Monotonicity of a Key Function Arised in Studies of Nematic Liquid Crystal Polymers

In our previous study of steady states and dynamics of two-dimensional nematic liquid crystal polymers under an imposed weak shear [1], we carried out multiscale asymptotic expansions [2, 3] to reveal the slow time evolution of the polymer orientation distribution. In the two-dimensional case, the orientation of a polymer rod is represented by angle θ [4–6]. Let ρ(θ, t) denote the probability density of the orientation of the polymer rod ensemble. Our multiscale asymptotic analysis on the Smoluchowski equation which governs the evolution of the probability density [1] demonstrates that, to the leading term, the probability density ρ(θ, t) behaves like a travelling wave with nonuniform velocity