On Stationary Solutions of the 2D Doi-Onsager Model

TL;DR

This paper analyzes the 2D Doi-Onsager model with various potential kernels, focusing on the classical case, and uses topological methods to establish solution uniqueness at low temperatures and explore bifurcation structures.

Contribution

It applies nonlinear functional analysis techniques to study the solution structure of the 2D Doi-Onsager model, providing new insights into solution uniqueness and bifurcations.

Findings

Uniqueness of trivial solution at low temperatures

Identification of local bifurcation points

Analysis specific to the classical Onsager kernel

Abstract

We study the 2D Doi--Onsager models with general potential kernel, with special emphasis on the classical Onsager kernel. Through application of topological methods from nonlinear functional analysis, in particular the Leray--Schauder degree theory, we obtain the uniqueness of the trivial solution for low temperatures as well as the local bifurcation structure of the solutions.