Local existence of smooth solutions to the FENE dumbbell model

Abstract

The author proves the local existence of smooth solutions to the finite extensible nonlinear elasticity (FENE) dumbbell model of polymeric flows in some weighted spaces if the non-dimensional parameter b > 2.

Keywords

Fokker-Planck equation / FENE model / Degenerate parabolic equations / Regularity

References

[1] Arnold A., Carrillo J. A., Manzini C.. Refined long-time asymptotics for some polymeric fluid flow models. Commun. Math. Sci., 2010, 8(3): 763-782

[2] Barrett J. W., Schwab C., Süli E.. Existence of global weak solutions for some polymeric flow models. Math. Models Methods Appl. Sci., 2005, 15(6): 939-983

[3] Barrett J. W., Süli E.. Existence of global weak solutions to kinetic models of dilute polymers. Multiscale Model Simul., 2007, 6: 506-546

[4] Barrett J. W., Süli E.. Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off. Math. Models Methods Appl. Sci., 2008, 18: 935-971

[5] Barrett J. W., Süli E.. Existence of equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers. Math. Models Methods Appl. Sci., 2011, 21(6): 1211-1289

[6] Bird R. B., Curtiss C., Armstrong R. C. Dynamics of Polymeric Liquids, Vol. 2, 1987, Wiley Interscience, New York: Kinetic Theory

[7] Constantin P., Foias C.. Navier-Stokes Equations, 1988, Chicago: The University of Chicago Press

[8] Doi M., Edwards S. F.. The Theory of Polymer Dynamics, 1986, Oxford: Oxford University Press

[9] E W. N., Li T. J., Zhang P. W.. Well-posedness for the dumbbell model of polymeric fluids. Comm. Math. Phys., 2004, 248(2): 409-427

[10] Friendman A.. Partial Differential Equation, Holt, 1969, New York: Rinehart and Winston

[11] He L. B., Zhang P.. L ² decay of solutions to a micro-macro model for polymeric fluids near equilibrium. SIAM J. Math. Anal., 2008, 40(5): 1905-1922

[12] Heywood J. G.. The Navier-Stokes equations: on the existence regularity and decay of solutions. Indiana Univ. Math. J., 1980, 29(5): 639-681

[13] Hong, J. X. and Yang, G., On the regularity of solutions to FENE models, preprint.

[14] Jourdain B., Lelièvre T.. Mathematical analysis of a stochastic differential equation arising in the micro-macro modelling of polymeric fluids, Probabilistic Methods in Fluids, 2003, River Edge, NJ: World Sci. Publ. 205-223

[15] Jourdain B., Lelièvre T., Le Bris C.. Existence of solution for a micro-macro model of polymeric fluid: the FENE model. J. Funct. Anal., 2004, 209(1): 162-193

[16] Jourdain B., Lelièvre T., Le Bris C. Long-time asymptotics of amultiscale model for polymeric fluid flows. Arch. Ration. Mech. Anal., 2006, 181: 97-148

[17] Lin F. H., Liu C., Zhang P.. On a micro-macro model for polymeric fluids near equilibrium. Comm. Pure Appl. Math., 2007, 60(6): 838-866

[18] Lin F. H., Zhang P.. The FENE dumbbell model near equilibrium. Acta Math. Sin. (Engl. Ser.), 2008, 24(4): 529-538

[19] Lin F. H., Zhang P., Zhang Z. F.. On the global existence of smooth solution to the 2-D FENE dumbbell model. Comm. Math. Phys., 2008, 277(2): 531-553

[20] Liu C., Liu H. L.. Boundary conditions for the microscopic FENE models. SIAM J. Appl. Math., 2008, 68(5): 1304-1315

[21] Liu H. L., Shin J.. Global well-posedness for the microscopic FENE model with a sharp boundary condition. J. Diff. Eq., 2012, 252(1): 641-662

[22] Liu, H. L. and Shin, J., The Cauchy-Dirichlet problem for the FENE dumbbell model of polymeric flows, Invent. Math., 2012, to appear. DOI: 10.1007/S00222-012-0399-y

[23] Majda A. J., Bertozzi A. L.. Vorticity and Incompressible Flow, 2002, Cambridge: Cambridge University Press

[24] Masmoudi N.. Well-posedness for the FENE dumbbell model of polymeric flows. Comm. Pure Appl. Math., 2008, 61(12): 1685-1714

[25] Masmoudi, N., Global existence of weak solutions to the FENE dumbbell model of polymeric flows, Invent. Math., 2012, to appear. DOI: 10.1007/s00222-012-0399-y

[26] Masmoudi N., Zhang P., Zhang Z. F.. Global well-posedness for 2D polymeric fluid models and growth estimate. Phys. D, 2008, 237(10–12): 1663-1675

[27] Nirenberg L.. On elliptic partial differential equation. Ann. Sc. Norm. Super. Pisa, 1959, 13: 115-162

[28] Oleǐnik O. A., Radkevič E. V.. Second Order Equations with Nonnegative Characteristic Form, 1973, Providence, RI: A. M. S.

[29] Öttinger H.. Stochastic Processes in Polymeric Liquids, 1996, Berlin, New York: Springer-Verlag

[30] Owens R. G., Phillips T. N.. Computational Rheology, 2002, London: Imperial College Press

[31] Renardy M.. An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Anal., 1991, 22(2): 313-327

[32] Schonbek M. E.. Existence and decay of polymeric flows. SIAM J. Math. Anal., 2009, 41(2): 564-587

[33] Zhang H., Zhang P. W.. Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ration. Mech. Anal., 2006, 181(2): 373-400