On a regularized family of models for the full Ericksen–Leslie system. (November 2015)

• Record Type: Journal Article
• Authors: Gal, Ciprian G.; Tebou, Louis
• Is Part Of: Nonlinear analysis. Volume 127(2015)
• Page Start: 312
• Page End: 351
• Publication Date: 2015-11
• Subjects: 35B44 -- 35D30 -- 35K45 -- 35Q30 -- 76A15; Navier–Stokes equations -- Euler equations -- Regularized Navier–Stokes -- Navier–Stokes-α -- Leray-α -- Modified-Leray-α -- Simplified Bardina -- Navier–Stokes–Voigt -- Ericksen–Leslie system -- Nematic liquid crystals -- Global attractor -- Convergence to equilibria; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.07.012
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider a general family of regularized systems for the full Ericksen–Leslie model for the hydrodynamics of liquid crystals in n -dimensional compact Riemannian manifolds. The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes- α -like equation, the Leray- α equation, the Modified Leray- α equation, the Simplified Bardina model, the Navier–Stokes–Voigt model and the Navier–Stokes equation) for the fluid velocity u suitably coupled with a parabolic equation for the director field d . We establish existence, stability and regularity results for this family. We also show the existence of a finite dimensional global attractor for our general model, and then establish sufficiently general conditions under which each trajectory converges to a single equilibrium by means of a Lojasiewicz–Simon inequality.