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Asymptotic states of a Smoluchowski equation. (English) Zbl 1086.76003
From the text: We study the high-concentration asymptotics of steady states of a Smoluchowski equation arising in the modeling of nematic liquid crystalline polymers. We study the Smoluchowski equation on the unit sphere with the Maier-Saupe potential. This choice of the potential allows us to investigate rigorously the asymptotics of the steady-state solutions for large values of the potential intensity, corresponding to large concentrations. We reduce the problem of finding steady state solutions of the Smoluchowski partial differential equation with Maier-Saupe potential to the finite-dimensional problem of finding the eigenvalues of a symmetric, traceless matrix. Linear combinations of these eigenvalues are critical points of a function associated with them. We find multiple steady solutions which are clustered in three distinct groups. As the concentration is increased, these steady solutions converge to uniform, prolate and oblate states. Furthermore, the methods of study allow us to devise asymptotic expansions for the steady states, expansions that are valid at high but finite concentrations.

MSC:
76A15 Liquid crystals
76T20 Suspensions
35Q35 PDEs in connection with fluid mechanics
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