Chen, Wenxiong; Li, Congming; Wang, Guofang On the stationary solutions of the 2D Doi-Onsager model. (English) Zbl 1217.34032 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 8, 2410-2425 (2010). This paper deals with the \(2D\) Doi-Onsager model\[ F = Z^{-1}e^{-U(F)} \text{ on } S^1 \tag{1} \]with the potential\[ U(F)(\phi) := U_l (F)(\phi) := \beta \int_{S^1} | \sin (\theta - \phi) |^l F(\theta) \;d \theta, \tag{2} \]where \(l\) is a positive integer, \(\beta\) is a parameter, \(F\) is a probability distribution function, i.e., \(F > 0, \;\int_{S^1} F =1\) and \(Z = \int_{S^1} e^{-U(F)} \;d \theta.\) First, the authors show that, for \(l=1,\) any pair of solutions \(F\) and \(U(F)\) of equations (1) and (2) are axially symmetric. If \(l\) is any odd positive integer, then, for any \(\beta > 0,\) the pair of equations (1) and (2) possesses non-constant solutions. In addition, in the case \(l=1,\) for any positive integer \(k\), there exists at least one pair of \(\frac{\pi}{k}\)-periodic non-constant solutions. Finally, if \(l=2k\) is an even integer, then for some appropriate values of \(\beta,\) the pair of equations (1) and (2) possesses solutions of the form \(A_0 + \sum_{m=0}^{k} (A_m \cos 2m\phi + B_m \sin 2m \phi)\) as well as axial symmetric solutions of the form \(A_0 + \sum_{m=0}^k A_m \cos 2m\phi.\) In the proof, the authors derive differential equations equivalent to (1) and (2) and, then, they use variational methods. Reviewer: Antonio Cañada Villar (Granada) Cited in 7 Documents MSC: 34B15 Nonlinear boundary value problems for ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations 34B60 Applications of boundary value problems involving ordinary differential equations 58E30 Variational principles in infinite-dimensional spaces Keywords:2D Doi-Onsager model; existence of solutions; multiplicity of solutions; variational methods; axial symmetry PDFBibTeX XMLCite \textit{W. Chen} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 8, 2410--2425 (2010; Zbl 1217.34032) Full Text: DOI References: [1] Constantin, P.; Kevrekidis, I.; Titi, E. S., Asymptotic states of a Smoluchowski equation, Arch. Ration. Mech. Anal., 174, 365-384 (2004) · Zbl 1086.76003 [2] Constantin, P.; Kevrekidis, I.; Titi, E. S., Remarks on a Smoluchowski equation, Discrete Contin. Dyn. 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