A Serrin criterion for compressible nematic liquid crystal flows

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.

The present paper is devoted to the compressible nematic liquid crystal flow in the whole space $ \mathbb{R}^N\,(N≥ 2)$. Here we concentrate on the incompressible limit in the $ L^p$ type critical Besov spaces setting. We first establish the existence of global solutions in the framework of $ L^p$ type critical spaces provided that the initial data are close to some equilibrium states. Based on the global existence, we then consider the incompressible limit problem in the ill prepared data case. We justify the low Mach number convergence to the incompressible flow of liquid crystals in proper function spaces. In addition, the accurate converge rates are obtained.

This paper is concerned with the strong solutions to the Cauchy problem of a simplified Ericksen-Leslie system of compressible nematic liquid crystals in two or three dimensions with vacuum as far field density. For strong solutions, some a priori decay rate (in large time) for the pressure, the spatial gradient of velocity field and the second spatial gradient of liquid crystal director field are obtained provided that the initial total energy is suitably small. Furthermore, with the help of the key decay rates, we establish the global existence and uniqueness of strong solutions (which may be of possibly large oscillations) in two spatial dimensions.

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.

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

On center singularity for compressible spherically symmetric 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.

Incompressible limit of the compressible nematic liquid crystal flow

We consider weak solutions to a two-dimensional simplified Ericksen-Leslie system of compressible flow of nematic liquid crystals. An initial-boundary value problem is first studied in a bounded domain. By developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity in the equations of angular momentum on the direction field, and adapting the standard three-level approximation scheme and the weak convergence arguments for the compressible Navier-Stokes equations, we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy. Then the Cauchy problem with large initial data is investigated, and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem, provided that the second component of initial data of the direction field satisfies some geometric angle condition.

In this work, efficient microfluidic flow rate tuning based on the asymmetric flow of nematic liquid crystal 5CB under a horizontal temperature gradient is studied. Rectangular microchannels with the width of 100 μm are fabricated through soft lithography and treated with homeotropic surface anchoring conditions. Polarized optical microscopy is applied to explore the unique optical anisotropic characteristics of the nematic liquid crystal. The asymmetric velocity profiles in the microchannel are obtained by particle tracking velocimetry. The effects of temperature, flow rate, and aspect ratio on the velocity profile and split ratio of the asymmetric flow are quantitatively studied for the first time, while the mechanism of the flow asymmetry of the nematic liquid crystal is discussed. The results show that the asymmetric flow of the nematic liquid crystal occurs after the horizontal temperature gradient is applied, with the velocity in the heated region markedly higher than its counterpart. The split ratio of the asymmetric flow increases with the increase in the temperature gradient and the decrease in the flow rate. The aspect ratio influences the asymmetric flow through approaches of average velocity and surface anchoring strength, while the former is more distinct. The impacts of temperature gradient, flow rate, and aspect ratio on the flow asymmetry of nematic liquid crystals are caused by the coupling between physical properties, velocity field, and director field. Microchannels based on the asymmetric flow characteristics of nematic liquid crystals can act as a novel kind of temperature-controlled microvalve to achieve efficient microfluidic flow tuning.

The present paper is dedicated to the study of the Cauchy problems for the three-dimensional compressible nematic liquid crystal flow. We obtain the global existence and the optimal decay rates of smooth solutions to the system under the condition that the initial data in lower regular spaces are close to the constant equilibrium state. Our main method is based on the spectral analysis and the smooth effect of dissipative operator.

Theoretical analysis of the stability of shear flow of nematic liquid crystals with a positive Leslie viscosity α3

Well-posedness of Entropy-Bounded Solutions to 3D Compressible Nematic Liquid Crystal Flows with Far Field Vacuum

Local existence of unique strong solution to non-isothermal model for incompressible nematic liquid crystals in 3D

Nonlinear stability for the three dimensional incompressible flow of nematic liquid crystals