Long-Time Behavior for a Hydrodynamic Model on Nematic Liquid Crystal Flows with Asymptotic Stabilizing Boundary Condition and External Force

In this paper, we consider a simplified Ericksen--Leslie model for the nematic liquid crystal flow. The evolution system consists of the Navier--Stokes equations coupled with a convective Ginzburg--Landau type equation for the averaged molecular orientation. We suppose that the Navier--Stokes equations are characterized by a no-slip boundary condition and a time-dependent external force $\mathbf{g}(t)$, while the equation for the molecular director is subject to a time-dependent Dirichlet boundary condition $\mathbf{h}(t)$. We show that in the two-dimensional case, each global weak solution converges to a single stationary state when $\mathbf{h}(t)$ and $\mathbf{g}(t)$ converge to a time-independent boundary datum $\mathbf{h}_\infty$ and $\mathbf{0}$, respectively. Estimates on the convergence rate are also obtained. In the three-dimensional case, we prove that global weak solutions are eventually strong so that results similar to the two-dimensional case can be proved. We also show the existence of globa...

[1]  Maurizio Grasselli,et al.  CONVERGENCE TO EQUILIBRIUM FOR A PARABOLIC–HYPERBOLIC PHASE-FIELD SYSTEM WITH NEUMANN BOUNDARY CONDITIONS , 2007 .

[2]  Elisabetta Rocca,et al.  On a 3D isothermal model for nematic liquid crystals accounting for stretching terms , 2011, 1107.3947.

[3]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[4]  Jishan Fan,et al.  Regularity criteria for a simplified Ericksen-Leslie system modeling the flow of liquid crystals , 2009 .

[5]  L. Simon.  Asymptotics for a class of non-linear evolution equations, with applications to geometric problems , 1983 .

[6]  Francisco Guillén-González,et al.  Reproductivity for a nematic liquid crystal model , 2006 .

[7]  R. Chill,et al.  Convergence to steady states in asymptotically autonomous semilinear evolution equations , 2003 .

[8]  F. Lin.  Nonlinear theory of defects in nematic liquid crystals; Phase transition and flow phenomena , 1989 .

[9]  Hao Wu,et al.  Long-time Behavior for Nonlinear Hydrodynamic System Modeling the Nematic Liquid Crystal Flows , 2009, 0904.0390.

[10]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[11]  Chun Liu,et al.  Simulations of singularity dynamics in liquid crystal flows: A C0 finite element approach , 2006, J. Comput. Phys..

[12]  Elisabetta Rocca,et al.  Global weak solution and blow-up criterion of the general Ericksen-Leslie system for nematic liquid crystal flows , 2012, 1212.0043.

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

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

[15]  Sen-Zhong Huang,et al.  Gradient Inequalities: With Applications to Asymptotic Behavior And Stability of Gradient-like Systems , 2006 .

[16]  Marko Antonio Rojas-Medar,et al.  Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model , 2009 .

[17]  Giulio Schimperna,et al.  Convergence to stationary solutions for a parabolic-hyperbolic phase-field system , 2006 .

[18]  Roger Temam,et al.  Navier–Stokes Equations and Nonlinear Functional Analysis: Second Edition , 1995 .

[19]  Jie Shen,et al.  On liquid crystal flows with free-slip boundary conditions , 2001 .

[20]  F. M. Leslie.  Theory of Flow Phenomena in Liquid Crystals , 1979 .

[21]  Alain Haraux,et al.  Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity , 2001 .

[22]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[23]  Jean Leray,et al.  Sur le mouvement d'un liquide visqueux emplissant l'espace , 1934 .

[24]  Francisco Guillén-González,et al.  Regularity and time-periodicity for a nematic liquid crystal model , 2009 .

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

[26]  Elisabetta Rocca,et al.  On the long-time behavior of some mathematical models for nematic liquid crystals , 2011 .

[27]  CHIN-YUAN LIN,et al.  Nonlinear evolution equations , 2000 .

[28]  S. Huang,et al.  Convergence in gradient-like systems which are asymptotically autonomous and analytic , 2001 .

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

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

[31]  N. Walkington,et al.  Mixed Methods for the Approximation of Liquid Crystal Flows , 2002 .

[32]  J. Ericksen.  Conservation Laws for Liquid Crystals , 1961 .

[33]  Maurizio Grasselli,et al.  Finite-dimensional global attractor for a system modeling the 2D nematic liquid crystal flow , 2011, 1210.2124.

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

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

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