Xianpeng Hu, Dehua Wang
The equations for the three-dimensional incompressible flow of liquid crystals are considered in a smooth bounded domain. The existence and uniqueness of the global strong solution with small initial data are established. It is also proved that when the strong solution exists, all the global weak solutions constructed in [16] must be equal to the unique strong solution.
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We study the motion of an incompressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler's equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a "physical condition", related to the fact that the pressure of a fluid has to be positive.
In this paper, we consider the Dirichlet problem of inhomogeneous incompressible nematic liquid crystal equations in bounded smooth domains of two or three dimensions. We prove the global existence and uniqueness of strong solutions with initial data being of small norm but allowed to have vacuum. More precisely, for two dimensional case, we only require that the basic energy $|\sqrt{ρ_0}u_0|_{L^2}^2+|\nabla d_0|_{L^2}^2$ is small, while for three dimensional case, we ask for the smallness of the production of the basic energy and the quantity $|\nabla u_0|_{L^2}^2+|\nabla^2d_0|_{L^2}^2$. Our …
The connection between the compressible flow of liquid crystals with low Mach number and the incompressible flow of liquid crystals is studied in a bounded domain. In particular, the convergence of weak solutions of the compressible flow of liquid crystals to the weak solutions of the incompressible flow of liquid crystals is proved when the Mach number approaches zero; that is, the incompressible limit is justified for weak solutions in a bounded domain.
We study the problem of the motion of the free surface of a liquid. We prove existence and stability for the linearized equations.
In this paper, we consider the non-isothermal model for incompressible flow of nematic liquid crystals in three dimensions and prove the local existence and uniqueness of the strong solution with periodic initial conditions on $ \mathbb{T}^3$ .
The present paper is dedicated to the global large solutions and incompressible limit for the compressible flow of liquid crystals under the assumption on almost constant density and large volume viscosity. The result is based on Fourier analysis and involved so-called critical Besov norm.
This paper is concerned with the incompressible limit of the compressible hydrodynamic flow of liquid crystals with periodic boundary conditions in R^N(N = 2, 3). It is rigorously shown that the local (and global) strong solution of the compressible system converges to the local (and global) strong solution of the incompressible system. Furthermore, the convergence rates are also obtained in some sense.
This article is devoted to the study of the so-called incompressible limit for solutions of the compressible liquid crystals system. We consider the problem in the whole space $\mathbb{R}^{\mathbb{N}}$ and a bounded domain of $\mathbb{R}^{\mathbb{N}}$ with Dirichlet boundary conditions. Here the number of dimension $\mathbb{N}=2$ or 3.
In this paper we construct a family of exact strong solutions to the two-dimensional incompressible liquid crystal equations with finite energy. The initial velocity is chosen to be rotationally symmetric and the image of the initial orientation of the liquid crystal is a non-trivial curve on the unit sphere. It turns out that this family of initial data evolves globally in time by liquid crystal flow and may shrink to a single point as time goes to infinity.
The initial boundary value problem for the three-dimensional incompressible flow of liquid crystals is considered in a bounded smooth domain. The existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small data. It is also proved that when the strong solution exists, a weak solution must be equal to the unique strong solution with the same data.
In this paper, we prove a priori estimates in Lagrangian coordinates for the equations of motion of an incompressible, inviscid, self-gravitating fluid with free boundary. The estimates show that on a finite time interval we control five derivatives of the fluid velocity and five and a half derivatives of the coordinates of the moving domain.
In this paper, we investigate the Cauchy problem for the incompressible nematic liquid crystal flows in three-dimensional whole space. First of all, we establish the global existence of solution by energy method under assumption of small initial data. Furthermore, the time decay rates of velocity and director are built when the initial data belongs to $L^1(\mathbb{R}^3)$ additionally. Finally, one also constructs the time convergence rates for the mixed space-time derivatives of velocity and director.
The initial-boundary value problem for the density-dependent incompressible flow of liquid crystals is studied in a three-dimensional bounded smooth domain. For the initial density away from vacuum, the existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small data. It is also proved that when the strong solution exists, a weak solution with the same data must be equal to the unique strong solution.
This paper concerns the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space $\mathbb{R}^{2}$ 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 initial orientation satisfies a geometric condition (see \eqref{eq1.3}). In particular, the initial data can be arbitrarily large and the initial density may contain vacuum st…
The Cauchy problem for the compressible flow of nematic liquid crystals in the framework of critical spaces is considered. We first establish the existence and uniqueness of global solutions provided that the initial data are close to some equilibrium states. This result improves the work by Hu and Wu [SIAM J. Math. Anal., 45 (2013), pp. 2678-2699] through relaxing the regularity requirement of the initial data in terms of the director field. We then consider the incompressible limit problem for ill prepared initial data. We prove that as the Mach number tends to zero, the global solution to t…
In this paper we derive LPS's criterion for the breakdown of classical solutions to the incompressible nematic liquid crystal flow, a simplified version of Ericksen-Leslie system modeling the hydrodynamic evolution of nematic liquid crystals in $\mathbb R^3$. We show that if $0<T<+\infty$} is the maximal time interval for the unique smooth solution $u\in C^\infty([0,T),\mathbb R^3)$, then $|u|+|\nabla d|\notin L^q([0,T],L^p(\mathbb R^3))$, where $p$ and $q$ safisfy the Ladyzhenskaya-Prodi-Serrin's condition: $\frac{3}{p}+\frac{2}{q}=1$ and $p\in(3,+\infty]$
We study the long-time behavior of global strong solutions to a hydrodynamic system for nonhomogeneous incompressible nematic liquid crystal flows driven by two types of external forces in a smooth bounded domain in $\mathbb{R}^2$. For arbitrary large regular initial data with the initial density being away from vacuum, we prove the decay of the velocity field for both cases. Furthermore, for the case with asymptotically autonomous external force, we can prove the convergence of the density function and the director vector as time goes to infinity. Estimates on convergence rate are also provid…
In this paper, the authors first establish the global well-posedness of strong solutions of the simplified Ericksen-Leslie model for nonhomogeneous incompressible nematic liquid crystal flows in two dimensions if the initial data satisfies some smallness condition. It is worth pointing out that the initial density is allowed to contain vacuum states and the initial velocity can be arbitrarily large. We also present a Serrin's type criterion, depending only on $\nabla d$, for the breakdown of local strong solutions. As a byproduct, the global strong solutions with large initial data are obtaine…
We study the Ericksen-Leslie's hyperbolic incompressible liquid crystal model. Under some constraints on the Leslie coefficients which ensure the basic energy law is dissipative, we prove the local-in-time existence and uniqueness of the classical solution to the system with finite initial energy. Furthermore, with an additional assumption on the coefficients which provides a damping effect, and the smallness of the initial energy, the unique global classical solution can be established.
In this paper, we investigate the density-dependent incompressible nematic liquid crystal flows in $n(n=2$ or $3)$ dimensional bounded domain. More precisely, we obtain the local existence and uniqueness of the solutions when the viscosity coefficient of fluid depends on density. Moreover, we establish blowup criterions for the regularity of the strong solutions in dimension two and three respectively. In particular, we build a blowup criterion just in terms of the gradient of density if the initial direction field satisfies some geometric configuration. For these results, the initial density …
In this paper, we consider the short time classical solution to a simplified hydrodynamic flow modeling incompressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at a finite time. More precisely, if $(u,d)$ is smooth up to time $T$ provided that $\int_0^T|\nabla\times u(t,\cdot)|_{BMO(\mathbb R^3)}+|\nabla d(t,\cdot)|^8_{L^4(\mathbb R^3)}dt<\infty $
In this paper, we prove a new type of energy estimates for the compressible Euler's equation with free boundary, with a boundary part and an interior part. These can be thought of as a generalization of the energies in Christodoulou and Lindblad [CL] to the compressible case and do not require the fluid to be irrotational. In addition, we show that our estimates are in fact uniform in the sound speed. As a consequence, we obtain convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Ebin [Eb] to whe…
Double Porosity Models for Liquid Filtration in Incompressible Poroelastic Media
本文基于均匀化理论，通过迭代均匀化方法推导了不可压缩多孔弹性介质中液体过滤的双孔隙度模型，得到了Biot-Terzaghi方程组及绝对刚性体的对应模型。
Correctness of Biot's model of in situ leaching for incompressible liquid and compressible solid components
本文研究了原地浸出稀有金属的数学模型，通过固定点定理解决自由边界问题，推导出宏观模型并证明其唯一性。
Incompressible limit of the Ericksen-Leslie hyperbolic liquid crystal model in compressible flow
本文在经典解框架下研究可压缩Ericksen-Leslie抛物-双曲液晶模型的不可压缩极限，通过引入惯性常数并建立关于Mach数一致的能量估计，证明了不可压缩系统全局经典解的存在性并给出收敛速率。
We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order.
Koch-Tataru theorem for 3D incompressible active nematic liquid crystals
本文基于Beris-Edwards框架研究了三维不可压缩活性向列液态晶体的流体动力学系统，通过Kato策略和Banach压缩原理，证明了在小初始数据下解的存在性和唯一性，首次在临界空间中获得该系统的适定性结果。
We consider a hydrodynamic system that models the Smectic-A liquid crystal flow. The model consists of the Navier-Stokes equation for the fluid velocity coupled with a fourth-order equation for the layer variable $\vp$, endowed with periodic boundary conditions. We analyze the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first prove that in 2D, the problem possesses a global attractor $\mathcal{A}$ in certain phase space. Then we establish the existence of an exponential attractor $\mathcal{M}$ which entails that the global att…
Thermal relaxation toward equilibrium and periodically pulsating spherical bubbles in an incompressible liquid
本文研究不可压缩液体中球形气泡因热效应的径向弛豫与周期脉动，通过细化线性化指数衰减估计，证明了小振幅周期扰动下非线性周期脉动解的存在性与稳定性。
Exponential stability of a free boundary problem with spherical symmetry for a gas bubble immersed in a bounded incompressible liquid
本文研究了有界不可压缩粘性液体中气泡的球对称自由边界问题，证明了平衡态在小扰动下的指数稳定性，并推导了小液体体积下的最优指数衰减率。
On a dryout point for a stationary incompressible thermal fluid with phase transition in a pipe
本文研究管道中定常不可压缩热流体的干涸点问题，通过求解一维 Navier-Stokes-Fourier 方程导出了干涸点存在的充要条件，并基于热力学确定了干涸点温度与气相密度。
We prove the global existence of finite energy weak solutions to the general liquid crystals system. The problem is studied in bounded domain of $R^3$ with Dirichlet boundary conditions and the whole space $R^3$.
We prove that the zero set of a minimizer of the modified Ericksen energy for nematic liquid crystals with variable degree of orientation consists locally of isolated points and a finite union of Hölder continuous curves with only finitely many crossings.
In this paper we establish the incompressible limit for the compressible free-boundary Euler equations with surface tension in the case of a liquid. Compared to the case without surface tension treated recently, the presence of surface tension introduces severe new technical challenges, in that several boundary terms that automatically vanish when surface tension is absent now contribute at top order. Combined with the necessity of producing estimates uniform in the sound speed in order to pass to the limit, such difficulties imply that neither the techniques employed for the case without surf…
Nonspherically symmetric equilibrium bubbles in a steadily rotating incompressible fluid
本文研究稳定旋转不可压缩流体中的平衡气泡，证明非平凡旋转流平衡解的存在性，构造非球对称角环面气泡，并将球形结果推广至轴对称纯方位角情形，且用 PINN 数值模拟气泡形状。
In this article, we consider the Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model without kinematic transport in three spatial dimensions, which is a nonlinear coupling of incompressible Navier-Stokes equations with wave map to $\mathbb{S}^2$. Global regularity for small and smooth initial data near the equilibrium is proved. The proof relies on the idea of space-time resonance.
In this paper, we investigate the uniform regularity and vanishing limit for the incompressible nematic liquid crystal flows in three dimensional bounded domain. It is shown that there exists a unique strong solution for the incompressible nematic liquid crystal flows with boundary condition in a finite time interval which is independent of the viscosity. The solution is uniformly bounded in a conormal Sobolev space. Finally, we also study the convergence rate of the viscous solutions to the inviscid ones.
Global well-posedness and decay rates for the three dimensional incompressible active liquid crystals
本文研究了在恒定活动性下三维不可压缩活性液晶的全局适定性及长时间行为，通过改进的对流算子估计，证明了小初始数据下的全局强解存在唯一性，并获得了混合衰减估计，揭示了高活动性下活性向列相趋于各向同性，且衰减率与活动性相关。
We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space $\mathbb{R}^{2}$ 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 $\|\sqrt{ρ_0}\mathbf{u}_0\|_{L^2}^2+\|\nabla\mathbf{d}_0\|_{L^2}^2$ is small. In particular, the initial density may contain vacuum states and even have compact…
We prove a continuation critereon for incompressible liquids with free surface boundary. We combine the energy estimates of Christodoulou and Lindblad with an analog of the estimate due to Beale, Kato, and Majda for the gradient of the velocity in terms of the vorticity, and use this to show solution can be continued so long as the second fundamental form and injectivity radius of the free boundary, the vorticity, and one derivative of the velocity on the free boundary remain bounded, assuming that the Taylor sign condition holds.
In this paper, we consider the Cauchy problem to the Ericksen-Leslie system of liquid crystals in $\mathbb R^3$. Global well-posedness of strong solutions are obtained under the condition that the product of $\|u_0\|_2+\|\nabla d_0\|_2$ and $\|\nabla u_0\|_2+\|\nabla^2d_0\|_2$ is suitably small. This result can be viewed as a supplement to the local existence and blow up criteria discussed in \cite{Hong1}.
Global existence for weak solutions to systems of nematic liquid crystals, with non-constant fluid density has been established by several authors. In this paper, we establish the regularity and uniqueness results for solutions to the density dependent nematic liquid crystals system.
In this paper, we prove the stability of half-degree point defect profiles in $\mathbb{R}^2$ for the nematic liquid crystal within Landau-de Gennes model.
In this paper, we prove the maximal $L_p$-$L_q$ regularity of the compressible and incompressible two phase flow with phase transition in the model problem case with the help of ${\mathcal R}$-bounded solution operators corresponding to generalized resolvent problem. The problem arises from the mathematical study of the motion of two-phase flows having gaseous phase and liquid phase separated by a sharp interface with phase transition. Using the result obtained in this paper, in \cite{S0} we proved the local well-posedness of free boundary problem for the compressible and incompressible two ph…
In this paper we provide a sufficient condition, in terms of the horizontal gradient of two horizontal velocity components and the gradient of liquid crystal molecular orientation field, for the breakdown of local in time strong solutions to the three-dimensional incompressible nematic liquid crystal flows. More precisely, let $T_{*}$ be the maximal existence time of the local strong solution $(u, d)$, then $T_{*}<+\infty$ if and only if \begin{align*} \int_{0}^{T_*} \big( \|\nabla_h u^h\|_{\dot{B}^0_{p,\frac{2p}{3}}}^q + \|\nabla d\|_{\dot{B}^0_{\infty,\infty}}^2 \big)dt = \infty\ \ \text{wit…
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.
In this paper we study the large time behavior of solutions to a nematic liquid crystals system in the whole space $\mathbb{R}^3$. The fluid under consideration has constant density and small initial data.
In this paper, we establish the uniquely existence of the global mild solution to the nematic liquid crystal equations in Besov-Morrey spaces. Some self-similarity and large time behavior of the global mild solution are also investigated.
We consider the existence of strong solution to liquid crystals system in critical Besov space,then give a criterion which is similar to Serrin's criterion on regularity of weak solution to Navier-Stokes equations.
In this note we prove the existence and uniqueness of local maximal smooth solution of the stochastic simplified Ericksen-Leslie systems modelling the dynamics of nematic liquid crystals under stochastic perturbations.
The study of hydrodynamics of liquid crystal leads to many fasci- nating mathematical problems, which has prompted various interesting works recently. This article reviews the static Oseen-Frank theory and surveys some recent progress on the existence, regularity, uniqueness, and large time asymp- totic of the hydrodynamic flow of nematic liquid crystals. We will also propose a few interesting questions for future investigations.
We study a PDE system describing the motion of liquid crystals by means of the $Q-$tensor description for the crystals coupled with the incompressible Navier-Stokes system. Using the method of Fourier splitting, we show that solutions of the system tend to the isotropic state at the rate $(1 + t)^{-3/2}$ as $t \to \infty$.
In this paper, we consider the short time strong solution to a simplified hydrodynamic flow modeling the compressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of both the maximum norm of the deformation tensor of velocity gradient and the square of maximum norm of gradient of liquid crystal director field.
A complex non-Newtonian fluid models the nematic liquid crystal flows confined in a bounded domain in $\mathbb{R}^3$ is considered. The system is a forced incompressible Navier-Stokes equation coupled with a parabolic type Q-tensor flows. Under suitable condition on the coefficients of the Landau-de Gennes energy functional, global strong solution for small initial data is proved.
Starting from Doi-Onsager equation for the liquid crystal, we first derive the Q-tensor equation by the Bingham closure. Then we derive the Ericksen-Leslie equation from the Q-tensor equation by taking the small Deborah number limit.
In this paper we obtain the wave equation modeling the nematic liquid-crystals in three space dimensions and study the lifespan of classical solution to Cauchy problem. The almost global existence to classical solution for small initial data will be presented.
In this paper, we mainly study a hydrodynamic system modeling the flow of nematic liquid crystals. In three dimensions, we first establish local well-posedness of the initial-boundary value problem of the system. Then, we prove the existence of global strong solution to the system with small initial-boundary condition.
We study the hydrodynamics of compressible flows of active liquid crystals in the Beris-Edwards hydrodynamics framework, using the Landau-de Gennes $Q$-tensor order parameter to describe liquid crystalline ordering. We prove the existence of global weak solutions for this active system in three space dimensions by the three-level approximations and weak convergence argument. New techniques and estimates are developed to overcome the difficulties caused by the active terms.
Singularity formation for full Ericksen-Leslie system of nematic liquid crystal flows in dimension two
该研究针对二维向列相液晶流的完整Ericksen-Leslie系统，证明了其泊肃叶层流会因原点几何效应产生奇性，推进了液晶流体动力学奇性理论研究。
On the hydrodynamics of hyperbolic liquid crystal elastomers
本文利用能量变分方法推导含惯性效应的液晶弹性体流体动力学，并证明完全拟线性双曲系统的大数据局部适定性及平衡态附近小初值三维全局正则性。
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