On the stationary solutions of the 2D Doi–Onsager model Original Research Article

In this paper, we study the 2D Doi–Onsager model equation(1) View the MathML sourceF=Z−1e−U(F),on S1 Turn MathJax on with the potential equation(2) View the MathML sourceU(F)(ϕ)≔Ul(F)(ϕ)≔β∫S1|sin(θ−ϕ)|lF(θ)dθ, Turn MathJax on where ll is a positive integer, ββ is a parameter, FF is a probability distribution function. We derive differential equations equivalent to (1) and (2) for all positive integers ll, then use the corresponding variational structure to obtain the existence of solutions, in particular, non-constant solutions. When l=1l=1, (1) and (2) are the original Onsager Models, and in this case, we prove that all solutions must be axially symmetric and there exist infinitely many such solutions.