10]

J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378.  Google Scholar

[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.  Google Scholar

[12]

P. G. de Gennes, The Physics of Liquid Crystals, $2^{nd}$ edition, Oxford University Press, Oxford, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Fluids, $2^{nd}$ edition, Gordon and Breach, New York, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

F. M. Leslie, Some contitutive equations for liquid crystals, Arch Ration. Mech. Anal., 28 (1968), 265-283. doi: 10.1007/BF00251810.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

P. L. Lions, Mathematical Topics in Fluid Dynamics, Vol. 1, Incompressible Models, Oxford Science Publication, Oxford, 1996.  Google Scholar

[28]

X. Liu and Z. Zhang, Existence of the flow of liquid crystal system, Chinese Ann. Math., 30 (2009), 1-20.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012.  Google Scholar

[32]

M. Struwe, On the evolution of harmonic maps of Riemannian surfaces, Comment. Math. Helv., 60 (1985), 558-581. doi: 10.1007/BF02567432.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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