WIAS Preprint No. 1865, (2013)

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



Authors

  • Feireisl, Eduard
  • Rocca, Elisabetta
  • Schimperna, Giulio
  • Zarnescu, Arghir

2010 Mathematics Subject Classification

  • 76A15 74G25 35D30

Keywords

  • nematic liquid crystal, Ball-Majumdar free energy, nonisothermal model, existence theorem

Abstract

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 $vu$ is coupled to a nonlinear convective parabolic equation describing the evolution of the Q-tensor $QQ$, 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 $vt$ 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 $f$ introduced by Ball and Majumdar. This potential gives rise to severe mathematical difficulties since it introduces, in the Q-tensor equation, a term which is at the same time singular in $QQ$ and degenerate in $vt$. To treat it a careful analysis of the properties of $f$, particularly of its blow-up rate, is carried out.

Appeared in

  • Ann. Mat. Pura Appl. IV. Ser., 194 (2015) pp. 1269--1299.

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