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.     

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.     

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.     

An Osgood type regularity criterion for the liquid crystal flows

In this paper, we prove an Osgood type regularity criterion for the model of liquid crystals, which says that the condition $$\sup_{2 \leq q< \infty} \int \nolimits_0^T \frac{\| \bar{S}_{q} \nabla {\bf u}(t)\|_{L^\infty}}{q \, {\rm \ln} \, q} {\rm d} t<\infty$$ implies the smoothness of the solution. Here, ${{\bar S_q=\sum\nolimits_{k=-q}^q \dot {\triangle}_k}}$ with ${\dot{\triangle}_k}$ being the frequency localization operator.     

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.     

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.     

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.     

On the temporal decay of solutions to the two‐dimensional nematic liquid crystal flows

We consider the temporal decay estimates for weak solutions to the two‐dimensional nematic liquid crystal flows, and we show that the energy norm of a global weak solution has non‐uniform decay under suitable conditions on the initial data. We also show the exact rate of the decay (uniform decay) of the energy norm of the global weak solution.     

Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model

In [3], L. Berselli showed that the regularity criterion ? u ∈ (0, T; L ^q (Ω)), for some q ∈ (3/2, + ∞], implies regularity for the weak solutions of the Navier–Stokes equations, being u the velocity field. In this work, we prove that such hypothesis on the velocity gradient is also sufficient to obtain regularity for a nematic Liquid Crystal model (a coupled system of velocity u and orientation crystals vector d ) when periodic boundary conditions for d are considered (without regularity hypothesis on d ). For Neumann and Dirichlet cases, the same result holds only for q ∈ [2, 3], whereas for q ∈ (3/2, 2) ∪ (3, + ∞] additional regularity hypothesis for d (either on ? d or Δ d ) must be imposed. On the other hand, when the Serrin's criterion u ∈ (0, T; L ^p (Ω)) with some p ∈ (3, + ∞] ([16]) for u is imposed, we can obtain regularity of the system only in the problem of periodic boundary conditions for d . When Neumann and Dirichlet cases for d are considered, additional regularity for d must be imposed for each p ∈ (3, + ∞] (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows

We prove the global existence and regularity of weak solution for the 2-D liquid crystal flows with the large initial velocity. The uniqueness of weak solution is also proved by using the Littlewood–Paley analysis.     

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.     

Integral equations for the generalized stokes operator: Applications to high reynolds number flows

This paper deals with a boundary integral equation for the generalized Stokes problem and its approximation by simpler integral equations when the Reynolds number tends to infinity. The two-dimensional case has been treated in [1]. This paper addresses the three-dimensional case. © 1994 John Wiley & Sons, Inc.