Recent developments of analysis for hydrodynamic flow of nematic liquid crystals

The study of hydrodynamics of liquid crystals leads to many fascinating mathematical problems, which has prompted various interesting works recently. This article reviews the static Oseen–Frank theory and surveys some recent progress on the existence, regularity, uniqueness and large time asymptotic of the hydrodynamic flow of nematic liquid crystals. We will also propose a few interesting questions for future investigations.

[1]  Apala Majumdar,et al.  Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond , 2008, 0812.3131.

[2]  Jean-Michel Coron,et al.  Harmonic maps with defects , 1986 .

[3]  Existence of minimal energy configurations of nematic liquid crystals with variable degree of orientation , 1990 .

[4]  F. Lin.  Static and moving defects in liquid crystals , 1991 .

[5]  R. Larson.  The Structure and Rheology of Complex Fluids , 1998 .

[6]  Z. Xin,et al.  Global Weak Solutions to Non-isothermal Nematic Liquid Crystals in 2D , 2013, 1307.2065.

[7]  Jinrui Huang,et al.  Incompressible Limit of the Compressible Nematic Liquid Crystal Flow , 2011, 1104.4377.

[8]  F. Lin,et al.  Static and dynamic theories of liquid crystals , 2001 .

[9]  Regularity of solutions of a degenerate elliptic variational problem , 1990 .

[10]  M. Frémond,et al.  A New Approach to Non-Isothermal Models for Nematic Liquid Crystals , 2011, 1104.1339.

[11]  F. Lin.  Some Analytical Issues for Elastic Complex Fluids , 2012 .

[12]  On Ericksen’s model for liquid crystals , 1994 .

[13]  P. Bauman,et al.  Analysis of Nematic Liquid Crystals with Disclination Lines , 2011, 1106.5031.

[14]  Global Solution to the Three-Dimensional Incompressible Flow of Liquid Crystals , 2009, 0906.4802.

[15]  F. C. Frank,et al.  I. Liquid crystals. On the theory of liquid crystals , 1958 .

[16]  Arghir Zarnescu,et al.  Refined approximation for minimizers of a Landau-de Gennes energy functional , 2010, 1006.5689.

[17]  Chun Liu,et al.  Time Evolution of Nematic Liquid Crystals with Variable Degree of Orientation , 2002, SIAM J. Math. Anal..

[18]  Xiaoli Li,et al.  Global strong solution to the density-dependent incompressible flow of liquid crystals , 2012, 1202.1011.

[19]  Z. Xin,et al.  Blow-up criteria of strong solutions to the Ericksen-Leslie system in $\Bbb R^3$ , 2013, 1303.4488.

[20]  Ping Sheng,et al.  Generalized hydrodynamic equations for nematic liquid crystals , 1998 .

[21]  Changyou Wang,et al.  WEAK SOLUTION TO COMPRESSIBLE HYDRODYNAMIC FLOW OF LIQUID CRYSTALS IN DIMENSION ONE , 2010 .

[22]  Arghir Zarnescu,et al.  Global Existence and Regularity for the Full Coupled Navier-Stokes and Q-Tensor System , 2010, SIAM J. Math. Anal..

[23]  James J. Feng,et al.  Closure approximations for the Doi theory: Which to use in simulating complex flows of liquid-crystalline polymers? , 1998 .

[24]  M. Yoneya,et al.  Physics of Liquid Crystals , 2014 .

[25]  Arghir Zarnescu,et al.  Orientability and Energy Minimization in Liquid Crystal Models , 2010, 1009.2688.

[26]  F. Lin,et al.  A remark on the map x/[x] , 1987 .

[27]  Gradient estimates and blow-up analysis for stationary harmonic maps , 1999, math/9905214.

[28]  Brian J. Edwards,et al.  Thermodynamics of flowing systems : with internal microstructure , 1994 .

[29]  Junyu Lin,et al.  Global Finite Energy Weak Solutions to the Compressible Nematic Liquid Crystal Flow in Dimension Three , 2014, SIAM J. Math. Anal..

[30]  Apala Majumdar,et al.  Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory , 2008, European Journal of Applied Mathematics.

[31]  F. Lin,et al.  Harmonic and quasi-harmonic spheres , 1999 .

[32]  Song Jiang,et al.  On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain , 2013, 1302.2793.

[33]  Zhifei Zhang,et al.  Rigorous Derivation from Landau-de Gennes Theory to Ericksen-Leslie Theory , 2013, SIAM J. Math. Anal..

[34]  Xiang Xu,et al.  On the General Ericksen–Leslie System: Parodi’s Relation, Well-Posedness and Stability , 2011, 1105.2180.

[35]  Arghir Zarnescu,et al.  Energy Dissipation and Regularity for a Coupled Navier–Stokes and Q-Tensor System , 2010, 1001.1032.

[36]  Chun Liu,et al.  Approximation of Liquid Crystal Flows , 2000, SIAM J. Numer. Anal..

[37]  N. David Mermin,et al.  Points, Lines and Walls: In Liquid Crystals, Magnetic Systems and Various Ordered Media. , 1982 .

[38]  Xiaoli Li,et al.  Global solution to the incompressible flow of liquid crystals , 2011, 1108.5477.

[39]  Daniel Coutand,et al.  Well-posedness of the full Ericksen-Leslie model of nematic liquid crystals , 2001 .

[40]  H. Brezis.  Sk—Valued maps with singularities , 1989 .

[41]  Chun Liu,et al.  Asymptotic behavior for a nematic liquid crystal model with different kinematic transport properties , 2009, 0901.1751.

[42]  Zhouping Xin,et al.  Blow-up Criteria of Strong Solutions to the Ericksen-Leslie System in ℝ3 , 2014 .

[43]  Fanghua Lin,et al.  Regularity and Existence of Global Solutions to the Ericksen–Leslie System in $${\mathbb{R}^2}$$R2 , 2013, 1305.5988.

[44]  J. J. Fenga.  A theory for flowing nematic polymers with orientational distortion , 2009 .

[45]  David Kinderlehrer,et al.  Mathematical Questions of Liquid Crystal Theory , 1987 .

[46]  Huanyao Wen,et al.  Strong solutions of the compressible nematic liquid crystal flow , 2011, 1104.5684.

[47]  Eduard Feireisl,et al.  Dynamics of Viscous Compressible Fluids , 2004 .

[48]  Apala Majumdar,et al.  Symmetry of Uniaxial Global Landau-de Gennes Minimizers in the Theory of Nematic Liquid Crystals , 2011, SIAM J. Math. Anal..

[49]  FANGHUA LIN,et al.  Regularity and Existence of Global Solutions to the Ericksen–Leslie System in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d , 2014, Communications in Mathematical Physics.

[50]  Tosio Kato,et al.  Remarks on the breakdown of smooth solutions for the 3-D Euler equations , 1984 .

[51]  Masao Doi,et al.  Constitutive Equation for Nematic Liquid Crystals under Weak Velocity Gradient Derived from a Molecular Kinetic Equation , 1983 .

[52]  Changyou Wang,et al.  On the rigidity of nematic liquid crystal flow on S2 , 2013, 1308.4000.

[53]  Changyou Wang,et al.  Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data , 2010, 1001.2296.

[54]  Yi Du,et al.  Regularity of the solutions to the liquid crystal equations with small rough data , 2014 .

[55]  Rugang Ye,et al.  Finite-time blow-up of the heat flow of harmonic maps from surfaces , 1992 .

[56]  Chun Liu,et al.  Existence of Solutions for the Ericksen-Leslie System , 2000 .

[57]  F. Lin,et al.  Global Existence of Weak Solutions of the Nematic Liquid Crystal Flow in Dimension Three , 2014, 1408.4146.

[58]  L. Simon.  Regularity Theory for Harmonic Maps , 1996 .

[59]  Zhouping Xin,et al.  Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in R2 , 2012 .

[60]  Modelling of Nematic Liquid Crystals in Electromagnetic Fields , 2009 .

[61]  Susanne Hertz,et al.  Theory and applications of liquid crystals , 1990 .

[62]  Dong Li,et al.  Remarks of global wellposedness of liquid crystal flows and heat flows of harmonic maps in two dimensions , 2012, 1205.1269.

[63]  Apala Majumdar,et al.  Nematic Liquid Crystals: From Maier-Saupe to a Continuum Theory , 2010 .

[64]  Alfred P. Sloanfellowship.  Well-posedness for the Navier-stokes Equations , 1999 .

[65]  Epifanio G. Virga,et al.  Variational Theories for Liquid Crystals , 2018 .

[66]  Zhifei Zhang,et al.  Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows , 2012 .

[67]  A boundary-value problem for nematic liquid crystals with a variable degree of orientation , 1991 .

[68]  F. Lin,et al.  Harmonic and quasi-harmonic spheres, part III. Rectifiablity of the parabolic defect measure and generalized varifold flows , 2002 .

[69]  O. Parodi,et al.  Stress tensor for a nematic liquid crystal , 1970 .

[70]  W. H. Steele.  Points , 1898, The Dental register.

[71]  Changyou Wang,et al.  Compressible hydrodynamic flow of liquid crystals in 1-D , 2011 .

[72]  Min-Chun Hong.  Global existence of solutions of the simplified Ericksen–Leslie system in dimension two , 2011 .

[73]  Karen Uhlenbeck,et al.  Boundary regularity and the Dirichlet problem for harmonic maps , 1983 .

[74]  Globally weak solutions to the flow of compressible liquid crystals system , 2012 .

[75]  Changyou Wang,et al.  Blow up Criterion for Nematic Liquid Crystal Flows , 2011, 1104.5683.

[76]  E. Fried,et al.  Statistical Foundations of Liquid-Crystal Theory II: Macroscopic Balance Laws , 2013, Archive for rational mechanics and analysis.

[77]  Harmonic maps into round cones and singularities of nematic liquid crystals , 1993 .

[78]  G. B. Jeffery.  The motion of ellipsoidal particles immersed in a viscous fluid , 1922 .

[79]  F. Lin,et al.  Nonparabolic dissipative systems modeling the flow of liquid crystals , 1995 .

[80]  Fanghua Lin,et al.  On nematic liquid crystals with variable degree of orientation , 1991 .

[81]  E Weinan,et al.  A Molecular Kinetic Theory of Inhomogeneous Liquid Crystal Flow and the Small Deborah Number Limit , 2006 .

[82]  Huanyao Wen,et al.  Blow Up Criterion for Compressible Nematic Liquid Crystal Flows in Dimension Three , 2011, 1104.5685.

[83]  Wei Wang,et al.  Well-Posedness of the Ericksen–Leslie System , 2012, 1208.6107.

[84]  Changyou Wang,et al.  Well-Posedness of Nematic Liquid Crystal Flow in $${L^{3}_{\rm uloc}(\mathbb{R}^3)}$$ , 2012, 1208.5965.

[85]  E. Feireisl,et al.  On a non-isothermal model for nematic liquid crystals , 2009 .

[86]  Chun Liu,et al.  On energetic variational approaches in modeling the nematic liquid crystal flows , 2008 .

[87]  F. Lin,et al.  Harmonic and quasi-harmonic spheres, part II , 2002 .

[88]  David Kinderlehrer,et al.  Existence and partial regularity of static liquid crystal configurations , 1986 .

[89]  Wei Wang,et al.  The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation , 2012, 1206.5480.

[90]  Steve Shkoller,et al.  WELL-POSEDNESS AND GLOBAL ATTRACTORS FOR LIQUID CRYSTALS ON RIEMANNIAN MANIFOLDS , 2001, math/0101203.

[91]  Wei Wang,et al.  From Microscopic Theory to Macroscopic Theory: a Systematic Study on Modeling for Liquid Crystals , 2013, 1305.4889.

[92]  A. Isihara,et al.  Theory of Liquid Crystals , 1972 .

[93]  D. Kinderlehrer,et al.  Stable defects of minimizers of constrained variational principles , 1988 .

[94]  Karen Uhlenbeck,et al.  Regularity of minimizing harmonic maps into the sphere , 1984 .

[95]  J. Ericksen.  Liquid crystals with variable degree of orientation , 1991 .

[96]  Regularity of solutions to the liquid crystal flows with rough initial data , 2013 .

[97]  F. Lin,et al.  The analysis of harmonic maps and their heat flows , 2008 .

[98]  J. Ericksen,et al.  Hydrostatic theory of liquid crystals , 1962 .

[99]  Fanghua Lin,et al.  Partial regularity of the dynamic system modeling the flow of liquid crystals , 1995 .

[100]  Fanghua Lin,et al.  Liquid Crystal Flows in Two Dimensions , 2010 .

[101]  Regularity and uniqueness for a class of solutions to the hydrodynamic flow of nematic liquid crystals , 2014, 1405.6737.

[102]  Dehua Wang,et al.  Global Weak Solution and Large-Time Behavior for the Compressible Flow of Liquid Crystals , 2011, 1108.4939.

[103]  F. Lin,et al.  On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals , 2010 .

[104]  J. Ball,et al.  Orientable and Non-Orientable Line Field Models for Uniaxial Nematic Liquid Crystals , 2008 .

[105]  Jing Li,et al.  Global Well-Posedness with Large Oscillations and Vacuum to the Three-Dimensional Equations of Compressible Nematic Liquid Crystal Flows , 2012, 1204.4966.

[106]  F. M. Leslie.  Some constitutive equations for liquid crystals , 1968 .

[107]  E. Fried,et al.  Statistical Foundations of Liquid-Crystal Theory. I: Discrete Systems of Rod-Like Molecules , 2012, Archive for rational mechanics and analysis.

[108]  Wendong Wang,et al.  Global existence of weak solution for the 2-D Ericksen–Leslie system , 2013, 1305.0622.

[109]  Wei Wang,et al.  From microscopic theory to macroscopic theory: dynamics of the rod-like liquid crystal molecules , 2013, 1305.4721.

[110]  L. Evans,et al.  Partial regularity for minimizers of singular energy functionals, with application to liquid crystal models , 2013, 1312.4471.