Global existence of weak solutions to the FENE dumbbell model of polymeric flows

Nader Masmoudi

doi:10.1007/s00222-012-0399-y

Inventiones mathematicae

February 2013, Volume 191, Issue 2, pp 427–500

Global existence of weak solutions to the FENE dumbbell model of polymeric flows

Authors
Authors and affiliations

Nader MasmoudiEmail author

Article

First Online:: 21 April 2012

Received:: 07 February 2011
Accepted:: 22 March 2012

DOI: 10.1007/s00222-012-0399-y

Cite this article as:: Masmoudi, N. Invent. math. (2013) 191: 427. doi:10.1007/s00222-012-0399-y

28 Citations
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Abstract

Systems coupling fluids and polymers are of great interest in many branches of sciences. One of the most classical models to describe them is the FENE (Finite Extensible Nonlinear Elastic) dumbbell model. We prove global existence of weak solutions to the FENE dumbbell model of polymeric flows. The main difficulty is the passage to the limit in a nonlinear term that has no obvious compactness properties. The proof uses many weak convergence techniques. In particular it is based on the control of the propagation of strong convergence of some well chosen quantity by studying a transport equation for its defect measure. In addition, this quantity controls a rescaled defect measure of the gradient of the velocity.

Mathematics Subject Classification

35Q30 82C31 76A05

References

1.
Alexandre, R., Villani, C.: On the Boltzmann equation for long-range interactions. Commun. Pure Appl. Math. 55(1), 30–70 (2002) MathSciNetMATHCrossRefGoogle Scholar
2.
Amann, H.: On the strong solvability of the Navier-Stokes equations. J. Math. Fluid Mech. 2(1), 16–98 (2000) MathSciNetMATHCrossRefGoogle Scholar
3.
Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158(2), 227–260 (2004) MathSciNetMATHCrossRefGoogle Scholar
4.
Arnold, A., Carrillo, J.A., Manzini, C.: Refined long-time asymptotics for some polymeric fluid flow models. Commun. Math. Sci. 8(3), 763–782 (2010) MathSciNetMATHGoogle Scholar
5.
Ball, J.M., Murat, F.: Remarks on Chacon’s biting lemma. Proc. Am. Math. Soc. 107(3), 655–663 (1989) MathSciNetMATHGoogle Scholar
6.
Barrett, J.W., Süli, E.: Existence of global weak solutions to some regularized kinetic models for dilute polymers. Multiscale Model. Simul. 6(2), 506–546 (2007) (electronic) MathSciNetMATHCrossRefGoogle Scholar
7.
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. 18(6), 935–971 (2008) MathSciNetMATHCrossRefGoogle Scholar
8.
Barrett, J.W., Süli, E.: Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: Finitely extensible nonlinear bead-spring chains. Math. Models Methods Appl. Sci. 21(6), 1211–1289 (2011) MathSciNetMATHCrossRefGoogle Scholar
9.
Barrett, J.W., Süli, E.: Existence and equilibration of global weak solutions to kinetic models for dilute polymers II: Hookean-type bead-spring chains. Math. Models Methods Appl. Sci. 22(5) (2012, to appear)
10.
Barrett, J.W., Schwab, C., Süli, E.: Existence of global weak solutions for some polymeric flow models. Math. Models Methods Appl. Sci. 15(6), 939–983 (2005) MathSciNetMATHCrossRefGoogle Scholar
11.
Bird, R.B., Amstrong, R., Hassager, O.: Dynamics of Polymeric Liquids, vol. 1. Wiley, New York (1977) Google Scholar
12.
Bird, R.B., Curtiss, C., Amstrong, R., Hassager, O.: Dynamics of Polymeric Liquids. Kinetic Theory, vol. 2. Wiley, New York (1987) Google Scholar
13.
Chemin, J.-Y.: Fluides parfaits incompressibles. Astérisque 230, 177 (1995) MathSciNetGoogle Scholar
14.
Chemin, J.-Y., Masmoudi, N.: About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal. 33(1), 84–112 (2001) (electronic) MathSciNetMATHCrossRefGoogle Scholar
15.
Chupin, L.: The FENE model for viscoelastic thin film flows. Methods Appl. Anal. 16(2), 217–261 (2009) MathSciNetMATHGoogle Scholar
16.
Chupin, L.: Fokker-Planck equation in bounded domain. Ann. Inst. Fourier (Grenoble) 60(1), 217–255 (2010) MathSciNetMATHCrossRefGoogle Scholar
17.
Constantin, P.: Nonlinear Fokker-Planck Navier-Stokes systems. Commun. Math. Sci. 3(4), 531–544 (2005) MathSciNetMATHGoogle Scholar
18.
Constantin, P., Masmoudi, N.: Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D. Commun. Math. Phys. 278(1), 179–191 (2008) MathSciNetMATHCrossRefGoogle Scholar
19.
Constantin, P., Sun, W.: Remarks on Oldroyd-B and related complex fluid models. Preprint, CMS (2010, to appear)
20.
Constantin, P., Fefferman, C., Titi, E.S., Zarnescu, A.: Regularity of coupled two-dimensional nonlinear Fokker-Planck and Navier-Stokes systems. Commun. Math. Phys. 270(3), 789–811 (2007) MathSciNetMATHCrossRefGoogle Scholar
21.
Crippa, G., De Lellis, C.: Estimates and regularity results for the DiPerna-Lions flow. J. Reine Angew. Math. 616, 15–46 (2008) MathSciNetMATHGoogle Scholar
22.
Degond, P., Liu, H.: Kinetic models for polymers with inertial effects. Netw. Heterog. Media 4(4), 625–647 (2009) MathSciNetMATHCrossRefGoogle Scholar
23.
Degond, P., Lemou, M., Picasso, M.: Viscoelastic fluid models derived from kinetic equations for polymers. SIAM J. Appl. Math. 62(5), 1501–1519 (2002) (electronic) MathSciNetMATHCrossRefGoogle Scholar
24.
Desjardins, B.: Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations. Differ. Integral Equ. 10(3), 577–586 (1997) MathSciNetMATHGoogle Scholar
25.
DiPerna, R.J., Lions, P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. (2) 130(2), 321–366 (1989) MathSciNetMATHCrossRefGoogle Scholar
26.
DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989) MathSciNetMATHCrossRefGoogle Scholar
27.
Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Oxford University Press, Oxford (1986) Google Scholar
28.
Du, Q., Liu, C., Yu, P.: FENE dumbbell model and its several linear and nonlinear closure approximations. Multiscale Model. Simul. 4(3), 709–731 (2005) (electronic) MathSciNetMATHCrossRefGoogle Scholar
29.
Li, W.E.T., Zhang, P.: Well-posedness for the dumbbell model of polymeric fluids. Commun. Math. Phys. 248(2), 409–427 (2004) MATHGoogle Scholar
30.
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992) Google Scholar
31.
Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics and Its Applications, vol. 26. Oxford University Press, Oxford (2004) MATHGoogle Scholar
32.
Fernández-Cara, E., Guillén, F., Ortega, R.R.: Some theoretical results for viscoplastic and dilatant fluids with variable density. Nonlinear Anal. 28(6), 1079–1100 (1997) MathSciNetMATHCrossRefGoogle Scholar
33.
Fernández-Cara, E., Guillén, F., Ortega, R.R.: Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 26(1), 1–29 (1998) MATHGoogle Scholar
34.
Fernández-Cara, E., Guillén, F., Ortega, R.R.: The mathematical analysis of viscoelastic fluids of the Oldroyd kind (2000)
35.
Gallez, X., Halin, P., Lielens, G., Keunings, R., Legat, V.: The adaptive Lagrangian particle method for macroscopic and micro-macro computations of time-dependent viscoelastic flows. Comput. Methods Appl. Mech. Eng. 180(3–4), 345–364 (1999) MathSciNetMATHCrossRefGoogle Scholar
36.
Giga, Y., Sohr, H.: Abstract L ^p estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102(1), 72–94 (1991) MathSciNetMATHCrossRefGoogle Scholar
37.
Grmela, M., Öttinger, H.C.: Dynamics and thermodynamics of complex fluids. I and II. Development of a general formalism. Phys. Rev. E (3) 56(6), 6620–6655 (1997) MathSciNetCrossRefGoogle Scholar
38.
Guillopé, C., Saut, J.-C.: Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal. 15(9), 849–869 (1990) MathSciNetMATHCrossRefGoogle Scholar
39.
Guillopé, C., Saut, J.-C.: Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type. RAIRO Modél. Math. Anal. Numér. 24(3), 369–401 (1990) MATHGoogle Scholar
40.
Hardy, G.H.: Notes on some points in the integral calculus, LX. an inequality between integrals. Messenger Math. 54, 150–156 (1925) Google Scholar
41.
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988). Reprint of the 1952 edn. Google Scholar
42.
He, L., Zhang, P.: L ² decay of solutions to a micro-macro model for polymeric fluids near equilibrium. SIAM J. Math. Anal. 40(5), 1905–1922 (2008/2009) MathSciNetCrossRefGoogle Scholar
43.
Jourdain, B., Lelièvre, T.: Mathematical analysis of a stochastic differential equation arising in the micro-macro modelling of polymeric fluids. In: Probabilistic Methods in Fluids, pp. 205–223. World Scientific, River Edge (2003) Google Scholar
44.
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(1), 162–193 (2004) MathSciNetMATHCrossRefGoogle Scholar
45.
Jourdain, B., Le Bris, C., Lelièvre, T., Otto, F.: Long-time asymptotics of a multiscale model for polymeric fluid flows. Arch. Ration. Mech. Anal. 181(1), 97–148 (2006) MathSciNetMATHCrossRefGoogle Scholar
46.
Keunings, R.: Simulation of viscoelastic fluid flow. In: Tucker, C.L. III (ed.) Fundamentals of Computer Modeling for Polymer Processing. Hanser Verlag, Munich (1989) Google Scholar
47.
Keunings, R.: On the Peterlin approximation for finitely extensible dumbbells. J. Non-Newton. Fluid Mech. 86, 85–100 (1997) CrossRefGoogle Scholar
48.
Kreml, O., Pokorný, M.: On the local strong solutions for the FENE dumbbell model. Discrete Contin. Dyn. Syst. Ser. S 3(2), 311–324 (2010) MathSciNetMATHCrossRefGoogle Scholar
49.
Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy Inequality. Vydavatelský Servis, Plzeň (2007). About its history and some related results MATHGoogle Scholar
50.
Le Bris, C., Lelièvre, T.: Multiscale modelling of complex fluids: a mathematical initiation. In: Multiscale Modeling and Simulation in Science. Lect. Notes Comput. Sci. Eng., vol. 66, pp. 49–137. Springer, Berlin (2009) CrossRefGoogle Scholar
51.
Lei, Z., Zhou, Y.: Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit. SIAM J. Math. Anal. 37(3), 797–814 (2005) (electronic) MathSciNetMATHCrossRefGoogle Scholar
52.
Lei, Z., Liu, C., Zhou, Y.: Global solutions for incompressible viscoelastic fluids. Arch. Ration. Mech. Anal. 188(3), 371–398 (2008) MathSciNetMATHCrossRefGoogle Scholar
53.
Lei, Z., Masmoudi, N., Zhou, Y.: Remarks on the blowup criteria for Oldroyd models. J. Differ. Equ. 248(2), 328–341 (2010) MathSciNetMATHCrossRefGoogle Scholar
54.
Leray, J.: Etude de diverses équations intégrales nonlinéaires et de quelques problèmes que pose l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933) MathSciNetMATHGoogle Scholar
55.
Leray, J.: Essai sur les mouvements plans d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934) MathSciNetMATHCrossRefGoogle Scholar
56.
Li, T., Zhang, P.: Mathematical analysis of multi-scale models of complex fluids. Commun. Math. Sci. 5(1), 1–51 (2007) MathSciNetMATHGoogle Scholar
57.
Lin, F.-H., Liu, C., Zhang, P.: On hydrodynamics of viscoelastic fluids. Commun. Pure Appl. Math. 58(11), 1437–1471 (2005) MathSciNetMATHCrossRefGoogle Scholar
58.
Lin, F.-H., Liu, C., Zhang, P.: On a micro-macro model for polymeric fluids near equilibrium. Commun. Pure Appl. Math. 60(6), 838–866 (2007) MathSciNetMATHCrossRefGoogle Scholar
59.
Lin, F.-H., Zhang, P., Zhang, Z.: On the global existence of smooth solution to the 2-D FENE dumbbell model. Commun. Math. Phys. 277(2), 531–553 (2008) MathSciNetMATHCrossRefGoogle Scholar
60.
Lions, P.-L.: Mathematical Topics in Fluid Mechanics, vol. 1. The Clarendon Press Oxford University Press, New York (1996). Incompressible models, Oxford Science Publications MATHGoogle Scholar
61.
Lions, P.-L.: Mathematical Topics in Fluid Mechanics, vol. 2. The Clarendon Press Oxford University Press, New York (1998). Compressible models, Oxford Science Publications MATHGoogle Scholar
62.
Lions, P.-L., Masmoudi, N.: On a free boundary barotropic model. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16(3), 373–410 (1999) MathSciNetMATHCrossRefGoogle Scholar
63.
Lions, P.-L., Masmoudi, N.: Global solutions for some Oldroyd models of non-Newtonian flows. Chin. Ann. Math., Ser. B 21(2), 131–146 (2000) MathSciNetMATHCrossRefGoogle Scholar
64.
Lions, P.-L., Masmoudi, N.: From the Boltzmann equations to the equations of incompressible fluid mechanics. II. Arch. Ration. Mech. Anal. 158(3), 195–211 (2001) MathSciNetCrossRefGoogle Scholar
65.
Lions, P.-L., Masmoudi, N.: Global existence of weak solutions to some micro-macro models. C. R. Math. Acad. Sci. Paris 345(1), 15–20 (2007) MathSciNetMATHCrossRefGoogle Scholar
66.
Liu, C., Liu, H.: Boundary conditions for the microscopic FENE models. SIAM J. Appl. Math. 68(5), 1304–1315 (2008) MathSciNetMATHCrossRefGoogle Scholar
67.
Liu, H., Shin, J.: Global well-posedness for the microscopic FENE model with a sharp boundary condition. Preprint (2010)
68.
Masmoudi, N.: Well-posedness for the FENE dumbbell model of polymeric flows. Commun. Pure Appl. Math. 61(12), 1685–1714 (2008) MathSciNetMATHCrossRefGoogle Scholar
69.
Masmoudi, N.: Global existence of weak solutions to macroscopic models of polymeric flows. J. Math. Pures Appl. (9) 96(5), 502–520 (2011) MathSciNetMATHGoogle Scholar
70.
Masmoudi, N.: Regularity of solutions to the FENE model in the polymer elongation variable R (2011, in preparation)
71.
Masmoudi, N.: Zero diffusion limit in the FENE model of polymeric flows (2011, in preparation)
72.
Masmoudi, N., Zhang, P., Zhang, Z.: Global well-posedness for 2D polymeric fluid models and growth estimate. Physica D 237(10–12), 1663–1675 (2008) MathSciNetMATHCrossRefGoogle Scholar
73.
Mischler, S.: Kinetic equations with Maxwell boundary conditions. Ann. Sci. Éc. Norm. Supér. (4) 43(5), 719–760 (2010) MathSciNetMATHGoogle Scholar
74.
Öttinger, H.C.: Stochastic Processes in Polymeric Fluids. Springer, Berlin (1996). Tools and examples for developing simulation algorithms MATHCrossRefGoogle Scholar
75.
Otto, F., Tzavaras, A.E.: Continuity of velocity gradients in suspensions of rod-like molecules. Commun. Math. Phys. 277(3), 729–758 (2008) MathSciNetMATHCrossRefGoogle Scholar
76.
Owens, R.G., Phillips, T.N.: Computational Rheology. Imperial College Press, London (2002) MATHCrossRefGoogle Scholar
77.
Renardy, M.: An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Anal. 22(2), 313–327 (1991) MathSciNetMATHCrossRefGoogle Scholar
78.
Renardy, M.: Mathematical Analysis of Viscoelastic Flows. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 73. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000) MATHCrossRefGoogle Scholar
79.
Schonbek, M.E.: Existence and decay of polymeric flows. SIAM J. Math. Anal. 41(2), 564–587 (2009) MathSciNetMATHCrossRefGoogle Scholar
80.
Solonnikov, V.A.: Estimates for solutions of a non-stationary linearized system of Navier-Stokes equations. Transl. Am. Math. Soc. 75, 1–116 (1968) MATHGoogle Scholar
81.
Temam, R.: Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1995) MATHCrossRefGoogle Scholar
82.
Zhang, H., Zhang, P.: Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ration. Mech. Anal. 181(2), 373–400 (2006) MathSciNetMATHCrossRefGoogle Scholar
83.
Zhang, L., Zhang, H., Zhang, P.: Global existence of weak solutions to the regularized Hookean dumbbell model. Commun. Math. Sci. 6(1), 85–124 (2008) MathSciNetGoogle Scholar

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Nader Masmoudi
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1.Courant InstituteNew York UniversityNew YorkUSA

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Global existence of weak solutions to the FENE dumbbell model of polymeric flows

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