Weak solutions for an initial–boundary Q-tensor problem related to liquid crystals. (January 2015)

• Record Type: Journal Article
• Authors: Guillén-González, Francisco; Rodríguez-Bellido, María Ángeles
• Is Part Of: Nonlinear analysis. Volume 112(2015)
• Page Start: 84
• Page End: 104
• Publication Date: 2015-01
• Subjects: 35A01 -- 35A02 -- 35B50 -- 35D30 -- 35K51 -- 35Q35 -- 76A15 -- 76D03 -- 76D05; Navier–Stokes equations -- Weak solution -- Uniqueness -- Maximum principle -- Symmetry -- Traceless; 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.2014.09.011
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The coupled Navier–Stokes and Q-tensor system is considered in a bounded three-dimensional domain under homogeneous Dirichlet boundary conditions for the velocity u and either nonhomogeneous Dirichlet or homogeneous Neumann boundary conditions for the tensor Q . The corresponding initial-value problem in the whole space R 3 was analyzed in Paicu and Zarnescu (2012). In this paper, three main results concerning weak solutions will be proved: the existence of global in time weak solutions (bounded up to infinite time), a uniqueness criteria and a maximum principle for Q . Moreover, we identify how to modify the system to deduce symmetry and traceless for Q, for any weak solution. The presence of a stretching term in the Q -system plays a crucial role in all the analysis.