# American Institute of Mathematical Sciences

June  2012, 5(2): 357-382. doi: 10.3934/krm.2012.5.357

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

 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084 2 CERMICS, École des Ponts ParisTech, 6 & 8, avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France, France

Received  July 2011 Revised  January 2012 Published  April 2012

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.
Citation: Lingbing He, Claude Le Bris, Tony Lelièvre. Periodic long-time behaviour for an approximate model of nematic polymers. Kinetic & Related Models, 2012, 5 (2) : 357-382. doi: 10.3934/krm.2012.5.357
##### References:
 [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 (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. [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. 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. 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. 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. 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., 19 (1981), 229. 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, (0244), 2002. [9] G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,", Fifth edition, (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. 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. doi: 10.1007/s00205-005-0411-4. [12] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Second edition, 112 (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. 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. [15] J. D. Meiss, "Differential Dynamical Systems,", Mathematical Modeling and Computation, 14 (2007). [16] I. Niven, H. S. Zuckerman and H. L. Montgomery, "An Introduction to the Theory of Numbers,", Fifth edition, (1991). [17] A.-S. Sznitman, Topics in propagation of chaos,, in, 1464 (1991), 165. [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.

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##### References:
 [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 (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. [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. 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. 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. 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. 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., 19 (1981), 229. 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, (0244), 2002. [9] G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,", Fifth edition, (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. 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. doi: 10.1007/s00205-005-0411-4. [12] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Second edition, 112 (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. 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. [15] J. D. Meiss, "Differential Dynamical Systems,", Mathematical Modeling and Computation, 14 (2007). [16] I. Niven, H. S. Zuckerman and H. L. Montgomery, "An Introduction to the Theory of Numbers,", Fifth edition, (1991). [17] A.-S. Sznitman, Topics in propagation of chaos,, in, 1464 (1991), 165. [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.
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