# 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 - A, 2016, 36 (1) : 371-402. doi: 10.3934/dcds.2016.36.371
##### References:
 [1] H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der Mathematischen Wissenschaften, (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. 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. 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. doi: 10.1007/s00220-007-0236-0. Google Scholar [6] R. Danchin, Fourior Analysis Methods for PDE's,, , (2005). Google Scholar [7] J. L. Ericksen, Hydrostatic theory of liquid crystal,, Arch. Rational Mech. Anal., 9 (1962), 371. Google Scholar [8] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I,, Arch. Rational Mech. Anal., 16 (1964), 269. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1006/aima.2000.1937. Google Scholar [17] F. Leslie, Theory of flow phenomenum in liquid crystals., In The Theory of Liquid Crystals, 4 (1979), 1. 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. 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. 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. 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., 2 (1996), 1. Google Scholar [23] F. Lin, J. Lin and C. Wang, Liquid crystal flow in two dimensions,, Arch. Rational Mech. Anal., 197 (2010), 297. 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. 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,, , (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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00220-008-0631-1. Google Scholar

show all references

##### References:
 [1] H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der Mathematischen Wissenschaften, (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. 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. 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. doi: 10.1007/s00220-007-0236-0. Google Scholar [6] R. Danchin, Fourior Analysis Methods for PDE's,, , (2005). Google Scholar [7] J. L. Ericksen, Hydrostatic theory of liquid crystal,, Arch. Rational Mech. Anal., 9 (1962), 371. Google Scholar [8] H. Fujita and T. Kato, On the Navier-Stokes initial value problem I,, Arch. Rational Mech. Anal., 16 (1964), 269. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1006/aima.2000.1937. Google Scholar [17] F. Leslie, Theory of flow phenomenum in liquid crystals., In The Theory of Liquid Crystals, 4 (1979), 1. 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. 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. 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. 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., 2 (1996), 1. Google Scholar [23] F. Lin, J. Lin and C. Wang, Liquid crystal flow in two dimensions,, Arch. Rational Mech. Anal., 197 (2010), 297. 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. 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,, , (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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00220-008-0631-1. Google Scholar
 [1] 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 [2] 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 [3] Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517 [4] Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195 [5] Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143 [6] 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 [7] Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148 [8] Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101 [9] Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35 [10] Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845 [11] Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315 [12] 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 [13] Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455 [14] Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609 [15] Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437 [16] Francisco Guillén-González, Mouhamadou Samsidy Goudiaby. Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4229-4246. doi: 10.3934/dcds.2012.32.4229 [17] Yang Liu, Sining Zheng, Huapeng Li, Shengquan Liu. Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3921-3938. doi: 10.3934/dcds.2017165 [18] Yinxia Wang. A remark on blow up criterion of three-dimensional nematic liquid crystal flows. Evolution Equations & Control Theory, 2016, 5 (2) : 337-348. doi: 10.3934/eect.2016007 [19] Hao Wu. Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 379-396. doi: 10.3934/dcds.2010.26.379 [20] M. Gregory Forest, Hongyun Wang, Hong Zhou. Sheared nematic liquid crystal polymer monolayers. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 497-517. doi: 10.3934/dcdsb.2009.11.497

2018 Impact Factor: 1.143