Nematic Liquid Crystal Flows Research Articles

Uniqueness of hydrodynamic flow of nematic liquid crystals in $\boldsymbol{L^n}$

In this paper, we consider the Cauchy problem of hydrodynamic flow of nematic liquid crystals. We get the uniqueness of the mild solution in $C([0,T);L^n)$.

Global Existence of Classical Solutions with Large Oscillations and Vacuum to the Three-Dimensional Compressible Nematic Liquid Crystal Flows

This paper is concerned with a simplified hydrodynamic flow modeling the three-dimensional motion of compressible, nematic liquid crystal materials. The authors establish the global existence of classical solution to the Cauchy problem with smooth initial data which are of small energy but possibly large oscillations with constant state as far-field condition which could be either vacuum or non-vacuum. The initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large, in particular, the initial density can even have compact support. As a byproduct, the large-time behavior of the solution is also studied.

On some large global solutions to the incompressible inhomogeneous nematic liquid crystal flows

ABSTRACTIn this paper, we investigate the Cauchy problem for the incompressible inhomogeneous nematic liquid crystal flows in with initial data in some critical Besov spaces. Under the assumptions that initial density is close to the equilibrium state and the smallness on the initial direction field and nonlinear smallness on the linearized velocity, we obtain the global existence of solutions by fully using the Fourier localization technique and the advantage of weighted function generated by heat kernel. The meanings of this kind of initial data are that we can provide an example of large highly oscillating data satisfying that nonlinear smallness assumption, despite each component of the initial data could be arbitrarily large.

On temporal decay of solution to the three‐dimensional compressible flow of nematic liquid crystal in Besov space

In this paper, we investigate large time behavior of global‐in‐time strong solution to the three‐dimensional compressible flow of nematic liquid crystal with low regularity assumptions on initial datum. More precisely, we show that the negative Besov space ‐norms (s ≥ 0) of solution are preserved along time evolution; by using this fact together with the conventional energy estimates in Besov space framework and the interpolation inequalities, we establish that, for the initial perturbation just small in homogeneous Besov space , the global‐in‐time strong solution to the Cauchy problem of the compressible flow of nematic liquid crystal has the following optimal temporal decay rate: urn:x-wiley:mma:media:mma5176:mma5176-math-0004 provided that we further assume that still belongs to . Here, is a constant unite vector. To illustrate our methods clearly, we also revisit the optimal temporal decay of solutions to the heat equation in the framework of Besov space.

Optimal Decay Rate for the Compressible Flow of Liquid Crystals in $$L^p$$ L p Type Critical Spaces

The purpose of this paper is to establish the optimal time decay rate of solutions for the compressible flow of nematic liquid crystals in $$L^p$$ type critical framework and any dimension $$N\ge 2$$ . Concretely, if the low frequencies of the initial data are in e.g. $$L^{p/2}(\mathbb {R}^N)$$ , we could obtain that the $$L^p$$ norm of the critical global solution $$(\rho -1,\mathbf{u})$$ decays like $$t^{-\frac{N}{p}+ \frac{N}{4}}$$ , while the $$L^p$$ norm of $$\mathbf{d}-\hat{\mathbf{d}}$$ decays like $$t^{-\frac{N}{2p}}$$ for $$t\rightarrow \infty $$ , exactly as shown in the work by Gao et al. (J Differ Equ 261:2334–2383, 2016) when $$p=2$$ and $$N=3$$ , for solutions with high Sobolev regularity. As a by-product, we derive an accurate description of the time decay rates, not only for Lebesgue spaces, but also for a family of Besov norms with negative or nonnegative regularity exponents, which improves the decay results in high Sobolev regularity. The main tools used here are the Littlewood–Paley theory and refined time weighted inequalities in Fourier space.

Dimension of singularities to the 3d simplified nematic liquid crystal flows

In this paper, we study the possible singular set of suitable weak solutions (u,d) to the 3d simplified nematic liquid crystal flows, and prove that the parabolic upper Minkowski dimension of the possible singular set is bounded by 9563. Moreover, if the suitable weak solution (u,d) of the 3d simplified nematic liquid crystal flows satisfies (u,∇d)∈Lw(0,T;Ls(R3))with3<w,s<∞,then the parabolic upper Minkowski dimension of the singular set is no greater than max{w,s}(2w+3s−1). If the suitable weak solution (u,d) satisfies (u,∇d)∈L∞(0,T;L3,∞(R3)),where L3,∞(R3) denotes the standard weak L3-Lebesgue space, then the parabolic upper Minkowski dimension of the singular set is bounded by 1.

Global regularity to the 3D incompressible nematic liquid crystal flows with vacuum

This paper is concerned with the 3D incompressible nematic liquid crystal equations in smooth bounded domains, where the velocity u and macroscopic molecular orientation d admit the Dirichlet and Neumann boundary condition respectively. Global-in-time strong solution is proved to exist when the basic energy is suitably small. The initial vacuum is allowed.

A note on a global strong solution to the 2D Cauchy problem of density-dependent nematic liquid crystal flows with vacuum

In Li–Liu–Zhong (Nonlinearity 30 (2017) 4062–4088), the authors proved the existence of a unique global strong solution to the Cauchy problem of 2D nonhomogeneous incompressible nematic liquid crystal flows with vacuum as far-field density provided the initial data density and the gradient of orientation decay not too slow at infinity, and the basic energy ‖ρ0u0‖L22+‖∇d0‖L22 is small. In this note, we aim at precisely describing this smallness condition.

Electrically modulated capillary filling imbibition of nematic liquid crystals.

The flow of nematic liquid crystals (NLCs) in the presence of an electric field is typically characterized by the variation in its rheological properties due to transition in its molecular arrangements. Here, we bring out a nontrivial interplay of a consequent alteration in the resistive viscous effects and driving electrocapillary interactions, toward maneuvering the capillary filling dynamics over miniaturized scales. Considering a dynamic interplay of the relevant bulk and interfacial forces acting in tandem, our results converge nicely to previously reported experimental data. Finally, we attempt a scaling analysis to bring forth further insight to the reported observations. Our analysis paves the way for the development of microfluidic strategies with previously unexplored paradigms of interaction between electrical and fluidic phenomenon, providing with an augmented controllability on capillary filling as compared to tthose reported to be achievable by the existing strategies. This, in turn, holds utilitarian scopes in improved designs of functional capillarities in electro-optical systems, electrorheological utilities, electrokinetic flow control, as well as in interfacing and imaging systems for biomedical microdevices.

On the Existence of Weak Solutions to the Steady Compressible Flow of Nematic Liquid Crystals

In this paper, we consider the existence of weak renormalized solutions for the steady compressible flow of nematic liquid crystals in a 3-D bounded domain with no-slip boundary condition. By using a three-level approximation scheme, we establish the existence of weak renormalized solutions under the hypothesis $$\gamma >1$$ for the adiabatic constant.

Global strong solutions for the incompressible nematic liquid crystal flows with density-dependent viscosity coefficient

This paper is concerned with an initial–boundary value problem of the incompressible nematic liquid crystal flows with density-dependent viscosity in a smooth bounded domain Ω⊂R3. The global well-posedness of strong solutions with large oscillations is established in vacuum, provided ‖∇u0‖L2+‖Δd0‖L2 is suitably small with arbitrary large initial density, which extended the local strong solution by Gao et al. [12] to be a global one.

Global low-energy weak solution and large-time behavior for the compressible flow of liquid crystals

In this paper, we consider the weak solution of the simplified Ericksen–Leslie system modeling compressible nematic liquid crystal flows in R3. When the initial data are of small energy and initial density is positive and essentially bounded, we prove the existence of a global weak solution in R3. The large-time behavior of a global weak solution is also established.

Blow-up of the Smooth Solution to the Compressible Nematic Liquid Crystal System

In this paper, we investigate the blow-up of the smooth solutions to a simplified Ericksen-Lesile system for compressible flows of nematic liquid crystals in different dimensional case. We prove that whether the smooth solution of the Cauchy problem or the initial-boundary problem to the nematic liquid crystal system will blow up in finite time. The main technique is the construction of the functional differential inequality.

On center singularity for compressible spherically symmetric nematic liquid crystal flows

The paper is concerned with a simplified system, proposed by Ericksen [6] and Leslie [20], modeling the flow of nematic liquid crystals. In the first part, we give a new Serrin's continuation principle for strong solutions of general compressible liquid crystal flows. Based on new observations, we establish a localized Serrin's regularity criterion for the 3D compressible spherically symmetric flows. It is proved that the classical solution loses its regularity in finite time if and only if, either the concentration or vanishing of mass forms or the norm inflammation of gradient of orientation field occurs around the center.

Global solution to the 3D inhomogeneous nematic liquid crystal flows with variable density

In this paper, we investigate the global existence and uniqueness of solution to the 3D inhomogeneous incompressible nematic liquid crystal flows with variable density in the framework of Besov spaces. It is proved that there exists a global and unique solution to the nematic liquid crystal flows if the initial data (ρ0−1,u0,n0−e3)∈M(B˙p,13p−1(R3))×B˙p,13p−1(R3)×B˙p,13p(R3) with 1≤p<6, and satisfies‖ρ0−1‖M(B˙p,13p−1)+‖u0‖B˙p,13p−1+‖n0−e3‖B˙p,13p≤cfor some small c>0 depending only on p. Here M(B˙p,13p−1(R3)) is the multiplier space of Besov space B˙p,13p(R3). Using the Lagrangian approach in Danchin and Mucha (2012, 2013) [7,8] is the key to our results.

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

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.

Incompressible limit for the compressible flow of liquid crystals in &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ L^p$\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; type critical Besov spaces

The present paper is devoted to the compressible nematic liquid crystal flow in the whole space $ \mathbb{R}^N\,(N≥ 2)$. Here we concentrate on the incompressible limit in the $ L^p$ type critical Besov spaces setting. We first establish the existence of global solutions in the framework of $ L^p$ type critical spaces provided that the initial data are close to some equilibrium states. Based on the global existence, we then consider the incompressible limit problem in the ill prepared data case. We justify the low Mach number convergence to the incompressible flow of liquid crystals in proper function spaces. In addition, the accurate converge rates are obtained.

Hydrodynamic instabilities at decay flow of nematic liquid crystals through the plane capillary of a variable gap

In this paper we consider the possibility to study hydrodynamic instabilities (HDI) in a specific type of a decay flow existing at a rise of a nematic liquid crystal (NLC) in the flat capillary of a variable gap. In this case the capillary can be replaced by a number of parallel channels of different gaps with different instant values of a pressure gradient, which induces a flow. The last feature makes such flow to be different from the previously studied decay flow [1] with a constant value of a pressure gradient, applied to the channels of different thickness.

Nonuniqueness of nematic liquid crystal flows in dimension three

For suitable initial and boundary data, we construct infinitely many weak solutions to the nematic liquid crystal flows in dimension three. These solutions are in the axisymmetric class with bounded energy and “backward bubbling” at a large time.

Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum

We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space with vacuum as far field density. It is proved that the 2D nonhomogeneous incompressible nematic liquid crystal flows admits a unique global strong solution provided the initial data density and the gradient of orientation decay not too slow at infinity, and the basic energy is small. In particular, the initial density may contain vacuum states and even have compact support. Moreover, the large time behavior of the solution is also investigated.