Nematic Liquid Crystal Flows Research Articles

Regularity criterion for 3D nematic liquid crystal flows in terms of finite frequency parts in dot{B}_{infty,infty }^{-1}

In this paper, we establish the regularity criterion for the weak solution of nematic liquid crystal flows in three dimensions when the L^{infty }(0,T;dot{B}_{infty,infty }^{-1})-norm of a suitable low frequency part of (u,nabla d) is bounded by a scaling invariant constant and the initial data (u_{0},nabla d_{0}). Our result refines the corresponding one in (Liu and Zhao in J. Math. Anal. Appl. 407:557-566, 2013) and that in (Ri in Nonlinear Anal. TMA 190:111619, 2020).

Local well-posedness of the inertial Qian\u2013Sheng\u2019s Q-tensor dynamical model near uniaxial equilibrium

We consider the inertial Qian–Sheng’s Q-tensor dynamical model for the nematic liquid crystal flow, which can be viewed as a system coupling the hyperbolic-type equations for the Q-tensor parameter with the incompressible Navier–Stokes equations for the fluid velocity. We prove the existence and uniqueness of local in time strong solutions to the system with the initial data near uniaxial equilibrium. The proof is mainly based on the classical Friedrich method to construct approximate solutions and the closed energy estimate.

Phase-field model for elastocapillary flows of liquid crystals

We propose a phase-field model to study interfacial flows of nematic liquid crystals that couple the capillary forces on the interface with the elastic stresses in the nematic phase. The theoretical model has two key ingredients: A tensor order parameter that provides a consistent description of the molecular and distortional elasticity, and a phase-field formalism that accurately represents the interfacial tension and the nematic anchoring stress by approximating a sharp-interface limit. Using this model, we carry out finite-element simulations of drop retraction in a surrounding fluid, with either component being nematic. The results are summarized by eight representative steady-state solutions in planar and axisymmetric geometries, each featuring a distinct configuration for the drop and the defects. The dynamics is dominated by the competition between the interfacial tension and the distortional elasticity in the nematic phase, mediated by the anchoring condition on the drop surface. As consequences of this competition, the steady-state drop deformation and the clearance between the defects and the drop surface both depend linearly on the elastocapillary number.

Hydrodynamics of Anisotropic Liquid Crystals in an Applied Magnetic Field

We consider a magnetic effect on hydrodynamic flows of nematic liquid crystals with unequal elastic constants in dimension two. Under a planar ansatz for the director field and velocity, we derive a reduced Ericksen--Leslie system which consists of a quasilinear parabolic equation and a Navier--Stokes equation involving critical nonlinearity due to elastic anisotropy. When unequal elastic constants are sufficiently close, we establish the global existence of smooth solutions to the reduced system and show convergence to an equilibrium as $t$ tends to $\infty$. A magnetic field-induced instability of the aligned state under the hydrodynamic flow is discussed.

Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics

<p style='text-indent:20px;'>We consider an unconditional fully discrete finite element scheme for a nematic liquid crystal flow with different kinematic transport properties. We prove that the scheme converges towards a unique critical point of the elastic energy subject to the finite element subspace, when the number of time steps go to infinity while the time step and mesh size are fixed. A Lojasiewicz type inequality, which is the key for getting the time asymptotic convergence of the whole sequence furnished by the numerical scheme, is also derived.</p>

Asymptotic behavior of two-dimensional stochastic nematic liquid crystal flows with multiplicative noise

Under non-periodic boundary conditions, we consider the long-term behavior of stochastic two-dimensional (2D) nematic liquid crystal flows with multiplicative noise. The structure of the corresponding stochastic model is different from that of the deterministic model. Noise destroys the basic balance law for the nematic liquid crystal flows, so we cannot employ the standard argument to obtain uniform a priori estimates of the solutions to this stochastic model. To overcome this problem, we use logarithmic energy estimates and the Itô formula in a Banach space to obtain uniform estimates that improve the previous result where the orientation field grows exponentially w.r.t. time t. In order to study the existence of a random attractor, we need to show that the solution is a stochastic flow. However, this is not obvious because of the particularly multiplicative noise in the orientation field. To show the flow property of the orientation field, we construct linear stochastic partial differential equations where the solutions are stochastic flows. After determining the relationship between these linear equations and the orientation field equation, we prove that each component of the orientation field is indeed a stochastic flow. The global well-posedness of the stochastic 2D nematic liquid crystal flows is only established for weak solutions, so the existence of the random attractor should be demonstrated in the weak solution space. Thus, using the standard method, we need to derive uniform a priori estimates in the strong solution space. However, the standard method fails because of the ill-posedness of the strong solution. We show that by proving the compactness property of the stochastic flow and using the regularity of the solutions, we can construct a compact absorbing ball in the weak solution space, which implies the existence of the random attractor. Finally, we show the existence of the invariant measure for the corresponding stochastic model, which is the first result for the long-term behavior of stochastic nematic liquid crystals under Dirichlet boundary conditions for the velocity field and Neumann boundary conditions for the orientation field.

Global well-posedness to the 3D Cauchy problem of compressible non-isothermal nematic liquid crystal flows with vacuum

We are concerned with global well-posedness of strong solutions to the Cauchy problem for compressible non-isothermal nematic liquid crystal flows with vacuum as far field density in R3. By using energy method, we establish the global existence and uniqueness of strong solutions provided that the quantity‖ρ0‖L∞+‖∇d0‖L3 is suitably small and the viscosity coefficients verify 3μ>λ. In particular, the initial density can even have compact support. When d is a constant vector and |d|=1, we also extend partially the corresponding result in Li (2020) where the global small solution of full compressible Navier–Stokes equations was obtained under the condition 2μ>λ. To our knowledge, the result in this paper could be viewed as the first one on the global existence of strong solutions to 3D Cauchy problem for compressible non-isothermal nematic liquid crystal flows with vacuum.

Strong Solutions to the Density-Dependent Incompressible Nematic Liquid Crystal Flows with Heat Effect

Strong Solutions to the Density-Dependent Incompressible Nematic Liquid Crystal Flows with Heat Effect

Global regularity to the 2D non-isothermal inhomogeneous nematic liquid crystal flows

ABSTRACT In this paper, we prove the global strong solutions for the Cauchy problem of two-dimensional (2D) incompressible non-isothermal nematic liquid crystal flows, if the initial orientation satisfies a geometric condition. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states. When d is a constant vector and , we also extend the corresponding result in Wang Y. [Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier–Stokes flows with vacuum. Discrete Contin Dyn Syst B. doi:10.3934/dcdsb.2020099.] to the whole space , where the global strong solution of 2D inhomogeneous incompressible heat conducting Navier–Stokes flows is established on bounded domain.

Pressure-driven changes to spontaneous flow in active nematic liquid crystals

.We consider the effects of a pressure gradient on the spontaneous flow of an active nematic liquid crystal in a channel, subject to planar anchoring and no-slip conditions on the boundaries of the channel. We employ a model based on the Ericksen-Leslie theory of nematics, with an additional active stress accounting for the activity of the fluid. By directly solving the flow equation, we consider an asymptotic solution for the director angle equation for large activity parameter values and predict the possible values of the director angle in the bulk of the channel. Through a numerical solution of the full nonlinear equations, we examine the effects of pressure on the branches of stable and unstable equilibria, some of which are disconnected from the no-flow state. In the absence of a pressure gradient, solutions are either symmetric or antisymmetric about the channel midpoint; these symmetries are changed by the pressure gradient. Considering the activity-pressure state space allows us to predict qualitatively the extent of each solution type and to show, for large enough pressure gradients, that a branch of non-trivial director angle solutions exists for all activity values.Graphical abstract

Global classical solutions to the 3D nematic liquid crystal flows with two directional viscosity

In this paper, we investigate the initial value problem for the three dimensional nematic liquid crystal flows with two directional viscosity. The main purpose of this paper is to establish global well-posedness of classical small solutions. The proof is mainly based on the energy estimate, anisotropic type Sobolev inequality and bootstrapping argument.

Note on global regular solution to the 3D liquid crystal equations

In this paper we provide a sufficient condition for regularity of solutions to the 3D nematic liquid crystal flows. More precisely, we prove that if the horizontal velocity component uh=(u1,u2) satisfies ∇uh∈Lq(0,T;Lp(R3)),3p+2q≤32,2≤p≤∞, then the solution (u,d) is regular.

Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows

In this paper, we consider the initial-boundary value problem to the non-isothermal incompressible liquid crystal system with both variable density and temperature. Global well-posedness of strong solutions is established for initial data being small perturbation around the equilibrium state. As the tools in the proof, we establish the maximal regularities of the linear Stokes equations and parabolic equations with variable coefficients and a rigid lemma for harmonic maps on bounded domains. This paper also generalizes the result in [ 5 ] to the inhomogeneous case.

Well-posedness for 3D nematic liquid crystal flows with damping

The three-dimensional nematic liquid crystal flows with damping are considered in this paper. The existence and uniqueness of strong solutions for the 3D nematic liquid crystal flows with damping are proved for betageq4 with any alpha>0.

Weak solutions of non-isothermal nematic liquid crystal flow in dimension three

For any smooth domain $$\Omega \subset {{\mathbb {R}}}^3$$, we establish the existence of a global weak solution $$(\mathbf{u},\mathbf{d}, \theta )$$ to the simplified, non-isothermal Ericksen–Leslie system modeling the hydrodynamic motion of nematic liquid crystals with variable temperature for any initial and boundary data $$(\mathbf{u}_0, \mathbf{d}_0, \theta _0)\in {\mathbf {H}}\times H^1(\Omega , {\mathbb {S}}^2)\times L^1(\Omega )$$, with $$ \mathbf{d}_0(\Omega )\subset {\mathbb {S}}_+^2$$ (the upper half sphere) and $$\text{ess inf}_\Omega \theta _0>0$$.

Global well‐posedness of the 3D incompressible nematic liquid crystal flows with density‐dependent viscosity coefficient

This paper deals with the 3D incompressible nematic liquid crystal flows with density‐dependent viscosity in bounded domain. The global well‐posedness of strong solutions are established in the vacuum cases, provided the assumption that is suitably small with large velocity. Furthermore, the exponential decay of the solution is also obtained.

Global Existence and Exponential Decay of Strong Solutions to the 2D Density-dependent Nematic Liquid Crystal Flows with Vacuum

This paper deals with the 2D incompressible nematic liquid crystal flows with density-dependent viscosity in bounded domain. The global well-posedness of strong solutions are established in the vacuum cases, provided the assumption that $\overline{\rho} + \|\nabla d_0\|_{L^2}$ is suitably small with large velocity, which extends the recent work [Discrete Contin. Dyn. Syst. 37 (2017), 4907--4922] and [Methods Appl. Anal. 22 (2015), 201--220] to the case of variable viscosity. Furthermore, the exponential decay of the solution is also obtained.

Global well-posedness for the three dimensional simplified inertial Ericksen-Leslie systems near equilibrium

We study a simplified inertial Ericksen-Leslie system for the nematic liquid crystal flow, which can be viewed as a system coupling Navier-Stokes equations and wave map equations. We prove the global existence of classical solution with initial data near equilibrium.

Strong solutions to the Cauchy problem of two-dimensional compressible non-isothermal nematic liquid crystal flows with vacuum and zero heat conduction

We study the Cauchy problem of nonisothermal compressible nematic liquid crystal flows involving zero heat conduction on the whole two-dimensional (2D) space with vacuum at infinity. For the initial density allowing vacuum states, we prove that there exists a local strong solution, provided that the initial data density and the gradient of orientation decay not too slowly at infinity. Our method relies on delicate weighted energy estimates and a Hardy-type inequality.

On the global existence of classical solutions for compressible nematic liquid crystal flows with vacuum

This paper deals with the Cauchy problem of compressible nematic liquid crystal flows in the whole space $$\mathbb {R}^3$$. We show that if, in addition, the conservation law of the total mass is satisfied (i.e., $$\rho _0\in L^1$$), then the global existence theorem with small density and $$L^3$$-norm of the gradient of $$d_0$$ holds for any $$\gamma >1$$. It is worth mentioning that the initial total energy can be arbitrarily large and the initial vacuum is allowed. Thus, the result obtained particularly extends the one due to Li et al. (J Math Fluid Mech 20:2105–2145, 2018), where the global well-posedness of classical solutions with small energy was proved.