Nonisothermal nematic liquid crystal flows with the Ball–Majumdar free energy

被引:0
作者
Eduard Feireisl
Giulio Schimperna
Elisabetta Rocca
Arghir Zarnescu
机构
[1] Czech Academy of Sciences,Institute of Mathematics
[2] Università di Pavia,Dipartimento di Matematica
[3] Weierstrass Institute for Applied Analysis and Stochastics,Dipartimento di Matematica
[4] Università di Milano,Pevensey III
[5] University of Sussex,undefined
来源
Annali di Matematica Pura ed Applicata (1923 -) | 2015年 / 194卷
关键词
Nematic liquid crystal; Ball–Majumdar free energy; Nonisothermal model; Existence theorem; 76A15; 74G25; 35D30; 35Q30;
D O I
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学科分类号
摘要
In this paper, we prove the existence of global-in-time weak solutions for an evolutionary PDE system modelling nonisothermal Landau–de Gennes nematic liquid crystal (LC) flows in three dimensions of space. In our model, the incompressible Navier–Stokes system for the macroscopic velocity u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{u}$$\end{document} is coupled to a nonlinear convective parabolic equation describing the evolution of the Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q$$\end{document}-tensor Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document}, namely a tensor-valued variable representing the normalized second-order moments of the probability distribution function of the LC molecules. The effects of the (absolute) temperature ϑ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta $$\end{document} are prescribed in the form of an energy balance identity complemented with a global entropy production inequality. Compared to previous contributions, we can consider here the physically realistic singular configuration potential f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f$$\end{document} introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q$$\end{document}-tensor equation, a term that is at the same time singular in Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}$$\end{document} and degenerate in ϑ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta $$\end{document}. To treat it, a careful analysis of the properties of f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f$$\end{document}, particularly of its blow-up rate, is carried out.
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页码:1269 / 1299
页数:30
相关论文
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