Axial Symmetry and Classification of Stationary Solutions of Doi-Onsager Equation on the Sphere with Maier-Saupe Potential

Abstract

This note serves to provide additional details for the proof of Lemma 3.6 in our paper [Liu, Zhang and Zhang, Comm. Math. Sci., 3(2005), pp.201-218]. Moreover, we will also present an alternative, yet simpler, proof based on arguments in [Wang, Zhang and Zhang, CPAM, 68(2015), no. 8, 1326-1398.In [1] we showed that in order to determine the number of solutions to the Doi-Onsager equation on the sphere with Maier-Saupe potential, it suffices to determine the number of zeros of B(η, α) in term of the intensity parameter α > 0, whereNote that, for η > 0, the mean value theorem gives 3e −η /

Characterization of stable kinetic equilibria of rigid, dipolar rod ensembles for coupled dipole–dipole and Maier–Saupe potentials

“…In this case, equation (79) with equilibrium probability density given in (78) reduces to the case of pure (non-dipolar) nematic polymers. From the theoretical results of pure nematic polymers [11,15,19], we know that for b > 7.5 there is only one family of rotation-equivalent prolate equilibrium states. In the analysis below we fix b = 8.…”

Stability of equilibria of nematic liquid crystalline polymers

“…It has been shown that in the absence of external field, all equilibrium states of nematic polymers are axisymmetric [8,24,35]. Without loss of generality, we assume the m 3 -axis is the axis of symmetry.…”

“…One mathematical advance in the understanding of the equilibrium states of the Doi model is the rigorous proof on that the equilibria of the Smoluchowski equation with the Maier-Saupe potential are uniaxial [8,24,35]. However, it seems more challenging to obtain a solid proof on the stability of the equilibria.…”

Inertial Manifolds for a Smoluchowski Equation on the Unit Sphere

“…For dissipative gradient systems, the dynamical behavior is characterized by a global attractor which is formed by the steady-states and their unstable manifolds, and all solutions approach a steady-state as t → ∞. Denoting u = ψ exp(V /2), the Lyapunov functional for the Smoluchowski equation is given by F(t) = S 2 ψ(m,t)logu(m,t) dm, which satisfies the equation This transcendental matrix equation is now well understood (see [5], [6], [8], [13], [18])). The isotropic steady stateψ = 1/4π persists for all potential intensities b > 0.…”

“…However, it is widely accepted that the Maier-Saupe potential affords sufficient degrees of freedom to capture the dynamics of the micro-micro interaction. In a recent development, the bifurcation diagram for the Onsager equation (and therefore also Smoluchowski equation) with the Maier-Saupe potential was confirmed rigorously (see [5], [6], [8], [13], [18], [19]). The equation undergoes two bifurcations.…”