November  2018, 23(9): 3879-3899. doi: 10.3934/dcdsb.2018115

On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals

1. 

NYU Shanghai, 1555 Century Avenue, Shanghai 200122, China

2. 

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China

* Corresponding author: W. Wang

Received  April 2017 Revised  December 2017 Published  November 2018 Early access  April 2018

Fund Project: Y. Liu is supported by NSF of China under Grant 11601334. W. Wang is supported by NSF of China under Grant 11501502 and "the Fundamental Research Funds for the Central Universities" 2016QNA3004.

This work is concerned with the solvability of a Navier-Stokes/Q-tensor coupled system modeling the nematic liquid crystal flow on a bounded domain in three dimensional Euclidian space with strong anchoring boundary condition for the order parameter. We prove the existence and uniqueness of local in time strong solutions to the system with an anisotropic elastic energy. The proof is based on mainly two ingredients: first, we show that the Euler-Lagrange operator corresponding to the Landau-de Gennes free energy with general elastic coefficients fulfills the strong Legendre condition. This result together with a higher order energy estimate leads to the well-posedness of the linearized system, and then a local in time solution of the original system which is regular in temporal variable follows via a fixed point argument. Secondly, the hydrodynamic part of the coupled system can be reformulated into a quasi-stationary Stokes type equation to which the regularity theory of the generalized Stokes system, and then a bootstrap argument can be applied to enhance the spatial regularity of the local in time solution.

Citation: Yuning Liu, Wei Wang. On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3879-3899. doi: 10.3934/dcdsb.2018115
References:
[1]

H. AbelsG. Dolzmann and Y. Liu, Well-posedness of a fully coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data, SIAM J. Math. Anal., 46 (2014), 3050-3077.  doi: 10.1137/130945405.

[2]

H. AbelsG. Dolzmann and Y. Liu, Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous dirichlet boundary conditions, Advances in Differential Equations, 21 (2016), 109-152. 

[3]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅱ, Comm. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160170104.

[4]

J. M. Ball and A. Majumdar, Nematic liquid crystals: From Maier-Saupe to a continuum theory, Molecular Crystals and Liquid Crystals, 525 (2010), 1-11. 

[5]

P. BaumanJ. Park and D. Phillips, Analysis of nematic liquid crystals with disclination lines, Arch. Ration. Mech. Anal., 205 (2012), 795-826.  doi: 10.1007/s00205-012-0530-7.

[6]

A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems, volume 36 of The Oxford Engineering Sciences Series. Oxford University Press, 1994.

[7]

Y. Y. CaiJ. Shen and X. Xu, A stable scheme and its convergence analysis for a 2D dynamic Qtensor model of nematic liquid crystals, Math. Models Methods Appl. Sci., 27 (2017), 1459-1488.  doi: 10.1142/S0218202517500245.

[8]

C. CavaterraE. RoccaH. Wu and X. Xu, Global strong solutions of the full Navier-Stokes and Q-tensor system for nematic liquid crystal flows in two dimensions, SIAM J. Math. Anal., 48 (2016), 1368-1399.  doi: 10.1137/15M1048550.

[9]

X. Chen and X. Xu, Existence and uniqueness of global classical solutions of a gradient flow of the Landau-de Gennes energy, Proceedings of the American Mathematical Society, 144 (2016), 1251-1263. 

[10]

Y. -Z. Chen and L. -C. Wu, Second Order Elliptic Equations and Elliptic Systems, volume 174 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1998. Translated from the 1991 Chinese original by Bei Hu.

[11]

M. DaiE. FeireislE. RoccaG. Schimperna and M. Schonbek, On asymptotic isotropy for a hydrodynamic model of liquid crystals, Asymptotic Analysis, 97 (2016), 189-210.  doi: 10.3233/ASY-151348.

[12]

F. De Anna, A global 2D well-posedness result on the order tensor liquid crystal theory, J. Differential Equations, 262 (2017), 3932-3979.  doi: 10.1016/j.jde.2016.12.006.

[13]

F. De Anna and A. Zarnescu, Uniqueness of weak solutions of the full coupled Navier-Stokes and Q-tensor system in 2D, Commun. Math. Sci., 14 (2016), 2127-2178.  doi: 10.4310/CMS.2016.v14.n8.a3.

[14]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, International Series of Monographs on Physics. Oxford University Press, Incorporated, 2nd edition, 1995.

[15]

E. FeireislE. RoccaG. Schimperna and A. Zarnescu, Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential, Comm. Math. Sci., 12 (2014), 317-343.  doi: 10.4310/CMS.2014.v12.n2.a6.

[16]

E. FeireislG. SchimpernaE. Rocca and A. Zarnescu, Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy, Annali di Matematica Pura ed Applicata (1923-), 194 (2015), 1269-1299.  doi: 10.1007/s10231-014-0419-1.

[17]

F. Guillén-González and M. Ángeles RodrÍguez-Bellido, Weak time regularity and uniqueness for a Q-tensor model, SIAM J. Math. Anal., 46 (2014), 3540-3567.  doi: 10.1137/13095015X.

[18]

F. Guillén-González and M. Á. Rodríguez-Bellido, Weak solutions for an initial-boundary Q-tensor problem related to liquid crystals, Nonlinear Analysis: Theory, Methods & Applications, 112 (2015), 84-104.  doi: 10.1016/j.na.2014.09.011.

[19]

J. Huang and S. Ding, Global well-posedness for the dynamical Q-tensor model of liquid crystals, Sci. China Math., 58 (2015), 1349-1366.  doi: 10.1007/s11425-015-4990-8.

[20]

G. IyerX. Xu and A. D. Zarnescu, Dynamic cubic instability in a 2D Q-tensor model for liquid crystals, Math. Models Methods Appl. Sci., 25 (2015), 1477-1517.  doi: 10.1142/S0218202515500396.

[21]

M. Paicu and A. Zarnescu, Global existence and regularity for the full coupled Navier-Stokes and Q-tensor system, SIAM J. Math. Anal., 43 (2011), 2009-2049.  doi: 10.1137/10079224X.

[22]

M. Paicu and A. Zarnescu, Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system, Arch. Ration. Mech. Anal., 203 (2012), 45-67.  doi: 10.1007/s00205-011-0443-x.

[23]

H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Birkhäuser Verlag, Basel, 2001.

[24]

V. A. Solonnikov, Lp-estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain, J. Math. Sci. (New York), 105 (2001), 2448-2484. Function theory and partial differential equations. doi: 10.1023/A:1011321430954.

[25]

W. WangP. Zhang and Z. Zhang, Rigorous derivation from Landau-de Gennes theory to Ericksen-Leslie theory, SIAM J. Math. Anal., 47 (2015), 127-158.  doi: 10.1137/13093529X.

[26]

M. Wilkinson, Strictly physical global weak solutions of a Navier-Stokes Q-tensor system with singular potential, Arch. Ration. Mech. Anal., 218 (2015), 487-526.  doi: 10.1007/s00205-015-0864-z.

[27]

J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987. Translated from the German by C. B. Thomas and M. J. Thomas.

show all references

References:
[1]

H. AbelsG. Dolzmann and Y. Liu, Well-posedness of a fully coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data, SIAM J. Math. Anal., 46 (2014), 3050-3077.  doi: 10.1137/130945405.

[2]

H. AbelsG. Dolzmann and Y. Liu, Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous dirichlet boundary conditions, Advances in Differential Equations, 21 (2016), 109-152. 

[3]

S. AgmonA. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. Ⅱ, Comm. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160170104.

[4]

J. M. Ball and A. Majumdar, Nematic liquid crystals: From Maier-Saupe to a continuum theory, Molecular Crystals and Liquid Crystals, 525 (2010), 1-11. 

[5]

P. BaumanJ. Park and D. Phillips, Analysis of nematic liquid crystals with disclination lines, Arch. Ration. Mech. Anal., 205 (2012), 795-826.  doi: 10.1007/s00205-012-0530-7.

[6]

A. N. Beris and B. J. Edwards, Thermodynamics of Flowing Systems, volume 36 of The Oxford Engineering Sciences Series. Oxford University Press, 1994.

[7]

Y. Y. CaiJ. Shen and X. Xu, A stable scheme and its convergence analysis for a 2D dynamic Qtensor model of nematic liquid crystals, Math. Models Methods Appl. Sci., 27 (2017), 1459-1488.  doi: 10.1142/S0218202517500245.

[8]

C. CavaterraE. RoccaH. Wu and X. Xu, Global strong solutions of the full Navier-Stokes and Q-tensor system for nematic liquid crystal flows in two dimensions, SIAM J. Math. Anal., 48 (2016), 1368-1399.  doi: 10.1137/15M1048550.

[9]

X. Chen and X. Xu, Existence and uniqueness of global classical solutions of a gradient flow of the Landau-de Gennes energy, Proceedings of the American Mathematical Society, 144 (2016), 1251-1263. 

[10]

Y. -Z. Chen and L. -C. Wu, Second Order Elliptic Equations and Elliptic Systems, volume 174 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1998. Translated from the 1991 Chinese original by Bei Hu.

[11]

M. DaiE. FeireislE. RoccaG. Schimperna and M. Schonbek, On asymptotic isotropy for a hydrodynamic model of liquid crystals, Asymptotic Analysis, 97 (2016), 189-210.  doi: 10.3233/ASY-151348.

[12]

F. De Anna, A global 2D well-posedness result on the order tensor liquid crystal theory, J. Differential Equations, 262 (2017), 3932-3979.  doi: 10.1016/j.jde.2016.12.006.

[13]

F. De Anna and A. Zarnescu, Uniqueness of weak solutions of the full coupled Navier-Stokes and Q-tensor system in 2D, Commun. Math. Sci., 14 (2016), 2127-2178.  doi: 10.4310/CMS.2016.v14.n8.a3.

[14]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, International Series of Monographs on Physics. Oxford University Press, Incorporated, 2nd edition, 1995.

[15]

E. FeireislE. RoccaG. Schimperna and A. Zarnescu, Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential, Comm. Math. Sci., 12 (2014), 317-343.  doi: 10.4310/CMS.2014.v12.n2.a6.

[16]

E. FeireislG. SchimpernaE. Rocca and A. Zarnescu, Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy, Annali di Matematica Pura ed Applicata (1923-), 194 (2015), 1269-1299.  doi: 10.1007/s10231-014-0419-1.

[17]

F. Guillén-González and M. Ángeles RodrÍguez-Bellido, Weak time regularity and uniqueness for a Q-tensor model, SIAM J. Math. Anal., 46 (2014), 3540-3567.  doi: 10.1137/13095015X.

[18]

F. Guillén-González and M. Á. Rodríguez-Bellido, Weak solutions for an initial-boundary Q-tensor problem related to liquid crystals, Nonlinear Analysis: Theory, Methods & Applications, 112 (2015), 84-104.  doi: 10.1016/j.na.2014.09.011.

[19]

J. Huang and S. Ding, Global well-posedness for the dynamical Q-tensor model of liquid crystals, Sci. China Math., 58 (2015), 1349-1366.  doi: 10.1007/s11425-015-4990-8.

[20]

G. IyerX. Xu and A. D. Zarnescu, Dynamic cubic instability in a 2D Q-tensor model for liquid crystals, Math. Models Methods Appl. Sci., 25 (2015), 1477-1517.  doi: 10.1142/S0218202515500396.

[21]

M. Paicu and A. Zarnescu, Global existence and regularity for the full coupled Navier-Stokes and Q-tensor system, SIAM J. Math. Anal., 43 (2011), 2009-2049.  doi: 10.1137/10079224X.

[22]

M. Paicu and A. Zarnescu, Energy dissipation and regularity for a coupled Navier-Stokes and Q-tensor system, Arch. Ration. Mech. Anal., 203 (2012), 45-67.  doi: 10.1007/s00205-011-0443-x.

[23]

H. Sohr, The Navier-Stokes Equations: An Elementary Functional Analytic Approach, Birkhäuser Verlag, Basel, 2001.

[24]

V. A. Solonnikov, Lp-estimates for solutions to the initial boundary-value problem for the generalized Stokes system in a bounded domain, J. Math. Sci. (New York), 105 (2001), 2448-2484. Function theory and partial differential equations. doi: 10.1023/A:1011321430954.

[25]

W. WangP. Zhang and Z. Zhang, Rigorous derivation from Landau-de Gennes theory to Ericksen-Leslie theory, SIAM J. Math. Anal., 47 (2015), 127-158.  doi: 10.1137/13093529X.

[26]

M. Wilkinson, Strictly physical global weak solutions of a Navier-Stokes Q-tensor system with singular potential, Arch. Ration. Mech. Anal., 218 (2015), 487-526.  doi: 10.1007/s00205-015-0864-z.

[27]

J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987. Translated from the German by C. B. Thomas and M. J. Thomas.

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