American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Qiao Liu, Ting Zhang, Jihong Zhao. Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 371-402. doi: 10.3934/dcds.2016.36.371
[2]

Xiaoli Li, Boling Guo. Well-posedness for the three-dimensional compressible liquid crystal flows. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1913-1937. doi: 10.3934/dcdss.2016078
[3]

Stefano Bosia. Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2012, 11 (2) : 407-441. doi: 10.3934/cpaa.2012.11.407
[4]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161
[5]

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
[6]

Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259
[7]

Yoshihiro Shibata. Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface. Evolution Equations & Control Theory, 2018, 7 (1) : 117-152. doi: 10.3934/eect.2018007
[8]

Gustavo Ponce, Jean-Claude Saut. Well-posedness for the Benney-Roskes/Zakharov- Rubenchik system. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 811-825. doi: 10.3934/dcds.2005.13.811
[9]

Hung Luong. Local well-posedness for the Zakharov system on the background of a line soliton. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2657-2682. doi: 10.3934/cpaa.2018126
[10]

Manas Bhatnagar, Hailiang Liu. Well-posedness and critical thresholds in a nonlocal Euler system with relaxation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5271-5289. doi: 10.3934/dcds.2021076
[11]

Akansha Sanwal. Local well-posedness for the Zakharov system in dimension d ≤ 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021147
[12]

Xujie Yang. Global well-posedness in a chemotaxis system with oxygen consumption. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021184
[13]

Zhaoyang Qiu, Yixuan Wang. Martingale solution for stochastic active liquid crystal system. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2227-2268. doi: 10.3934/dcds.2020360
[14]

Haibo Cui, Qunyi Bie, Zheng-An Yao. Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1395-1410. doi: 10.3934/dcdsb.2018156
[15]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2699-2723. doi: 10.3934/dcds.2020382
[16]

Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341
[17]

Hartmut Pecher. Local well-posedness for the Klein-Gordon-Zakharov system in 3D. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1707-1736. doi: 10.3934/dcds.2020338
[18]

Jiawei Chen, Zhongping Wan, Liuyang Yuan. Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 567-581. doi: 10.3934/naco.2013.3.567
[19]

Michele Colturato. Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics. Evolution Equations & Control Theory, 2018, 7 (2) : 217-245. doi: 10.3934/eect.2018011
[20]

Qi Wang, Ling Jin, Zengyan Zhang. Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2105-2134. doi: 10.3934/dcds.2020108

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (172)
  • HTML views (73)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy