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Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework
1. | Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081 |
2. | Department of Mathematics, Zhejiang University, Hangzhou 310027 |
3. | College of Science, Northwest A&F University, Yangling, Shaanxi 712100 |
References:
[1] |
H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Berlin-Heidelberg: Springer, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
M. Cannone, Y. Meyer and F. Planchon, Solutions sutosimilaires éequations de Naveir-Stokes, Séminaire Équations aux Dérivées Partielles de l'École Polytecnique, 1993-1994. |
[3] |
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. |
[4] |
J. Y. Chemin and N. Lerner, Flot de damps de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differ. Equ., 121 (1995), 314-328.
doi: 10.1006/jdeq.1995.1131. |
[5] |
J. Y. Chemin and P. Zhang, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Commun. Math. Phys., 272 (2007), 529-566.
doi: 10.1007/s00220-007-0236-0. |
[6] |
R. Danchin, Fourior Analysis Methods for PDE's, http://perso-math.univ-mlv.fr/users/danchin.raphael/courschine.pdf 2005. |
[7] |
J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. |
[8] |
H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal., 16 (1964), 269-351.
doi: 10.1007/BF00276188. |
[9] |
G. Gui and P. Zhang, Stability to the global large solutions of 3-D Navier-Stokes equations, Advances in Math., 225 (2010), 1248-1284.
doi: 10.1016/j.aim.2010.03.022. |
[10] |
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, arXiv:1305.1395v1. |
[11] |
J. Hineman and C. Wang, Well-posedness of nematic liquid crystal flow in $L_{loc}^3 (\mathbbR^{3})$, Arch. Rational Mech. Anal., 210 (2013), 177-218.
doi: 10.1007/s00205-013-0643-7. |
[12] |
M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Cal. Var., 40 (2011), 15-36.
doi: 10.1007/s00526-010-0331-5. |
[13] |
J. Huang, M. Paicu and P. Zhang, Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-lipschitz velocity, Arch. Rational Mech. Anal., 209 (2013), 631-682.
doi: 10.1007/s00205-013-0624-x. |
[14] |
T. Huang and C. Wang, Blow up criterion for nematic liquid crystal flows, Comm. Partial Differ. Equ., 37 (2012), 875-884.
doi: 10.1080/03605302.2012.659366. |
[15] |
T. Kato, Strong $L^p$-solutions of the Navier-Stokes equations in $R^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[16] |
H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Advances in Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[17] |
F. Leslie, Theory of flow phenomenum in liquid crystals. In The Theory of Liquid Crystals, London-New York: Academic Press, 4 (1979), 1-81. |
[18] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall/CRC, 2002.
doi: 10.1201/9781420035674. |
[19] |
X. Li and D. Wang, Global solution to the incompressible flow of liquid crystal, J. Differ. Equ., 252 (2012), 745-767.
doi: 10.1016/j.jde.2011.08.045. |
[20] |
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. |
[21] |
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. |
[22] |
F. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Disc. Contin. Dyn. Syst., A, 2 (1996), 1-23. |
[23] |
F. Lin, J. Lin and C. Wang, Liquid crystal flow in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336.
doi: 10.1007/s00205-009-0278-x. |
[24] |
F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chinese Annal. Math., 31 (2010), 921-938.
doi: 10.1007/s11401-010-0612-5. |
[25] |
F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three, arXiv:1408.4146v1 [math.AP] 18 Aug, 2014.
doi: 10.1002/cpa.21583. |
[26] |
J. Lin and S. Ding, On the well-posedness for the heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals in critical spaces, Math. Meth. Appl. Sci., 35 (2012), 158-173.
doi: 10.1002/mma.1548. |
[27] |
Q. Liu and J. Zhao, A regularity criterion for the solution of the nematic liquid crystal flows in terms of $\dotB_{\infty,\infty}^{-1}$-norm, J. Math. Anal. Appl., 407 (2013), 557-566.
doi: 10.1016/j.jmaa.2013.05.048. |
[28] |
Q. Liu, T. Zhang and J. Zhao, Global solutions to the 3D incompressible nematic liquid crystal system, J. Differ. Equ., 258 (2015), 1519-1547.
doi: 10.1016/j.jde.2014.11.002. |
[29] |
M. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoamericana, 21 (2005), 179-235.
doi: 10.4171/RMI/420. |
[30] |
M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Commun. Math. Phys., 307 (2011), 713-759.
doi: 10.1007/s00220-011-1350-6. |
[31] |
M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.
doi: 10.1016/j.jfa.2012.01.022. |
[32] |
W. Tan and Z. Yin, Global existence in critical space for liquid crystal flows in $\mathbb{R}^N2$, preprint. |
[33] |
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. |
[34] |
T. Zhang, Global wellposedness problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Commun. Math. Phys., 287 (2009), 211-224.
doi: 10.1007/s00220-008-0631-1. |
show all references
References:
[1] |
H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, 343. Berlin-Heidelberg: Springer, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
M. Cannone, Y. Meyer and F. Planchon, Solutions sutosimilaires éequations de Naveir-Stokes, Séminaire Équations aux Dérivées Partielles de l'École Polytecnique, 1993-1994. |
[3] |
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. |
[4] |
J. Y. Chemin and N. Lerner, Flot de damps de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differ. Equ., 121 (1995), 314-328.
doi: 10.1006/jdeq.1995.1131. |
[5] |
J. Y. Chemin and P. Zhang, On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Commun. Math. Phys., 272 (2007), 529-566.
doi: 10.1007/s00220-007-0236-0. |
[6] |
R. Danchin, Fourior Analysis Methods for PDE's, http://perso-math.univ-mlv.fr/users/danchin.raphael/courschine.pdf 2005. |
[7] |
J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378. |
[8] |
H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Rational Mech. Anal., 16 (1964), 269-351.
doi: 10.1007/BF00276188. |
[9] |
G. Gui and P. Zhang, Stability to the global large solutions of 3-D Navier-Stokes equations, Advances in Math., 225 (2010), 1248-1284.
doi: 10.1016/j.aim.2010.03.022. |
[10] |
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, arXiv:1305.1395v1. |
[11] |
J. Hineman and C. Wang, Well-posedness of nematic liquid crystal flow in $L_{loc}^3 (\mathbbR^{3})$, Arch. Rational Mech. Anal., 210 (2013), 177-218.
doi: 10.1007/s00205-013-0643-7. |
[12] |
M. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two, Cal. Var., 40 (2011), 15-36.
doi: 10.1007/s00526-010-0331-5. |
[13] |
J. Huang, M. Paicu and P. Zhang, Global well-posedness of incompressible inhomogeneous fluid systems with bounded density or non-lipschitz velocity, Arch. Rational Mech. Anal., 209 (2013), 631-682.
doi: 10.1007/s00205-013-0624-x. |
[14] |
T. Huang and C. Wang, Blow up criterion for nematic liquid crystal flows, Comm. Partial Differ. Equ., 37 (2012), 875-884.
doi: 10.1080/03605302.2012.659366. |
[15] |
T. Kato, Strong $L^p$-solutions of the Navier-Stokes equations in $R^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[16] |
H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Advances in Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[17] |
F. Leslie, Theory of flow phenomenum in liquid crystals. In The Theory of Liquid Crystals, London-New York: Academic Press, 4 (1979), 1-81. |
[18] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall/CRC, 2002.
doi: 10.1201/9781420035674. |
[19] |
X. Li and D. Wang, Global solution to the incompressible flow of liquid crystal, J. Differ. Equ., 252 (2012), 745-767.
doi: 10.1016/j.jde.2011.08.045. |
[20] |
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. |
[21] |
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. |
[22] |
F. Lin and C. Liu, Partial regularities of the nonlinear dissipative systems modeling the flow of liquid crystals, Disc. Contin. Dyn. Syst., A, 2 (1996), 1-23. |
[23] |
F. Lin, J. Lin and C. Wang, Liquid crystal flow in two dimensions, Arch. Rational Mech. Anal., 197 (2010), 297-336.
doi: 10.1007/s00205-009-0278-x. |
[24] |
F. Lin and C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chinese Annal. Math., 31 (2010), 921-938.
doi: 10.1007/s11401-010-0612-5. |
[25] |
F. Lin and C. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three, arXiv:1408.4146v1 [math.AP] 18 Aug, 2014.
doi: 10.1002/cpa.21583. |
[26] |
J. Lin and S. Ding, On the well-posedness for the heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals in critical spaces, Math. Meth. Appl. Sci., 35 (2012), 158-173.
doi: 10.1002/mma.1548. |
[27] |
Q. Liu and J. Zhao, A regularity criterion for the solution of the nematic liquid crystal flows in terms of $\dotB_{\infty,\infty}^{-1}$-norm, J. Math. Anal. Appl., 407 (2013), 557-566.
doi: 10.1016/j.jmaa.2013.05.048. |
[28] |
Q. Liu, T. Zhang and J. Zhao, Global solutions to the 3D incompressible nematic liquid crystal system, J. Differ. Equ., 258 (2015), 1519-1547.
doi: 10.1016/j.jde.2014.11.002. |
[29] |
M. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoamericana, 21 (2005), 179-235.
doi: 10.4171/RMI/420. |
[30] |
M. Paicu and P. Zhang, Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces, Commun. Math. Phys., 307 (2011), 713-759.
doi: 10.1007/s00220-011-1350-6. |
[31] |
M. Paicu and P. Zhang, Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system, J. Funct. Anal., 262 (2012), 3556-3584.
doi: 10.1016/j.jfa.2012.01.022. |
[32] |
W. Tan and Z. Yin, Global existence in critical space for liquid crystal flows in $\mathbb{R}^N2$, preprint. |
[33] |
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. |
[34] |
T. Zhang, Global wellposedness problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space, Commun. Math. Phys., 287 (2009), 211-224.
doi: 10.1007/s00220-008-0631-1. |
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