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

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.