Xueke Pu, Meng Wang, Wendong Wang
In this paper, we consider the Landau-Lifshitz equation of the ferromagnetic spin chain from $\R^2$ to the unit sphere $S^2$ under the general Oseen-Frank energy. We obtain global existence and uniqueness of weak solutions for large energy data; moreover, the number of singular points is finite.
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We study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of $Γ$-development we recover Landau-de Gennes corrections to the Oseen-Frank energy. We provide an explicit characterisation of minimizing $Q$-tensors at this order in terms of optimal Oseen-Frank directors and observe the emerging biaxiality. We apply our results to…
We study the relationship between Onsager's molecular theory and the Oseen-Frank theory for nematic liquid crystals. Under the molecular setting, we consider the free energy that includes the effects of nonlocal molecular interactions. By imposing the strong anchoring boundary condition on the second moment of the number density function, we prove the existence of global minimizers for the free energy. Moreover, when the re-scaled interaction distance tends to zero, the corresponding global minimizers will converge to a uniaxial distribution whose orientation is described by a minimizer of Ose…
In this article, we prove the existence of measure-valued solutions to the Ericksen-Leslie system equipped with the Oseen-Frank energy. We introduce the concept of generalized gradient Young measures. Via a Galerkin approximation, we show the existence of weak solutions to a regularized system and attain measure-valued solutions for vanishing regularization. Additionally, it is shown that the measure-valued solution fulfills an energy inequality.
In the first part of this paper, we establish global existence of solutions of the liquid crystal (gradient) flow for the well-known Oseen-Frank model. The liquid crystal flow is a prototype of equations from the Ericksen-Leslie system in the hydrodynamic theory and generalizes the heat flow for harmonic maps into the 2-sphere. The Ericksen-Leslie system is a system of the Navier-Stokes equations coupled with the liquid crystal flow. In the second part of this paper, we also prove global existence of solutions of the Ericksen-Leslie system for a general Oseen-Frank model in $\Bbb R^2$.
We investigate the Ericksen-Leslie system for the Oseen-Frank model with unequal Frank elastic constants in $\mathbb{R}^3$. To generalize the result of Hineman-Wang \cite{HW}, we prove existence of solutions to the Ericksen-Leslie system with initial data having small $L^3_{uloc}$-norm. In particular, we use a new idea to obtain a local $L^3$-estimate through interpolation inequalities and a covering argument, which is different from the one in \cite{HW}. Moreover, for uniqueness of solutions, we find a new way to remove the restriction on the Frank elastic constants by using the rotation inva…
We analyze the Ericksen-Leslie system equipped with the Oseen-Frank energy in three space dimensions. Recently, the author introduced the concept of measure-valued solutions to this system and showed the global existence of these generalized solutions. In this paper, we show that suitable measure-valued solutions, which fulfill an associated energy inequality, enjoy the weak-strong uniqueness property, i. e. the measure-valued solution agrees with a strong solution if the latter exists. The weak-strong uniqueness is shown by a relative energy inequality for the associated nonconvex energy func…
Boundary regularity for Oseen-Frank minimizers with arbitrary positive splay, twist, and bend constants
针对Oseen-Frank能量的全局极小化子，在无小各向异性假设下证明其边界正则性，解决了Lin和Liu问题(c)的边界正则性部分。
Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond
我们研究了三维区域中单轴边界条件下连续Landau-De Gennes能量泛函的全局极小。我们分析了物理相关的小弹性常数的限制，并表明，全球极小收敛强烈，在$W^{1,2}$，一个全球极小的Oseen-Frank理论预测的单轴液晶恒定序参数。此外，收敛是均匀的内部的域，远离奇点的限制Oseen-Frank全球极小。我们得到了Landau-De Gennes全局极小值的特征值的收敛速率和特征向量的正则性的结果。我们还研究了双轴性和单轴性之间的相互作用，Landau-De Gennes的整体能量极小，并获得各种相关的数量，如双轴性参数和大小的可容许强双轴区域的估计。
In this work we recover the Oseen-Frank theory of nematic liquid crystals as a $Γ$-limit of a particular mean-field free energy as the sample size becomes infinitely large. The Frank constants are not necessarily all equal. Our analysis takes place in a broader framework however, also providing results for more general systems such as biaxial or polar molecules. We start from a mesoscopic model describing a competition between entropy and a non-local pairwise interactions between molecules. We assume the interaction potential is separable so that the energy can be reduced to a model involving …
Systems of elongated molecules, doped with small amounts of molecules lacking mirror symmetry can form macroscopically twisted cholesteric liquid crystal phases. The aim of this work is to rigorously derive the Oseen-Frank model of cholesterics from a more fundamental model concerned with pairwise molecular interactions. A non-local mean-field model of the two-species nematic host/chiral dopant mixture is proposed, and it is shown that Oseen-Frank's elastic free energy for cholesteric liquid crystals can be obtained in a simultaneously large-domain and dilute-dopant asymptotic regime. By techn…
We analyze the Ericksen-Leslie system equipped with the Oseen-Frank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measure-valued solutions to the considered system and showed global existence as well as weak-strong uniqueness of these generalized solutions. In this paper, we show that the expectation of the measure valued solution is a dissipative solution. The concept of a dissipative solution itself relies on an inequality instead of an equality, but is described by functions instead of parametrized me…
Zero-homogeneous and $O(2)$-equivariant critical points of the Oseen-Frank energy with multiple Frank constants
本文证明了具有多个Frank常数的Oseen-Frank能量存在零齐次且$O(2)$等变的临界点，这些点具有度1拓扑缺陷，并在轴外扰动下极小化。
Dimensional Reduction and emergence of defects in the Oseen-Frank model for nematic liquid crystals
本文研究薄平板域内Oseen-Frank向列相液晶模型的厚度趋于零极限，严格证明有限个拓扑缺陷点的涌现，其位置由重整化能决定，对应二维Ginzburg-Landau泛函的渐近行为。
Variational principles for the interaction of liquid crystals and electric fields in the Oseen--Frank model
研究单轴向列型液晶与外部电场相互作用，通过变分原理将问题重述为双极小化问题，用矢量势代替标量静电势，量化静电反作用解耦，证明物理极小值收敛到极限调和映射，利于理论分析与数值模拟。
We consider the relationship between three continuum liquid crystal theories: Oseen-Frank, Ericksen and Landau-de Gennes. It is known that the function space is an important part of the mathematical model and by considering various function space choices for the order parameters $s$, ${\bf n}$, and ${\bf Q}$, we establish connections between the variational formulations of these theories. We use these results to derive a version of the Oseen-Frank theory using special functions of bounded variation. This proposed model can describe both orientable and non-orientable defects. Finally we study a…
We study a class of Landau-de Gennes energy functionals in the asymptotic regime of small elastic constant $L>0$. We revisit and sharpen the results in [18] on the convergence to the limit Oseen-Frank functional. We examine how the Landau-de Gennes global minimizers are approximated by the Oseen-Frank ones by determining the first order term in their asymptotic expansion as $L\to 0$. We identify the appropriate functional setting in which the asymptotic expansion holds, the sharp rate of convergence to the limit and determine the equation for the first order term. We find that the equation has…
This paper concerns the uniqueness of weak solutions to the Cauchy problem to the Ericksen-Leslie system of liquid crystal models in $\mathbb R^2$, with both general Leslie stress tensors and general Oseen-Frank density. It is shown here that such a system admits a unique weak solution provided that the Frank coefficients are close to some positive constant, which solves an interesting open problem. One of the main ideas of our proof is to perform suitable energy estimates at level one order lower than the natural basic energy estimates for the Ericksen-Leslie system.
From microscopic theory to macroscopic theory: a systematic study on static modeling for liquid crystals
本文提出一种系统性的液晶建模方法，连接微观与宏观理论，提出基于Onsager分子理论的Q张量理论，并推导出Oseen-Frank理论及向列相的特征。
Within the framework of the generalized Landau-de Gennes theory, we identify a $Q$-tensor-based energy that reduces to the four-constant Oseen-Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the $Q$-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimization problem is well-posed for a significantly wider ch…
Uniaxial nematic liquid crystals are modelled in the Oseen-Frank theory through a unit vector field $n$. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which $n$ should be equivalent to -$n$. This symmetry is preserved in the constrained Landau-de Gennes theory that works with the tensor $Q=s\big(n\otimes n- \frac{1}{3} Id\big)$.We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural …
We consider an energy functional motivated by the celebrated $K_{13}$ problem in the Oseen-Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an additional surface term. It is known that this energy is unbounded from below and our aim has been to study the local minimizers. We show that even having a critical point in a suitable energy space imposes severe restrictions on the boundary conditions. Having suitable boundary conditions makes the energy functional bounded and in this case we study the partial regularity …
In the first part of this paper, we will consider minimizing configurations of the Oseen-Frank energy functional $E(n, m)$ for a biaxial nematics $(n, m):Ω\to \mathbb S^2\times \mathbb S^2$ with $n\cdot m=0$ in dimension three, and establish that it is smooth off a closed set of $1$-dimension Hausdorff measure zero. In the second part, we will consider a simplified Ericksen-Leslie system for biaxial nematics $(n, m)$ in a two dimensional domain and establish the existence of a unique global weak solution $(u, n, m)$ that is smooth off at most finitely many singular times for any initial and bo…
We consider $\mathbb{S}^2$-valued maps on a domain $Ω\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy with vertical penalization in $Ω$ and horizontal penalization on $\partialΩ$. We first show the global minimality of universal constant configurations in a specific range of the physical parameters using a Poincaré-type inequality. Then, we prove that any energy minimizer takes its values into a fixed meridian of the sphere $\mathbb{S}^2$, and deduce uniqueness of minimizers up to the action of the appropriate symmetry group. We also prove a comparison principle for minim…
Energy-variational structure in evolution equations
本文利用偏微分方程系统的能量变分结构，证明一类改进测度值解可等价表述为能量变分解，并通过四类演化方程说明该结构具有普遍性。
On the asymptotic limit for the dynamic isotropic-nematic phase transition with anisotropic elasticity
该研究基于朗道-德热纳Q张量理论，严格证明当各向异性弹性常数L₂为负时，液晶各向同性-向列相转变动力学的渐近极限，验证了德热纳的相关论断。
We study the small Deborah number limit of the Doi-Onsager equation for the dynamics of nematic liquid crystals without hydrodynamics. This is a Smoluchowski-type equation that characterizes the evolution of a number density function, depending upon both particle position $x\in \mathbb{R}^d(d=2,3)$ and orientation vector $m\in\mathbb{S}^2$ (the unit sphere). We prove that, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge strongly to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat f…
The Landau-de Gennes energy in nematic liquid crystals depends on four elastic constants $L_1$, $L_2$, $L_3$, $L_4$. In the case of $L_4\neq 0$, Ball and Majumdar (Mol. Cryst. Liq. Cryst., 2010) found an example that the original Landau-de Gennes energy functional in physics does not satisfy a coercivity condition, which causes a problem in mathematics to establish existence of energy minimizers. At first, we introduce a new Landau-de Gennes energy density with $L_4\neq 0$, which is equivalent to the original Landau-de Gennes density for uniaxial tensors and satisfies the coercivity condition …
Phase transition of anisotropic Ginzburg--Landau equation
本文研究了各向异性 Ginzburg--Landau 方程在小参数趋于零时的有效几何运动，证明解收敛于平均曲率流下的锐界面极限，并利用调制能量法与弱收敛方法揭示其与液晶 Oseen--Frank 模型的联系。
Orientation Waves in a Director Field With Rotational Inertia
从液晶连续介质理论出发，研究了取向波在具有转动惯量和势能的指向矢场中的传播。有两种类型的波，我们称之为张开波和扭曲波。弱非线性展波由二次非线性Hunter-Saxton方程描述。在这里，我们表明，弱非线性扭曲波描述了一个新的立方非线性，完全可积的渐近方程。这个方程提供了一个令人惊讶的亨特-萨克斯顿方程作为平流方程的代表。有一个类似的表示卡马萨-霍尔姆方程。利用该渐近方程分析了具有扭转波初值的一维指向矢场方程的初值问题。
A Landau-de Gennes Type Theory for Cholesteric-Helical Smectic-Smectic C* Liquid Crystal Phase Transitions
本文提出一种改进的Landau-de Gennes理论，用于研究温度驱动的胆甾型-螺旋取向相-取向C*相转变，并通过数学分析和数值模拟验证了相变过程。
For the Landau-de Gennes functional on 3D domains, \begin{equation*} I_{\varepsilon}(Q,Ω):=\int_Ω\left\{\frac{1}{2}|\nabla Q|^2+\frac{1}{\varepsilon^2}\left( -\frac{a^2}{2}\mathrm{tr}(Q^2)-\frac{b^2}{3}\mathrm{tr}(Q^3)+\frac{c^2}{4}[\mathrm{tr}(Q^2)]^2 \right) \right\}\,dx, \end{equation*} it is well-known that under suitable boundary conditions, the global minimizer $Q_\varepsilon$ converges strongly in $H^1(Ω)$ to a uniaxial minimizer $Q_*=s_+(n_*\otimes n_*-\frac{1}{3}\mathrm{Id})$ up to some subsequence $\varepsilon_n\rightarrow\infty$ , where $n_*\in H^1(Ω,\mathbb{S}^2)$ is a minimizing h…
We consider energy minimizing configurations of a nematic liquid crystal around a spherical colloid particle, in the context of the Landau-de Gennes model. The nematic is assumed to occupy the exterior of a ball of radius r_0, satisfy homeotropic weak anchoring at the surface of the colloid, and approach a uniform uniaxial state at infinity. We study the minimizers in two different limiting regimes: for balls which are small compared to the characteristic length scale r_0<<L, and for large balls, r_0>>L. The relationship between the radius and the anchoring strength W is also relevant. For sma…
Multiscale Analysis of a Landau--de Gennes Model for Nematic thin-film composites
该研究在Landau–de Gennes Q张量框架下，首次严格处理向列相薄膜的同步均匀化与降维极限，锚定强度作为标度参数产生不同有效模型，有效弹性响应依赖于标度比ρ。
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