Orientability and asymptotic convergence of Q-tensor flow of biaxial nematic liquid crystals

• Published in: Calculus of variations and partial differential equations 2022-10, Vol.61 (5), Article 173
• Main Authors: Huang, Jinrui, Lin, Junyu
• Format: Article
• Language: English
• Subjects: Analysis; Arches; Asymptotic properties; Calculus of Variations and Optimal Control; Optimization; Control; Convergence; Crystal structure; Flow mapping; Gradient flow; Heat transmission; Liquid crystals; Mathematical analysis; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Nematic crystals; Systems Theory; Tensors; Theoretical
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-022-02272-x

In recent paper, we will consider two contents on the maximal biaxial nematic liquid crystals. In the first part, we get an orientability issue, that is if Q ∈ W 1 , p ( Ω , N ) , p ≥ 2 , then there is ( n , m ) ∈ M with n , m ∈ W 1 , p ( Ω ) , such that Q = r ( n ⊗ n - m ⊗ m ) . Unlike S 2 , the set M is not simple connect. Our orientability result extends to maximal biaxial nematics from earlier conclusions corresponding to uniaxial nematics in Ball and Zarnescu (Arch Rational Mech Anal 202:493–535, 2011). In the second part, we study an asymptotic convergence of approximate solutions Q ϵ of the Q -tensor flow in R 3 as the parameter ϵ goes to zero. The limiting direction map ( n ,  m ) satisfies a gradient flow, which is different from the heat flow of harmonic map that takes value into S 2 or M . A partial regularity of this gradient flow is also derived. We extend the works in Ball and Zarnescu (Arch Rational Mech Anal 202:493–535, 2011) and Wang et al. (Arch Rational Mech Anal 225:663–683, 2017) to the maximal biaxial nematic liquid crystals.