2009
DOI: 10.1016/j.na.2008.10.092
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Regularity and time-periodicity for a nematic liquid crystal model

Abstract: In this paper two main results are obtained for a nematic liquid crystal model with timedependent boundary Dirichlet data for the orientation of the crystal molecules. First, the initial-boundary problem is considered, obtaining the existence of global in time (up to infinity time) weak solution, the existence of global regular solution for viscosity coefficient big enough, and the weak/strong uniqueness. Second, using these previous results and the existence of time-periodic weak solutions proved in [2], the … Show more

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Cited by 26 publications
(49 citation statements)
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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%
“…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%
“…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%
“…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%