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November  2017, 37(11): 5521-5539. doi: 10.3934/dcds.2017240

The 3D liquid crystal system with Cannone type initial data and large vertical velocity

Renhui Wan

School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China

Received  February 2017 Revised  June 2017 Published  July 2017

The hydrodynamic theory of the nematic liquid crystals was established by Ericksen [4] and Leslie [8]. In this paper, based on a new technique, we obtain global well-posedness to a simplified model introduced by Lin [9] in the critical Besov space with Cannone type initial data and large vertical velocity, which improves the main result in [15]. In addition, the small condition on $u_0$ is independent of another small condition on $d_0-\bar{d}_0$, which is quite different from the previous works [15,16].

Keywords: Liquid crystal system, Cannone, well-posedness.
Mathematics Subject Classification: Primary: 76A15; Secondary: 35B65.
Citation: Renhui Wan. The 3D liquid crystal system with Cannone type initial data and large vertical velocity. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5521-5539. doi: 10.3934/dcds.2017240
References:
[1]

H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, in: Grundlehren der mathematischen Wissenschaften, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoam., 13 (1997), 515-541.  doi: 10.4171/RMI/229.

[3]

R. Danchin, Local theory in critical spaces for the compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.  doi: 10.1081/PDE-100106132.

[4]

J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378.  doi: 10.1007/BF00253358.

[5]

Y. Hao and X. Liu, The existence and blow-up criterion of liquid crystals system in critical Besov space, Commun. Pure Appl. Anal., 13 (2014), 225-236. 

[6]

M. 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.

[7]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.

[8]

F. Leslie, Theory of flow phenomenum in liquid crystals. In The Theory of Liquid Crystals, London-New York: Academic Press, 4 (1979), 1–81.

[9]

F. 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.

[10]

F. 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.

[11]

F. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst. A, 2 (1996), 1-22. 

[12]

F. Lin, J. Lin and C. Wang, Liquid crystal flow in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.  doi: 10.1007/s00205-009-0278-x.

[13]

F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chinese Ann. Math., 31 (2010), 921-938.  doi: 10.1007/s11401-010-0612-5.

[14]

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), 1532-1571.  doi: 10.1002/cpa.21583.

[15]

Q. Liu, T. Zhang and J. Zhao, Global solutions to the 3D incompressible nematic liquid crystal system, J. Differential Equations, 258 (2015), 1519-1547.  doi: 10.1016/j.jde.2014.11.002.

[16]

Q. Liu, T. Zhang and J. Zhao, Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework, Discrete Contin. Dyn. Syst., 36 (2016), 371-402.  doi: 10.3934/dcds.2016.36.371.

[17]

C. 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.

[18]

C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity, J. Math. Phys., 56 (2015), 091512, 18pp. doi: 10.1063/1.4931467.

show all references

References:
[1]

H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, in: Grundlehren der mathematischen Wissenschaften, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.

[2]

M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoam., 13 (1997), 515-541.  doi: 10.4171/RMI/229.

[3]

R. Danchin, Local theory in critical spaces for the compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.  doi: 10.1081/PDE-100106132.

[4]

J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378.  doi: 10.1007/BF00253358.

[5]

Y. Hao and X. Liu, The existence and blow-up criterion of liquid crystals system in critical Besov space, Commun. Pure Appl. Anal., 13 (2014), 225-236. 

[6]

M. 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.

[7]

H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.

[8]

F. Leslie, Theory of flow phenomenum in liquid crystals. In The Theory of Liquid Crystals, London-New York: Academic Press, 4 (1979), 1–81.

[9]

F. 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.

[10]

F. 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.

[11]

F. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Discrete Contin. Dyn. Syst. A, 2 (1996), 1-22. 

[12]

F. Lin, J. Lin and C. Wang, Liquid crystal flow in two dimensions, Arch. Ration. Mech. Anal., 197 (2010), 297-336.  doi: 10.1007/s00205-009-0278-x.

[13]

F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chinese Ann. Math., 31 (2010), 921-938.  doi: 10.1007/s11401-010-0612-5.

[14]

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), 1532-1571.  doi: 10.1002/cpa.21583.

[15]

Q. Liu, T. Zhang and J. Zhao, Global solutions to the 3D incompressible nematic liquid crystal system, J. Differential Equations, 258 (2015), 1519-1547.  doi: 10.1016/j.jde.2014.11.002.

[16]

Q. Liu, T. Zhang and J. Zhao, Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework, Discrete Contin. Dyn. Syst., 36 (2016), 371-402.  doi: 10.3934/dcds.2016.36.371.

[17]

C. 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.

[18]

C. Zhai and T. Zhang, Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity, J. Math. Phys., 56 (2015), 091512, 18pp. doi: 10.1063/1.4931467.

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