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

Incompressible limit of the compressible nematic liquid crystal flow

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

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.

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.

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 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.

This paper is concerned with global existence of large solutions to the initial-boundary value problem of the planar magnetohydrodynamic compressible flow. Under the assumptions that viscosity and heat conductivity coefficients are constants, magnetic diffusion is a function of the specific volume, we obtain the global existence of strong solutions. Some new methods are developed to deal with the complex interaction between the hydrodynamic and magnetodynamics effects.

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.

We revisit the 2D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic (MHD) equations in $\mathbb{R}^2$. For the initial density allowing vacuum at infinity, we derive the global existence and uniqueness of strong solutions provided that the initial density and the initial magnetic decay not too slowly at infinity. In particular, the initial data can be arbitrarily large. This improves our previous work where the initial density has non-vacuum states at infinity. The result could also be viewed as an extension of the study in L{\"u}-Xu-Zhong for the inhomogeneous case to the full inhomogeneous situation. The method is based on delicate spatial weighted estimates and the structural characteristic of the system under consideration. As a byproduct, we get the global existence of strong solutions to the 2D Cauchy problem for nonhomogeneous heat conducting Navier-Stokes equations with vacuum at infinity.

We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for the incompressible MHD equations in bounded smooth domains of $\mathbb R^3$ under some suitable smallness conditions. The initial density is allowed to have vacuum, in particular, it can vanish in a set of positive Lebessgue measure. More precisely, under the assumption that the production of the quantities $\|\sqrt\rho_0u_0\|_{L^2(\Omega)}^2+\|H_0\|_{L^2(\Omega)}^2$ and $\|\nabla u_0\|_{L^2(\Omega)}^2+\|\nabla H_0\|_{L^2(\Omega)}^2$ is suitably small, with the smallness depending only on the bound of the initial density and the domain, we prove that there is a unique strong solution to the Dirichlet problem of the incompressible MHD system.

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.

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

We consider the Cauchy problem for the compressible non‐isothermal nematic liquid crystal equations in . The existence of global strong solutions is established when the initial mass of the flow and the ‐norm of the spatial derivative of the initial macroscopic average of the nematic liquid crystal orientation field are suitably small. The results show that the strong solution may have large oscillations and can contain vacuum states.

We are concerned with the Cauchy problem of compressible non-isothermal nematic liquid crystal flows in R2 with zero density at infinity. By weighted energy estimates and a Hardy-type inequality, we derive the local existence and uniqueness of strong solutions, provided that the initial density and gradient of the orientation decay not too slowly at infinity. The novelty of this paper is that we allow vacuum at infinity.

Global existence and time behaviour of strong solution to a compressible non-Newtonian fluid