Abstract
In this paper, we investigate the long-time behavior of some micro-macro models for polymeric fluids (Hookean model and FENE model), in various settings (shear flow, general bounded domain with homogeneous Dirichlet boundary conditions on the velocity, general bounded domain with non-homogeneous Dirichlet boundary conditions on the velocity). We use both probabilistic approaches (coupling methods) and analytic approaches (entropy methods).
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Ané, C., Blachère, S., ChafaÏ, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques. SMF, 2000
Arnold, A., Markowich, P., Toscani, G., Unterreiter, A.: On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Comm. Partial Differential Equations 26, 43–100 (2001)
Arnold, A., Unterreiter, A.: Entropy decay of discretized Fokker-Planck equations I: Temporal semidiscretization. Comput. Math. Appl. 46, 1683–1690 (2003)
Barrett, J.W., Schwab, C., Süli, E.: Existence of global weak solutions for some polymeric flow models. Math. Models Methods Appl. Sci. 15, 939–983 (2005)
Bird, R.B., Armstrong, R.C., Hassager, O.: Dynamics of polymeric liquids, Volume 1. Wiley Interscience, 1987
Bird, R.B., Curtiss, C.F., Armstrong, R.C., Hassager, O.: Dynamics of polymeric liquids, Volume 2. Wiley Interscience, 1987
Constantin, P., Kevrekidis, I., Titi, E.S.: Asymptotic states of a Smoluchowski equation. Arch. Ration. Mech. Anal. 174, 365–384 (2004)
Constantin, P., Kevrekidis, I., Titi, E.S.: Remarks on a Smoluchowski equation. Discrete Contin. Dyn. Syst. 11, 101–112 (2004)
Degond, P., Lemou, M., Picasso, M.: Viscoelastic fluid models derived from kinetic equations for polymers. SIAM J. Appl. Math. 62, 1501–1519 (2002)
Desvillettes, L., Villani, C.: On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math. 54, 1–42 (2001)
Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. International Series of Monographs on Physics. Oxford University Press, Oxford, 1988
Du, Q., Liu, C., Yu, P.: FENE dumbbell model and its several linear and nonlinear closure approximations. Multiscale Model. Simul. 4, 709–731 (2005)
E, W., Li, T.J., Zhang, P.W.: Well-posedness for the dumbbell model of polymeric fluids. Comm. Math. Phys. 248, 409–427 (2004)
Evans, L. C.: A survey of entropy methods for partial differential equations. Bull. Amer. Math. Soc. 41, 409–438 (2004)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Springer-Verlag, 1977
Jourdain, B., Le Bris, C., Lelièvre, T.: On a variance reduction technique for micro-macro simulations of polymeric fluids. J. Non-Newtonian Fluid Mech. 122, 91–106 (2004)
Jourdain, B., Le Bris, C., Lelièvre, T.: An elementary argument regarding the long-time behaviour of the solution to a stochastic differential equation. An. Univ. Craiova Ser. Math. Comput. Sci. 32, 39–47 (2005)
Jourdain, B., Lelièvre, T.: Mathematical analysis of a stochastic differential equation arising in the micro-macro modelling of polymeric fluids. Probabilistic Methods in Fluids Proceedings of the Swansea 2002 Workshop. Davies, I.M., Jacob, N., Truman, A., Hassan, O., Morgan, K., Weatherill, N.P. (eds.), World Scientific, 2003, pp. 205–223
Jourdain, B., Lelièvre, T., Le Bris, C.: Numerical analysis of micro-macro simulations of polymeric fluid flows: a simple case. Math. Models Methods in Appl. Sci. 12, 1205–1243 (2002)
Jourdain, B., Lelièvre, T., Le Bris, C.: Existence of solution for a micro-macro model of polymeric fluid: the FENE model. J. Funct. Anal. 209, 162–193 (2004)
Li, T., Zhang, H., Zhang, P.W.: Local existence for the dumbbell model of polymeric fluids. Comm. Partial Differential Equations 29, 903–923 (2004)
Lindvall, T.: Lectures on the coupling method. Wiley, 1992
Luo, C., Zhang, H., Zhang, P.W.: The structure of equilibrium solutions of one dimensional Doi equation. Nonlinearity 18, 379–389 (2005)
Markowich, P.A., Villani, C.: On the trend to equilibrium for the Fokker-Planck equation: an interplay between physics and functional analysis. Mat. Contemp. 19, 1–29 (2000)
Mattingly, J.C., Stuart, A.M., Higham, D.J.: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stochastic Process. Appl. 101, 185–232 (2002)
Odasso, C.: Ergodicity for the stochastic complex Ginzburg-Landau equations. Ann. IH Poincaré-PR, in press
Öttinger, H.C.: Stochastic Processes in Polymeric Fluids. Springer, 1995
Otto, F.: Suspensions of rod-like molecules: transient growth of perturbations of homogeneous flows. Preprint, 2004
Renardy, M.: An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Anal. 22, 313–327 (1991)
Snyders, J., Zakai, M.: On nonnegative solutions of the equation AD+DA'=-C. SIAM J. Appl. Math. 18, 704–714 (1970)
Temam, R.: Navier-Stokes Equations. Revised edition, North-Holland, 1979
Triebel, H.: Interpolation theory, function spaces, differential operators. North- Holland, 1978
Zhang, H., Zhang, P.W.: Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ration. Mech. Anal. In press
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Jourdain, B., Le Bris, C., Lelièvre, T. et al. Long-Time Asymptotics of a Multiscale Model for Polymeric Fluid Flows. Arch. Rational Mech. Anal. 181, 97–148 (2006). https://doi.org/10.1007/s00205-005-0411-4
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DOI: https://doi.org/10.1007/s00205-005-0411-4