Compressible Nematic Liquid Crystal Flows Research Articles

Notes on Liouville‐type theorems for the 3D compressible nematic liquid crystal equations

In this note, we investigate steady compressible nematic liquid crystal flows in . We establish Liouville‐type theorems for smooth solutions that fulfill specific conditions within Lorentz spaces.

Energy conservation for the weak solutions to the 3D compressible nematic liquid crystal flow

Energy conservation for the weak solutions to the 3D compressible nematic liquid crystal flow

Global well-posedness to the Cauchy problem of 2D compressible nematic liquid crystal flows with large initial data and vacuum

Global well-posedness to the Cauchy problem of 2D compressible nematic liquid crystal flows with large initial data and vacuum

A blow‐up criterion for the compressible nematic liquid crystal flows in three dimensions

In this paper, we consider the strong solutions under three different types of boundary problems to a compressible nematic liquid crystals system in three dimensions. We establish the blow‐up criterion in terms of the density and the gradient of direction field, provided the shear viscosity and bulk viscosity coefficients of the fluid satisfy an additional relation (i.e., ). In particular, the initial vacuum is allowed.

A new blowup criterion of strong solutions to the two‐dimensional equations of compressible nematic liquid crystal flows

In this paper, we concern the Cauchy problem of two‐dimensional (2D) compressible nematic liquid crystal flows with vacuum as far‐field density. Under a geometric condition for the initial orientation field, we establish a blowup criterion in terms of the integrability of the density for strong solutions to the compressible nematic liquid crystal flows. This criterion generalizes previous results of compressible nematic liquid crystal flows with vacuum, which concludes the initial boundary problem and Cauchy problem.

Global Strong Solutions to the Compressible Nematic Liquid Crystal Flows with Large Oscillations and Vacuum in 2D Bounded Domains

Global Strong Solutions to the Compressible Nematic Liquid Crystal Flows with Large Oscillations and Vacuum in 2D Bounded Domains

Global well-posedness to the compressible nematic liquid crystal flows with large oscillations and vacuum in 3D exterior domains

This paper aims at extending our previous work (Y. Liu and X. Zhong, arXiv:2204.06227) to the exterior problem. We show the global well-posedness of strong solutions to a simplified Ericksen–Leslie model of compressible nematic liquid crystal flows in a three-dimensional exterior domain when the initial total energy is sufficiently small. In order to obtain necessary a priori estimates on the boundary, some new analytic techniques are used.

Global existence of strong solutions with large oscillations and vacuum to the compressible nematic liquid crystal flows in 3D bounded domains

Global existence of strong solutions with large oscillations and vacuum to the compressible nematic liquid crystal flows in 3D bounded domains

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.

Optimal time-decay rates of the 3D compressible nematic liquid crystal flows with discontinuous initial data and large oscillations

The global existence of low-energy weak solutions for the 3D compressible nematic liquid crystal flows with discontinuous initial data and large oscillations has been proved by Wu and Tan (2018) under the assumptions that the initial energy is small and the initial density has positive lower and upper bounds. However, up to now, the time-decay rate of these solutions has remained an open problem since the solutions have low regularity, and particularly the density has no regularity. We resolve this problem by proving time-decay rates of the solutions in Lr-norm with 2≤r≤∞. Moreover, if additionally the initial data satisfies some low-frequency assumption, the optimal lower bound decay rates of solution are also obtained. Therefore, our decay rates are optimal in this sense.

Existence and uniqueness of solution to one-dimensional compressible biaxial nematic liquid crystal flows

Existence and uniqueness of solution to one-dimensional compressible biaxial nematic liquid crystal flows

Optimal decay rates of higher–order derivatives of solutions for the compressible nematic liquid crystal flows in $ \mathbb R^3 $

<abstract><p>In this paper, we are concerned with optimal decay rates of higher–order derivatives of the smooth solutions to the $ 3D $ compressible nematic liquid crystal flows. The main novelty of this paper is three–fold: First, under the assumptions that the initial perturbation is small in $ H^N $–norm $ (N\geq3) $ and bounded in $ L^1 $–norm, we show that the highest–order spatial derivatives of density and velocity converge to zero at the $ L^2 $–rates is $ (1+t)^{-\frac{3}{4}-\frac{N }{2 }} $, which are the same as ones of the heat equation, and particularly faster than the $ L^2 $–rate $ (1+t)^{-\frac{1}{4}-\frac{N }{2 }} $ in [J.C. Gao, et al., J. Differential Equations, 261: 2334-2383, 2016]. Second, if the initial data satisfies some additional low frequency assumption, we also establish the lower optimal decay rates of solution as well as its all–order spatial derivatives. Therefore, our decay rates are optimal in this sense. Third, we prove that the lower bound of the time derivatives of density, velocity and macroscopic average converge to zero at the $ L^2 $–rate is $ (1+t)^{-\frac{5}{4}} $. Our method is based on low-frequency and high-frequency decomposition and energy methods.</p></abstract>

Global solution to the compressible non-isothermal nematic liquid crystal equations with constant heat conductivity and vacuum

This paper considers the initial-boundary value problem of the one-dimensional full compressible nematic liquid crystal flow problem. The initial density is allowed to touch vacuum, and the viscous and heat conductivity coefficients are kept to be positive constants. Global existence of strong solutions is established for any H^{2} initial data in the Lagrangian flow map coordinate, which holds for both vacuum and non-vacuum case. The key difficulty is caused by the lack of the positive lower bound of the density. To overcome such difficulty, it is observed that the ratio of frac{rho _{0(y)}}{rho (t,y)} is proportional to the time integral of the upper bound of temperature and vector director field, along the trajectory. Density weighted Sobolev type inequalities are constructed for both temperature and director field in terms of frac{rho _{0(y)}}{rho (t,y)} and small dependence on their dissipation estimates. Besides this, to deal with cross terms arising due to liquid crystal flow, higher order priori estimates are established by using effective viscous flux.

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.

Global small solutions of the compressible nematic liquid crystal flow in a bounded domain

We study the 3D compressible nematic liquid crystal flow in a bounded domain. We first prove a regularity criterion, and then use a bootstrap argument to prove the global well-posedness of strong solutions provided that $\|\rho _0\|_{L^\infty }+\|\nabla d

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.

Incompressible limit of the compressible nematic liquid crystal flows in a bounded domain with perfectly conducting boundary

In this paper, we study the asymptotic behavior of the regular solution to a simplified Ericksen–Leslie model for the compressible nematic liquid crystal flow in a bounded smooth domain in R2 as the Mach number tends to zero. The evolution system consists of the compressible Navier–Stokes equations coupled with the transported heat flow for the averaged molecular orientation. We suppose that the Navier–Stokes equations are characterized by a Navier's slip boundary condition, while the transported heat flow is subject to Neumann boundary condition. By deriving a differential inequality with certain decay property, the low Mach limit of the solutions is verified for all time, provided that the initial data are well-prepared.

Low Mach number limit of global solutions to 3‐D compressible nematic liquid crystal flows with Dirichlet boundary condition

In this paper, we consider the uniform estimates of strong solutions in the Mach number ϵ and t ∈ [0,∞) for the compressible nematic liquid crystal flows in a 3‐D bounded domain , provided the initial data are small enough and the density is close to the constant state. Here, we consider the case that the velocity field satisfies the Dirichlet boundary condition. Based on the uniform estimates, we obtain the global convergence of the compressible nematic liquid crystal system to the incompressible nematic liquid crystals system as the Mach number tends to zero.

Incompressible limit for compressible nematic liquid crystal flows in a bounded domain

ABSTRACTIn this paper, we consider the incompressible limit of strong solution for the compressible nematic liquid crystal flows in a 3D bounded domain where the the velocity field satisfies the Dirichlet boundary condition. We establish the uniform estimates with respect to the Mach number for the strong solutions with large initial data in a short time interval. Consequently, we obtain the convergence of the compressible nematic liquid crystal system to the incompressible nematic liquid crystals system as the Mach number tends to zero.

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.