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Periodic long-time behaviour for an approximate model of nematic polymers

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  • We study the long-time behaviour of a nonlinear Fokker-Planck equation, which models the evolution of rigid polymers in a given flow, after a closure approximation. The aim of this work is twofold: first, we propose a microscopic derivation of the classical Doi closure, at the level of the kinetic equation ; second, under specific assumptions on the parameters and the initial condition, we prove convergence of the solution to the Fokker-Planck equation to a particular periodic solution in the long-time limit.
    Mathematics Subject Classification: Primary: 35B40, 76A15.

    Citation:

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  • [1]

    C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, "Sur les Inégalités de Sobolev Logarithmiques," Panoramas et Synthèses, 10, Société Mathématique de France, 2000.

    [2]

    A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Part. Diff. Eq., 26 (2001), 43-100.

    [3]

    J.-P. Bartier, J. Dolbeault, R. Illner and M. Kowalczyk, A qualitative study of linear drift-diffusion equations with time-dependent or degenerate coefficients, Math. Models and Methods in Applied Sciences, 17 (2007), 327-362.doi: 10.1142/S0218202507001942.

    [4]

    G. Ciccotti, T. Lelièvre and E. Vanden-Eijnden, Projection of diffusions on submanifolds: Application to mean force computation, Commun. Pur. Appl. Math., 61 (2008), 371-408.doi: 10.1002/cpa.20210.

    [5]

    P. Constantin, I. Kevrekidis and E. S. Titi, Asymptotic states of a Smoluchowski equation, Archive Rational Mech. Analysis, 174 (2004), 365-384.doi: 10.1007/s00205-004-0331-8.

    [6]

    P. Constantin, I. Kevrekidis and E. S. Titi, Remarks on a Smoluchowski equation, Disc. and Cont. Dyn. Syst., 11 (2004), 101-112.doi: 10.3934/dcds.2004.11.101.

    [7]

    M. Doi, Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases, J. Polym. Sci., Polym. Phys. Ed., 19 (1981), 229-243.doi: 10.1002/pol.1981.180190205.

    [8]

    J. Dolbeault, D. Kinderlehrer and M. Kowalczyk, The flashing ratchet: Long time behavior and dynamical systems interpretation, Technical Report 0244, CEREMADE, 2002. Available from: http://www.ceremade.dauphine.fr/preprints/CMD/2002-44.pdf.

    [9]

    G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers," Fifth edition, The Clarendon Press, Oxford University Press, New York, 1979.

    [10]

    M. Hitsuda and I. Mitoma, Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions, J. Multivariate Anal., 19 (1986), 311-328.doi: 10.1016/0047-259X(86)90035-7.

    [11]

    B. Jourdain, C. Le Bris, T. Lelièvre and F. Otto, Long-time asymptotics of a multiscale model for polymeric fluid flows, Archive for Rational Mechanics and Analysis, 181 (2006), 97-148.doi: 10.1007/s00205-005-0411-4.

    [12]

    Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Second edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998.

    [13]

    C. Le Bris, T. Lelièvre and E. Vanden-Eijnden, Analysis of some discretization schemes for constrained stochastic differential equations, C. R. Math. Acad. Sci. Paris, 346 (2008), 471-476.doi: 10.1016/j.crma.2008.02.016.

    [14]

    J. H. Lee, M. G. Forest and R. Zhou, Alignment and Rheo-oscillator criteria for sheared nematic polymer films in the monolayer limit, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 339-356.

    [15]

    J. D. Meiss, "Differential Dynamical Systems," Mathematical Modeling and Computation, 14, SIAM, Philadelphia, PA, 2007.

    [16]

    I. Niven, H. S. Zuckerman and H. L. Montgomery, "An Introduction to the Theory of Numbers," Fifth edition, John Wiley & Sons, Inc., New York, 1991.

    [17]

    A.-S. Sznitman, Topics in propagation of chaos, in "École d'Été de Probabilités de Saint-Flour XIX-1989," Lecture Notes in Math., 1464, Springer, Berlin, (1991), 165-251.

    [18]

    H. Zhang and P.-W. Zhang, A theoretical and numerical study for the rod-like model of a polymeric fluid, Journal of Computational Mathematics, 22 (2004), 319-330.

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