A Serrin criterion for compressible nematic liquid crystal flows

This paper is concerned with a simplified system, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals. We establish a blowup criterion for three‐dimensional compressible nematic liquid crystal flows, which is analogous to the well‐known Serrin's blowup criterion for three‐dimensional incompressible viscous flows. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

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.     

Blow up criterion for three‐dimensional nematic liquid crystal flows with partial viscosity

In this paper, we investigate the Cauchy problem for the three‐dimensional nematic liquid crystal flows with partial viscosity, and a blow up criterion of smooth solutions is established. This result is analogous to the celebrated Beale‐Kato‐Majda breakdown criterion for the incompressible Euler equations. Copyright © 2012 John Wiley & Sons, Ltd.     

A regularity condition of strong solutions to the two‐dimensional equations of compressible nematic liquid crystal flows

This paper is concerned with the short time strong solutions for Cauchy problem to a simplified Ericksen–Leslie system of compressible nematic liquid crystals in two dimensions with vacuum as far field density. We establish a blow‐up criterion for possible breakdown of such solutions at a finite time, which is analogous to the well‐known Serrin's blow‐up criterion for the incompressible Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Time‐periodic solution to the compressible nematic liquid crystal flows in periodic domain

In this paper, we consider the time‐periodic solution to a simplified version of Ericksen‐Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals with a time‐periodic external force in a periodic domain in . By using an approach of parabolic regularization and combining with the topology degree theory, we establish the existence of the time‐periodic solution to the model under some smallness and symmetry assumptions on the external force. Then, we give the uniqueness of the periodic solution of this model.     

On temporal decay estimates for the compressible nematic liquid crystal flow in

We use a general energy method recently developed by [Guo Y, Wang Y. Decay of dissipative equations and negative sobolev spaces. Commun. Partial Differ. Equ. 2012;37:2165–2208.] to prove the global existence and temporal decay rates of solutions to the three-dimensional compressible nematic liquid crystal flow in the whole space. In particular, the negative Sobolev norms of solutions are shown to be preserved along time evolution, and then the optimal decay rates of the higher order spatial derivatives of solutions are obtained by energy estimates and the interpolation inequalities.     

A blow-up criterion for two dimensional compressible viscous heat-conductive flows

We establish a blow-up criterion in terms of the upper bound of the density and temperature for the strong solution to 2D compressible viscous heat-conductive flows. The initial vacuum is allowed.     

BKM's criterion for the 3D nematic liquid crystal flows via two velocity components and molecular orientations

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 _?<+∞ if and only if where u ^h =(u ¹,u ²), ?_h =(? ₁,? ₂). This result can be regarded as the generalization of the well‐known Beale‐Kato‐Majda (BKM) type criterion and is even new for the three‐dimensional incompressible Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Blowup criterion for viscous, compressible micropolar fluids with vacuum

We prove that the maximum norm of velocity gradients controls the possible breakdown of smooth (strong) solutions for the 3-dimensional viscous, compressible micropolar fluids. More precisely, if a solution of the system is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without the bound of the velocity gradients as the critical time approaches. Our result is a generalization of Huang et al. (2011) [13] from viscous barotropic flows to the viscous, compressible micropolar fluids. In addition, initial vacuum states are also allowed in our result.     

A family of global classical solutions to 3D incompressible nematic liquid crystal flows

In this paper we construct a family of finite energy smooth solutions to the three-dimensional incompressible nematic liquid crystal flows. We achieve this by choosing the steady state Beltrami flows which have infinite energies as the initial data and using a special cut-off technique.     

奇数维可压缩液晶流方程解的逐点估计

首先, 本文利用标准的能量估计方法得到高维(3 维及以上) 的液晶流方程组小初值经典解的整体存在性. 然后, 本文运用Green 函数方法, 得到奇数维情形(3 维及以上) 该解的逐点估计. 该结果表明, 密度ρ和动量m同Navier-Stokes 方程组一样满足一般Huygens 原理, 而单位向量场d则没有这种现象, 其有着与热方程的解类似的时空估计.     

Existence of the periodic solution to the nematic liquid crystal equations in weighted Sobolev spaces

This paper is concerned with the time-periodic solution to the simplified incompressible nematic liquid crystal equation. We prove the existence of the time-periodic solution of this equation with small external forces g₁ and g₂, satisfying the T-periodic conditions g_j(t)=g_j(t+T) for j=1,2 in weighted Sobolev spaces.     

On decay estimates of the 3D nematic liquid crystal flows in critical Besov spaces

In this paper, we develop the energy argument in homogeneous Besov space framework to study the large time behavior of global‐in‐time strong solutions to the Cauchy problem of the three‐dimensional incompressible nematic liquid crystal flows with low regularity assumptions on initial data. More precisely, if the small initial data with 1 < p < ∞ and further assume that with 1 < q≤p and , then the global‐in‐time strong solution (u,d) to the nematic liquid crystal flows admits the following temporal decay rate: Here, is a constant unit vector. The highlight of our argument is to show that the ‐norms (with ) of solution are preserved along time evolution. Copyright © 2016 John Wiley & Sons, Ltd.     

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 ? 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.     

On multi-dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain

We study the global existence of weak solutions to a multi-dimensional simplified Ericksen–Leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$, where N=2 or 3$N=2\text{or}3$. By exploiting a maximum principle, Nirenberg?s interpolation inequality and a smallness condition imposed on the N  -th component of initial direction field _d0${\mathbf{d}}_0$ to overcome the difficulties induced by the supercritical nonlinearity |∇d|²d${|\mathrm{\nabla }\mathbf{d}|}^2\mathbf{d}$ in the equations of angular momentum, and then adapting a modified three-dimensional approximation scheme and the weak convergence arguments for the compressible Navier–Stokes equations, we establish the global existence of weak solutions to the initial-boundary problem with large initial energy and without any smallness condition on the initial density and velocity.