# American Institute of Mathematical Sciences

January  2016, 36(1): 371-402. doi: 10.3934/dcds.2016.36.371

## 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

Received  September 2014 Revised  April 2015 Published  June 2015

In this paper, we consider the well-posedness of the Cauchy problem of the 3D incompressible nematic liquid crystal system with initial data in the critical Besov space $\dot{B}^{\frac{3}{p}-1}_{p,1}(\mathbb{R}^{3})\times \dot{B}^{\frac{3}{q}}_{q,1}(\mathbb{R}^{3})$ with $1< p<\infty$, $1\leq q<\infty$ and \begin{align*} -\min\{\frac{1}{3},\frac{1}{2p}\}\leq \frac{1}{q}-\frac{1}{p}\leq \frac{1}{3}. \end{align*} In particular, if we impose the restrictive condition $1< p<6$, we prove that there exist two positive constants $C_{0}$ and $c_{0}$ such that the nematic liquid crystal system has a unique global solution with initial data $(u_{0},d_{0}) = (u^{h}_{0}, u^{3}_{0}, d_{0})$ which satisfies \begin{align*} ((1+\frac{1}{\nu\mu})\|d_{0}-\overline{d}_{0}\|_{\dot{B}^{\frac{3}{q}}_{q,1}}+ \frac{1}{\nu}\|u_{0}^{h}\|_{\dot{B}^{\frac{3}{p}-1}_{p,1}}) \exp\left\{\frac{C_{0}}{\nu^{2}}(\|u_{0}^{3}\|_{\dot{B}^{\frac{3}{p}-1}_{p,1}}+\frac{1}{\mu})^{2}\right\}\leq c_{0}, \end{align*} where $\overline{d}_{0}$ is a constant vector with $|\overline{d}_{0}|=1$. Here $\nu$ and $\mu$ are two positive viscosity constants.
Citation: 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
##### References:
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Tataru, Well-posedness for the Navier-Stokes equations, Advances in Math., 157 (2001), 22-35. doi: 10.1006/aima.2000.1937.  Google Scholar [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. Google Scholar [18] P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall/CRC, 2002. doi: 10.1201/9781420035674.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] M. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoamericana, 21 (2005), 179-235. doi: 10.4171/RMI/420.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] W. Tan and Z. Yin, Global existence in critical space for liquid crystal flows in $\mathbbR^N$,, preprint., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [2] M. Cannone, Y. Meyer and F. Planchon, Solutions sutosimilaires éequations de Naveir-Stokes,, Séminaire Équations aux Dérivées Partielles de l'École Polytecnique, (): 1993.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] R. Danchin, Fourior Analysis Methods for PDE's, http://perso-math.univ-mlv.fr/users/danchin.raphael/courschine.pdf 2005. Google Scholar [7] J. L. Ericksen, Hydrostatic theory of liquid crystal, Arch. Rational Mech. Anal., 9 (1962), 371-378.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [18] P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall/CRC, 2002. doi: 10.1201/9781420035674.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] M. Paicu, Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoamericana, 21 (2005), 179-235. doi: 10.4171/RMI/420.  Google Scholar [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.  Google Scholar [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.  Google Scholar [32] W. Tan and Z. Yin, Global existence in critical space for liquid crystal flows in $\mathbbR^N$,, preprint., ().   Google Scholar [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.  Google Scholar [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.  Google Scholar
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