A Blow-up Criterion for 2D Compressible Nematic Liquid Crystal Flows in Terms of Density

This paper is concerned with the Cauchy problem for compressible nematic liquid crystal flows in the two-dimensional space (2D). We establish a blow-up criterion in terms of the density only, provided the macroscopic average of the nematic liquid crystal orientation field satisfies a geometric condition. In particular, the initial vacuum is allowed and the compatibility condition is removed.     

A blowup criterion for the compressible nematic liquid crystal flows in dimension two

In this paper, we establish a blowup criterion for the two-dimensional compressible nematic liquid crystal flows. The criterion is given in terms of the density and the gradient of direction field, where the later satisfies the Serrin-type blowup criterion. For this result, we do not need the initial density to be positive.     

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.     

A free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension

In this paper, we consider the free boundary problem for a simplified version of Ericksen–Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one. We obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum.     

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.     

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.     

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.     

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.     

可压缩液晶系统强解的破裂准则

研究可压缩液晶方程组强解的破裂准则,建立了一种仅依据于速度梯度的破裂准则,此种准则类似于理想可压缩流情形的Beale-Kato-Majda准则和由Huang和Xin得到的可压缩Navier-Stokes方程组情形的准则.证明用到能量不等式和高阶能量不等式.主要困难是初始密度含有真空.     

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.     

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 blow-up criterion for 3-D non-resistive compressible heat-conductive magnetohydrodynamic equations with initial vacuum

In this paper, we prove a blow-up criterion of strong solutions to the 3-D viscous and non-resistive magnetohydrodynamic equations for compressible heat-conducting flows with initial vacuum. This blow-up criterion depends only on the gradient of velocity and the temperature, which is similar to the one for compressible Navier-Stokes equations.     

Blowup of smooth solution for non‐isentropic magnetohydrodynamic equations without heat conductivity

In this paper, we establish a blow‐up criterion for the three‐dimentional viscous, compressible magnetohydrodynamic flows. It is shown that for the Cauchy problem and the initial‐boundary‐value problem with initial density allowed to vanish, the strong or smooth solution for the three‐dimentional magnetohydrodynamic flows exists globally if the density, temperature, and magnetic field is bounded from above. Copyright © 2016 John Wiley & Sons, Ltd.     

Global regularity for the Cauchy problem of three-dimensional compressible magnetohydrodynamics equations

In this paper, we study the Cauchy problem for the three-dimensional compressible magnetohydrodynamics equations. We establish a blowup criterion for global regularity of strong solutions, which depends only on density and magnetic field. In addition, the initial data can be arbitrarily large and contain vacuum states. The proof is based on the new a priori estimates for three-dimensional compressible magnetohydrodynamics equations.     

Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum

We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on the above-mentioned results, we prove that we cannot get a global classical solution, no matter how small the initial data are, as long as the initial mass density is of compact support. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one-dimensional case.     

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.     

Global classical solutions to the compressible micropolar viscous fluids with large oscillations and vacuum

In this paper, we consider the three dimensional Cauchy problem of the compressible micropolar viscous flows. We prove the existence of unique global classical solution for smooth initial data with small initial energy but possibly large oscillations and the initial density may allowed to contain the interior and far field vacuum states. Furthermore, the large time behavior of the solution is obtained as well.     

Some uniform estimates and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows

This paper concerns the global existence and the large time behavior of strong and classical solutions to the two-dimensional (2D) Stokes approximation equations for the compressible flows. We consider the unique global strong solution or classical solution to the 2D Stokes approximation equations for the compressible flows together with the space-periodicity boundary condition or the no-stick boundary condition or Cauchy problem for arbitrarily large initial data. First, we prove that the density is bounded from above independent of time in all these cases. Secondly, we show that for the space-periodicity boundary condition or the no-stick boundary condition, if the initial density contains vacuum at least at one point, then the global strong (or classical) solution must blow up as time goes to infinity.     

GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR FOR THE STRONG SOLUTIONS IN H2 TO THE 3D COMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS

In this paper, we investigate the global existence and long time behavior of strong solutions for compressible nematic liquid crystal flows in three-dimensional whole space. The global existence of strong solutions is obtained by the standard energy method under the condition that the initial datas are close to the constant equilibrium state in H²-framework. If the initial datas in L¹-norm are finite additionally, the optimal time decay rates of strong solutions are established. With the help of Fourier splitting method, one also establishes optimal time decay rates for the higher order spatial derivatives of director.     

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.