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

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

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

Lingbing He ^1, , Claude Le Bris ^2, and Tony Lelièvre ^2,

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

Abstract
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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.

Keywords: Entropy method, nematic polymer., Doi closure, periodic solution, liquid crystal.

Mathematics Subject Classification: Primary: 35B40, 76A1.

Citation: Lingbing He, Claude Le Bris, Tony Lelièvre. Periodic long-time behaviour for an approximate model of nematic polymers. Kinetic and 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, 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.

show all references

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, 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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