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Eulerian dynamics with a commutator forcing Ⅱ: Flocking
The 3D liquid crystal system with Cannone type initial data and large vertical velocity
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China |
The hydrodynamic theory of the nematic liquid crystals was established by Ericksen [
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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