Ericksen Leslie System Research Articles

Local and global existence for the Ericksen - Leslie problem in unbounded domains

The work deals with the Ericksen-Leslie System for nematic liquid crystals on the space RN with N≥3, on R+3 and on Ω⊆R3 exterior domain with sufficiently smooth boundary. The crystal orientation is described by unit vector v that is a small perturbation of a fixed constant vector η. We prove through a combination of energy method with dispersive a priori estimates a local existence and global existence for small initial data by a contraction argument. In particular, we obtain the following regularity of the liquid velocity u and of the crystal orientation vu∈L∞((0,T);Hs(Ω)),∇u∈L2((0,T);Hs(Ω)),∇v∈L∞((0,T);Hs(Ω)),∇2v∈L2((0,T);Hs(Ω)) for s>N2−1 if Ω=RN and s∈(12,1] if Ω=R+3 or in the exterior case, asking low regularity assumptions on u0 and v0.

Singularity Formation for Full Ericksen–Leslie System of Nematic Liquid Crystal Flows in Dimension Two

Singularity Formation for Full Ericksen–Leslie System of Nematic Liquid Crystal Flows in Dimension Two

Global well-posedness for Ericksen-Leslie system with zero viscosity

We study the global well-posedness of the simplified Ericksen-Leslie system with zero viscosity on the periodic domain T 2 \mathbb {T}^2 . Precisely, we prove the global existence and uniqueness of smooth solution to Ericksen-Leslie system if the initial data ( u 0 , ∇ d 0 ) (u_0,\nabla d_0) is small in H 4 ( T 2 ) × H 4 ( T 2 ) H^4(\mathbb {T}^2)\times H^4(\mathbb {T}^2) . Furthermore, we derive the time decay estimate of ∇ d \nabla d in H 1 ( T 2 ) H^1(\mathbb {T}^2) .

Long-time behavior of solution for the reformulated non-autonomous Ericksen-Leslie system

Long-time behavior of solution for the reformulated non-autonomous Ericksen-Leslie system

Error Analysis of a Pressure Penalty Scheme for the Reformulated Ericksen-Leslie System with Variable Density

Error Analysis of a Pressure Penalty Scheme for the Reformulated Ericksen-Leslie System with Variable Density

Global Well-Posedness for the Cauchy Problems of the 3d Simplified Ericksen–Leslie System with Fujita–Kato Type Initial Data

Global Well-Posedness for the Cauchy Problems of the 3d Simplified Ericksen–Leslie System with Fujita–Kato Type Initial Data

Long‐time dynamics of Ericksen–Leslie system on 𝕊2

In this paper, we study the long‐time behavior of full Ericksen–Leslie system modeling the hydrodynamics of nematic liquid crystals between two dimensional unit spheres. Under a weaker assumption for Leslie's coefficients, we give the key energy inequality for the global weak solution. At last, inspired by the conditions on the simplified system, we establish several sufficient conditions which guarantee the uniform convergence of the system in and spaces as time tends to infinity under small initial data.

Uniqueness of global weak solutions for the general Ericksen–Leslie system with Ginzburg–Landau penalization in {mathbb {T}}^2

The Ericksen–Leslie system is a fundamental hydrodynamic model that describes the evolution of incompressible liquid crystal flows of nematic type. In this paper, we prove the uniqueness of global weak solutions to the general Ericksen–Leslie system with a Ginzburg–Landau type approximation in a two dimensional periodic domain. The proof is based on some delicate energy estimates for the difference of two weak solutions within a suitable functional framework that is less regular than the usual one at the natural energy level, combined with the Osgood lemma involving a specific double-logarithmic type modulus of continuity. We overcome the essential mathematical difficulties arising from those highly nonlinear terms in the Leslie stress tensor and in particular, the lack of maximum principle for the director equation due to the stretching effect of the fluid on the director field. Our argument makes full use of the coupling structure as well as the dissipative nature of the system, and relies on some techniques from harmonic analysis and paradifferential calculus in the periodic setting.

Blow-up criteria of the simplified Ericksen–Leslie system

In this paper, we establish scaling invariant blow-up criteria for a classical solution to the simplified Ericksen–Leslie system in terms of the positive part of the second eigenvalue of the strain matrix and orientation field in mixed-norm Lebesgue spaces. Our result may be also regarded as an extension or improvement of the corresponding results obatined by Neustupa and Penel (Trends in Partial Differential Equations of Mathematical Physics, pp. 197–212, 2005), Miller (Arch. Ration. Mech. Anal. 235(1):99–139, 2020) and Huang and Wang (Commun. Partial Differ. Equ. 37(5):875–884, 2012).

A fully-decoupled discontinuous Galerkin method for the nematic liquid crystal flows with SAV approach

In this paper, we consider the numerical approximation of a hydrodynamic Ericksen–Leslie system, which describes the macroscopic continuum of liquid crystals. A numerical challenge for solving the governing system is how to construct a suitable numerical scheme so that it is unconditionally energy stable at the discrete level. We propose a novel linear, decoupled fully discrete scheme, which is based on the design of the scalar auxiliary variable (SAV) for the nonlinear potential and combined with the discontinuous Galerkin (DG) for spatial discretization, the pressure-projection method for the Navier–Stokes equations, and implicit–explicit (IMEX) approaches for the highly nonlinear and coupling terms. We rigorously prove the unconditional energy stability and optimal error estimates of the proposed scheme, especially the optimal error bounds for the pressure. Finally, several numerical examples are performed to numerically demonstrate the accuracy, energy stability, and applicability of the proposed scheme.

Nematic Liquid Crystal Flow with Partially Free Boundary

We study a simplified Ericksen-Leslie system modeling the flow of nematic liquid crystals with partially free boundary conditions. It is a coupled system between the Navier-Stokes equation for the fluid velocity with a transported heat flow of harmonic maps, and both of these parabolic equations are critical for analysis in two dimensions. The boundary conditions are physically natural and they correspond to the Navier slip boundary condition with zero friction for the velocity field and a Plateau-Neumann type boundary condition for the map. In this paper we construct smooth solutions of this coupled system that blow up in finite time at any finitely many given points on the boundary or in the interior of the domain.

On singularities of Ericksen-Leslie system in dimension three

In this paper, we consider the initial and boundary value problem of Ericksen-Leslie system modeling nematic liquid crystal flows in dimension three. Two examples of singularity at finite time are constructed. The first example is constructed in a special axisymmetric class with suitable axisymmetric initial and boundary data, while the second example is constructed for initial data with small energy but nontrivial topology. A counter example of maximum principle to the system is constructed by utilizing the Poiseuille flow in dimension one.

Error estimates of a sphere-constraint-preserving numerical scheme for Ericksen-Leslie system with variable density

In this paper, a new first-order, sphere-constraint-preserving numerical scheme is constructed for the Ericksen-Leslie equations with variable density. Firstly, by denoting the orientation field vector $ \pmb{d} $ in the polar coordinate, we rewrite the Ericksen-Leslie system into an equivalent system such that the sphere constraint $ \lvert \pmb{d}\rvert = 1 $ can be still preserved at the discrete level. Secondly, we propose a first-order discretization scheme in time for the equivalent new system in which the numerical velocity and modified pressure are determined by a generalized Stokes problem at each time step. Then, the unconditional energy stability and the rigorous error estimates in time are both derived. Finally, some numerical simulations in two dimensions are provided to demonstrate the stability of energy and accuracy of the presented scheme.

An energy stable finite difference scheme for the Ericksen–Leslie system with penalty function and its optimal rate convergence analysis

An energy stable finite difference scheme for the Ericksen–Leslie system with penalty function and its optimal rate convergence analysis

Global Well-Posedness for the Compressible Nematic Liquid Crystal Flows

In this paper, we prove the unique existence of global strong solutions and decay estimates for the simplified Ericksen–Leslie system describing compressible nematic liquid crystal flows in RN, 3≤N≤7. Firstly, we rewrite the system in Lagrange coordinates, and secondly, we prove the global well-posedness for the transformed system, which is the main task in this paper. The proof is based on the maximal Lp-Lq regularity and the Lp-Lq decay estimates to the linearized problem.

Liouville Theorems for a Stationary and Non-stationary Coupled System of Liquid Crystal Flows in Local Morrey Spaces

We consider here the simplified Ericksen–Leslie system on the whole space $$\mathbb {R}^{3}$$ . This system deals with the incompressible Navier–Stokes equations strongly coupled with a harmonic map flow which models the dynamical behavior for nematic liquid crystals. For both, the stationary (time independent) case and the non-stationary (time dependent) case, using the fairly general framework of a kind of local Morrey spaces, we obtain some a priori conditions on the unknowns of this coupled system to prove that they vanish identically. This results are known as Liouville-type theorems. As a bi-product, our theorems also improve some well-known results on Liouville-type theorems for the particular case of classical Navier–Stokes equations.

Global weak solutions to the stochastic Ericksen–Leslie system in dimension two

<p style='text-indent:20px;'>We establish the global existence of weak martingale solutions to the simplified stochastic Ericksen–Leslie system modeling the nematic liquid crystal flow driven by Wiener-type noises on the two-dimensional bounded domains. The construction of solutions is based on the convergence of Ginzburg–Landau approximations. To achieve such a convergence, we first utilize the concentration-cancellation method for the Ericksen stress tensor fields based on a Pohozaev type argument, and then the Skorokhod compactness theorem, which is built upon uniform energy estimates.</p>

Existence of Sobolev regular solutions for the incompressible flow of liquid crystals in three dimensions

<abstract><p>This paper considers a simplified three dimensional Ericksen-Leslie System for nematic liquid crystal flows in the unbounded domain $ \Omega: = \mathbb R^+\times \mathbb R^2 $ or the smooth bounded domain $ \Omega $. The hydrodynamic system consists of the Navier-Stokes type equations for the fluid velocity coupled with a convective Ginzburg-Landau type equation for the averaged molecular orientation. We first establish the global existence of Sobolev regular solution with finite energies in Sobolev space $ H^{s}(\Omega)\times H^{s}(\Omega) $, where the index $ s $ of the Sobolev space can be any large fixed integer, but $ s\neq+\infty $. Then we give an asymptotic expansions of a family of Sobolev regularity solutions for such system in $ \Omega $.</p></abstract>

Large Ericksen number limit for the 2D general Ericksen–Leslie system

In this paper, we address the issue of the so-called large Ericksen number limit for the general Ericksen–Leslie system of liquid crystals. By the standard nondimensionalization, we first prove that there exists a unique local smooth solution to the limiting system. Then we provide a qualitative estimate for the difference between the solutions of the general Ericksen–Leslie system and the limiting system, which corresponds to the partial decoupling effect in the limit of large Ericksen number.

The zero inertia limit from hyperbolic to parabolic Ericksen-Leslie system of liquid crystal flow

The original Ericksen-Leslie consists the conservation laws of the nematic liquid crystals, which includes the inertial effect. Thus the full Ericksen-Leslie system has hyperbolic feature. In this paper we study the zero inertia limit that is from the hyperbolic to parabolic Ericksen-Leslie's liquid crystal flow. By introducing an initial layer and constructing an energy norm and energy dissipation functional depending on the solutions of the limiting system, we derive a global in time uniform energy bound to the remainder system under the small size of the initial data. This work justifies the validity of the well-known parabolic Ericksen-Leslie system without inertial term.