Incompressible Nematic Liquid Crystal Flows Research Articles

A remark on global strong solution of two‐dimensional inhomogeneous nematic liquid crystal flows in a bounded domain

AbstractWe study an initial boundary value problem of two‐dimensional (2D) inhomogeneous incompressible nematic liquid crystal flows with nonnegative density. It is proved that there exists a unique global strong solution provided that the initial orientation satisfies a geometric condition.

Strong Solutions to the Density-Dependent Incompressible Nematic Liquid Crystal Flows with Heat Effect

Strong Solutions to the Density-Dependent Incompressible Nematic Liquid Crystal Flows with Heat Effect

Global regularity to the 2D non-isothermal inhomogeneous nematic liquid crystal flows

ABSTRACT In this paper, we prove the global strong solutions for the Cauchy problem of two-dimensional (2D) incompressible non-isothermal nematic liquid crystal flows, if the initial orientation satisfies a geometric condition. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states. When d is a constant vector and , we also extend the corresponding result in Wang Y. [Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier–Stokes flows with vacuum. Discrete Contin Dyn Syst B. doi:10.3934/dcdsb.2020099.] to the whole space , where the global strong solution of 2D inhomogeneous incompressible heat conducting Navier–Stokes flows is established on bounded domain.

Global well‐posedness of the 3D incompressible nematic liquid crystal flows with density‐dependent viscosity coefficient

This paper deals with the 3D incompressible nematic liquid crystal flows with density‐dependent viscosity in bounded domain. The global well‐posedness of strong solutions are established in the vacuum cases, provided the assumption that is suitably small with large velocity. Furthermore, the exponential decay of the solution is also obtained.

Global Existence and Exponential Decay of Strong Solutions to the 2D Density-dependent Nematic Liquid Crystal Flows with Vacuum

This paper deals with the 2D incompressible nematic liquid crystal flows with density-dependent viscosity in bounded domain. The global well-posedness of strong solutions are established in the vacuum cases, provided the assumption that $\overline{\rho} + \|\nabla d_0\|_{L^2}$ is suitably small with large velocity, which extends the recent work [Discrete Contin. Dyn. Syst. 37 (2017), 4907--4922] and [Methods Appl. Anal. 22 (2015), 201--220] to the case of variable viscosity. Furthermore, the exponential decay of the solution is also obtained.

On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum

This paper deals with the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible nematic liquid crystal flows. The global well-posedness of strong solutions with large velocity is established provided that \begin{document}$ \|\rho_0\|_{L^\infty}+\|\nabla d_0\|_{L^3} $\end{document} is suitably small. In particular, the initial density may contain vacuum states and even have compact support. Furthermore, the large time behavior of the solution is also obtained.

Global existence and large-time behavior of strong solutions to the Cauchy problem of the incompressible nematic liquid crystal flow

Global existence and large-time behavior of strong solutions to the Cauchy problem of the incompressible nematic liquid crystal flow

Optimal distributed control of a 2D simplified Ericksen–Leslie system for the nematic liquid crystal flows

We study the initial boundary value problem of a simplified Ericksen–Leslie system modeling the incompressible nematic liquid crystal flows in two dimensions of space, where the equations of the velocity field are characterized by a time-dependent external force g(t) and a no-slip boundary condition, and the equations for the molecular orientation are subjected to a time-dependent Dirichlet boundary condition h(t). Based on the recently addressed well-posedness and regularity results of the system, we present a rigorous proof to show the existence of optimal distributed controls, the control-to-state operator is Fréchet differentiable and first-order necessary optimality conditions for an associated optimal control problem.

Global well-posedness of the two dimensional Beris–Edwards system with general Laudau–de Gennes free energy

In this paper, we consider the Beris–Edwards system for incompressible nematic liquid crystal flows. The system under investigation consists of the Navier–Stokes equations for the fluid velocity u coupled with an evolution equation for the order parameter Q-tensor. One important feature of the system is that its elastic free energy takes a general form and in particular, it contains a cubic term that possibly makes it unbounded from below. In the two dimensional periodic setting, we prove that if the initial L∞-norm of the Q-tensor is properly small, then the system admits a unique global weak solution. The proof is based on the construction of a specific approximating system that preserves the L∞-norm of the Q-tensor along the time evolution.

Existence and uniqueness of mild solutions to the incompressible nematic liquid crystal flow

In this paper, we consider the incompressible nematic liquid crystal flow in the whole space, and obtain the existence of global mild solutions with small initial data in the scaling invariant space. The main tool we have used is the implicit function theorem. Furthermore, we establish the asymptotic stability of solutions when the time goes to infinity. And in view of our construction of solutions in the weak Lp-spaces, we finally derive the existence of self-similar solutions provided the initial data are small homogeneous functions.

On some large global solutions to the incompressible inhomogeneous nematic liquid crystal flows

ABSTRACTIn this paper, we investigate the Cauchy problem for the incompressible inhomogeneous nematic liquid crystal flows in with initial data in some critical Besov spaces. Under the assumptions that initial density is close to the equilibrium state and the smallness on the initial direction field and nonlinear smallness on the linearized velocity, we obtain the global existence of solutions by fully using the Fourier localization technique and the advantage of weighted function generated by heat kernel. The meanings of this kind of initial data are that we can provide an example of large highly oscillating data satisfying that nonlinear smallness assumption, despite each component of the initial data could be arbitrarily large.

Global regularity to the 3D incompressible nematic liquid crystal flows with vacuum

This paper is concerned with the 3D incompressible nematic liquid crystal equations in smooth bounded domains, where the velocity u and macroscopic molecular orientation d admit the Dirichlet and Neumann boundary condition respectively. Global-in-time strong solution is proved to exist when the basic energy is suitably small. The initial vacuum is allowed.

A note on a global strong solution to the 2D Cauchy problem of density-dependent nematic liquid crystal flows with vacuum

In Li–Liu–Zhong (Nonlinearity 30 (2017) 4062–4088), the authors proved the existence of a unique global strong solution to the Cauchy problem of 2D nonhomogeneous incompressible nematic liquid crystal flows with vacuum as far-field density provided the initial data density and the gradient of orientation decay not too slow at infinity, and the basic energy ‖ρ0u0‖L22+‖∇d0‖L22 is small. In this note, we aim at precisely describing this smallness condition.

Global strong solutions for the incompressible nematic liquid crystal flows with density-dependent viscosity coefficient

This paper is concerned with an initial–boundary value problem of the incompressible nematic liquid crystal flows with density-dependent viscosity in a smooth bounded domain Ω⊂R3. The global well-posedness of strong solutions with large oscillations is established in vacuum, provided ‖∇u0‖L2+‖Δd0‖L2 is suitably small with arbitrary large initial density, which extended the local strong solution by Gao et al. [12] to be a global one.

Global solution to the 3D inhomogeneous nematic liquid crystal flows with variable density

In this paper, we investigate the global existence and uniqueness of solution to the 3D inhomogeneous incompressible nematic liquid crystal flows with variable density in the framework of Besov spaces. It is proved that there exists a global and unique solution to the nematic liquid crystal flows if the initial data (ρ0−1,u0,n0−e3)∈M(B˙p,13p−1(R3))×B˙p,13p−1(R3)×B˙p,13p(R3) with 1≤p<6, and satisfies‖ρ0−1‖M(B˙p,13p−1)+‖u0‖B˙p,13p−1+‖n0−e3‖B˙p,13p≤cfor some small c>0 depending only on p. Here M(B˙p,13p−1(R3)) is the multiplier space of Besov space B˙p,13p(R3). Using the Lagrangian approach in Danchin and Mucha (2012, 2013) [7,8] is the key to our results.

Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum

We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space with vacuum as far field density. It is proved that the 2D nonhomogeneous incompressible nematic liquid crystal flows admits a unique global strong solution provided the initial data density and the gradient of orientation decay not too slow at infinity, and the basic energy is small. In particular, the initial density may contain vacuum states and even have compact support. Moreover, the large time behavior of the solution is also investigated.

Blow-up criterion of classical solutions for the incompressible nematic liquid crystal flows

In this paper, we consider the short time classical solution to a simplified hydrodynamic flow modeling incompressible, nematic liquid crystal materials in R3. We establish a criterion for possible breakdown of such solutions at a finite time. More precisely, if (u,d) is smooth up to time T provided that ∫0T||∇ × u(t,⋅)||BMO(R3)+||∇ d(t,⋅)||L4(R3)8dt < ∞.

Global solution to the nematic liquid crystal flows with heat effect

The temperature-dependent incompressible nematic liquid crystal flows in a bounded domain Ω⊂RN (N=2,3) are studied in this paper. Following Danchin's method in [7], we use a localization argument to recover the maximal regularity of Stokes equation with variable viscosity, by which we first prove the local existence of a unique strong solution, then extend it to a global one provided that the initial data is a sufficiently small perturbation around the trivial equilibrium state. This paper also generalizes Hu–Wang's result in [21] to the non-isothermal case.

Optimal Boundary Control of a Simplified Ericksen–Leslie System for Nematic Liquid Crystal Flows in 2D

In this paper, we investigate an optimal boundary control problem for a two dimensional simplified Ericksen--Leslie system modelling the incompressible nematic liquid crystal flows. The hydrodynamic system consists of the Navier--Stokes equations for the fluid velocity coupled with a convective Ginzburg--Landau type equation for the averaged molecular orientation. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the molecular orientation is subject to a time-dependent Dirichlet boundary condition that corresponds to the strong anchoring condition for liquid crystals. We first establish the existence of optimal boundary controls. Then we show that the control-to-state operator is Fr\'echet differentiable between appropriate Banach spaces and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two

This paper is devoted to the study of the initial-boundary value problem for density-dependent incompressible nematic liquid crystal flows with vacuum in a bounded smooth domain of \begin{document}$\mathbb{R}^2$\end{document} . The system consists of the Navier-Stokes equations, describing the evolution of an incompressible viscous fluid, coupled with various kinematic transport equations for the molecular orientations. Assuming the initial data are sufficiently regular and satisfy a natural compatibility condition, the existence and uniqueness are established for the global strong solution if the initial data are small. We make use of a critical Sobolev inequality of logarithmic type to improve the regularity of the solution. Our result relaxes the assumption in all previous work that the initial density needs to be bounded away from zero.