Regularity and time-periodicity for a nematic liquid crystal model

Climent-Ezquerra, Blanca; Guillén-González, Francisco; Moreno-Iraberte, M. Jesus

doi:10.1016/j.na.2008.10.092

Cited by 26 publications

(49 citation statements)

References 8 publications

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“…For incompressible fluids a wellknown model, known as Cahn-Hilliard fluid, consists of the classical Navier-Stokes equations suitably coupled with a convective Cahn-Hilliard equation (see [33,34], also [6,17,37,42,48,54,61] and references therein). In related contexts there have also been considered models in which the Cahn-Hilliard equation is replaced by the (convective) Allen-Cahn equation (see, e.g., [9,25,26,29,63,67]) or, in the case of liquid crystals, by the convective Ginzburg-Landau equation (see [43], also [22,23,44,47] and references therein). Denoting by u = (u 1 , u 2 , u 3 ) the velocity field and by φ the order parameter, the Cahn-Hiliard-Navier-Stokes and the Allen-CahnNavier-Stokes systems can be written in a unified form.…”

Section: Introductionmentioning

confidence: 99%

Trajectory attractors for binary fluid mixtures in 3D

Gal

Grasselli

2010

Chin. Ann. Math. Ser. B

119

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Two different models for the evolution of incompressible binary fluid mixtures in a three-dimensional bounded domain are considered. They consist of the 3D incompressible Navier-Stokes equations, subject to time-dependent external forces and coupled with either a convective Allen-Cahn or Cahn-Hilliard equation. Such systems can be viewed as generalizations of the Navier-Stokes equations to two-phase fluids. Using the trajectory approach, the authors prove the existence of the trajectory attractor for both systems.

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Section: Introductionmentioning

confidence: 99%

Trajectory attractors for binary fluid mixtures in 3D

Gal

Grasselli

2010

Chin. Ann. Math. Ser. B

119

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“…Since a, b ∈ L 1 (0, T ), imposing ν large enough and using the argument done in [4] we can deduce the existence of global in time strong solution.…”

Section: Prodi's Strong Estimatesmentioning

confidence: 93%

Weak Time Regularity and Uniqueness for a $Q$-Tensor Model

Guillén-González¹,

Rodríguez-Bellido²

2014

SIAM J. Math. Anal.

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The coupled Navier-Stokes and Q-Tensor system is one of the models used to describe the behavior of the nematic liquid crystals. The existence of weak solutions and a uniqueness criteria have been already studied (see [11] for a Cauchy problem in the whole R 3 and [7] for a initial-boundary problem in a bounded domain Ω).Nevertheless, results on strong regularity have only been treated in [11] for a Cauchy problem in the whole R 3 .In this paper, imposing Dirichlet or Neumann boundary conditions, we show the existence and uniqueness of a local in time weak solution with weak regularity for the time derivative of the velocity and the tensor variables (u, Q). Moreover, we gives a regularity criteria implying that this solution is global in time. Note that the regularity furnished by the weak regularity for (u, Q) and the weak regularity for (∂ t u, ∂ t Q) is not equivalent to the strong regularity.Finally, when large enough viscosity is imposed, we obtain the existence (and uniqueness) of global in time strong solution. In fact, if non-homogeneous Dirichlet condition for Q is imposed, the strong regularity needs to be obtained together with the weak regularity for (∂ t u, ∂ t Q).

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“…and ∇ · (μ 4 D(u)) = μ 4 2 Δu (since ∇ · u = 0), (1.4), (1.5) and (1.6) can be rewritten as the following partial differential equation (PDE) system in (0, T ) × Ω:…”

Section: Reformulation Of Nematic Modelmentioning

confidence: 99%

“…• In [4], firstly, the initial-boundary problem (1.15) is considered, obtaining the existence of global in time (up to infinity time) weak solutions, the existence of global regular solutions for big enough viscosity coefficient and the weak/strong uniqueness. Here an elliptic lifting function is used again.…”

Section: Nematic Problemmentioning

confidence: 99%

A review of mathematical analysis of nematic and smectic-A liquid crystal models

Climent-Ezquerra¹,

Guillén-González²

2013

Eur. J. Appl. Math

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We review the mathematical analysis of some uniaxial, liquid crystal phases. First, we state the models for the two different studied phases: nematic and smectic-A liquid crystals. The spatial and temporal profiles of the liquid crystal configurations will be described by means of strongly nonlinear parabolic partial differential systems, which are presented at the same time. Then, we will state some results about existence, regularity, time-periodicity and stability of solutions at infinite time for both models.It is our aim to show that, although nematic and smectic-A phases have different physical properties and are modeled by different nonlinear parabolic problems, there exists a common mathematical machinery to rewrite the models and to obtain the analytical results.

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Regularity and time-periodicity for a nematic liquid crystal model

Cited by 26 publications

References 8 publications

Trajectory attractors for binary fluid mixtures in 3D

Trajectory attractors for binary fluid mixtures in 3D

Weak Time Regularity and Uniqueness for a $Q$-Tensor Model

A review of mathematical analysis of nematic and smectic-A liquid crystal models

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