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Inventiones mathematicae

, Volume 191, Issue 2, pp 427–500 | Cite as

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

  • Nader Masmoudi
Article
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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 
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References

  1. 1.
    Alexandre, R., Villani, C.: On the Boltzmann equation for long-range interactions. Commun. Pure Appl. Math. 55(1), 30–70 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Amann, H.: On the strong solvability of the Navier-Stokes equations. J. Math. Fluid Mech. 2(1), 16–98 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158(2), 227–260 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 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) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ball, J.M., Murat, F.: Remarks on Chacon’s biting lemma. Proc. Am. Math. Soc. 107(3), 655–663 (1989) MathSciNetzbMATHGoogle Scholar
  6. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 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) Google Scholar
  10. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bird, R.B., Amstrong, R., Hassager, O.: Dynamics of Polymeric Liquids, vol. 1. Wiley, New York (1977) Google Scholar
  12. 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. 13.
    Chemin, J.-Y.: Fluides parfaits incompressibles. Astérisque 230, 177 (1995) MathSciNetGoogle Scholar
  14. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Chupin, L.: The FENE model for viscoelastic thin film flows. Methods Appl. Anal. 16(2), 217–261 (2009) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Chupin, L.: Fokker-Planck equation in bounded domain. Ann. Inst. Fourier (Grenoble) 60(1), 217–255 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Constantin, P.: Nonlinear Fokker-Planck Navier-Stokes systems. Commun. Math. Sci. 3(4), 531–544 (2005) MathSciNetzbMATHGoogle Scholar
  18. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Constantin, P., Sun, W.: Remarks on Oldroyd-B and related complex fluid models. Preprint, CMS (2010, to appear) Google Scholar
  20. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Crippa, G., De Lellis, C.: Estimates and regularity results for the DiPerna-Lions flow. J. Reine Angew. Math. 616, 15–46 (2008) MathSciNetzbMATHGoogle Scholar
  22. 22.
    Degond, P., Liu, H.: Kinetic models for polymers with inertial effects. Netw. Heterog. Media 4(4), 625–647 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 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) MathSciNetzbMATHGoogle Scholar
  25. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Oxford University Press, Oxford (1986) Google Scholar
  28. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Li, W.E.T., Zhang, P.: Well-posedness for the dumbbell model of polymeric fluids. Commun. Math. Phys. 248(2), 409–427 (2004) zbMATHGoogle Scholar
  30. 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. 31.
    Feireisl, E.: Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics and Its Applications, vol. 26. Oxford University Press, Oxford (2004) zbMATHGoogle Scholar
  32. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 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) zbMATHGoogle Scholar
  34. 34.
    Fernández-Cara, E., Guillén, F., Ortega, R.R.: The mathematical analysis of viscoelastic fluids of the Oldroyd kind (2000) Google Scholar
  35. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  36. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 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. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 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) zbMATHGoogle Scholar
  40. 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. 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. 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. 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. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  45. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  46. 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. 47.
    Keunings, R.: On the Peterlin approximation for finitely extensible dumbbells. J. Non-Newton. Fluid Mech. 86, 85–100 (1997) CrossRefGoogle Scholar
  48. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy Inequality. Vydavatelský Servis, Plzeň (2007). About its history and some related results zbMATHGoogle Scholar
  50. 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. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Lei, Z., Liu, C., Zhou, Y.: Global solutions for incompressible viscoelastic fluids. Arch. Ration. Mech. Anal. 188(3), 371–398 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Lei, Z., Masmoudi, N., Zhou, Y.: Remarks on the blowup criteria for Oldroyd models. J. Differ. Equ. 248(2), 328–341 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  54. 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) MathSciNetzbMATHGoogle Scholar
  55. 55.
    Leray, J.: Essai sur les mouvements plans d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934) MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Li, T., Zhang, P.: Mathematical analysis of multi-scale models of complex fluids. Commun. Math. Sci. 5(1), 1–51 (2007) MathSciNetzbMATHGoogle Scholar
  57. 57.
    Lin, F.-H., Liu, C., Zhang, P.: On hydrodynamics of viscoelastic fluids. Commun. Pure Appl. Math. 58(11), 1437–1471 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  58. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  59. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  60. 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 zbMATHGoogle Scholar
  61. 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 zbMATHGoogle Scholar
  62. 62.
    Lions, P.-L., Masmoudi, N.: On a free boundary barotropic model. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16(3), 373–410 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  63. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  64. 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. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Liu, C., Liu, H.: Boundary conditions for the microscopic FENE models. SIAM J. Appl. Math. 68(5), 1304–1315 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Liu, H., Shin, J.: Global well-posedness for the microscopic FENE model with a sharp boundary condition. Preprint (2010) Google Scholar
  68. 68.
    Masmoudi, N.: Well-posedness for the FENE dumbbell model of polymeric flows. Commun. Pure Appl. Math. 61(12), 1685–1714 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Masmoudi, N.: Global existence of weak solutions to macroscopic models of polymeric flows. J. Math. Pures Appl. (9) 96(5), 502–520 (2011) MathSciNetzbMATHGoogle Scholar
  70. 70.
    Masmoudi, N.: Regularity of solutions to the FENE model in the polymer elongation variable R (2011, in preparation) Google Scholar
  71. 71.
    Masmoudi, N.: Zero diffusion limit in the FENE model of polymeric flows (2011, in preparation) Google Scholar
  72. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Mischler, S.: Kinetic equations with Maxwell boundary conditions. Ann. Sci. Éc. Norm. Supér. (4) 43(5), 719–760 (2010) MathSciNetzbMATHGoogle Scholar
  74. 74.
    Öttinger, H.C.: Stochastic Processes in Polymeric Fluids. Springer, Berlin (1996). Tools and examples for developing simulation algorithms zbMATHCrossRefGoogle Scholar
  75. 75.
    Otto, F., Tzavaras, A.E.: Continuity of velocity gradients in suspensions of rod-like molecules. Commun. Math. Phys. 277(3), 729–758 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Owens, R.G., Phillips, T.N.: Computational Rheology. Imperial College Press, London (2002) zbMATHCrossRefGoogle Scholar
  77. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  78. 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) zbMATHCrossRefGoogle Scholar
  79. 79.
    Schonbek, M.E.: Existence and decay of polymeric flows. SIAM J. Math. Anal. 41(2), 564–587 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  80. 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) zbMATHGoogle Scholar
  81. 81.
    Temam, R.: Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1995) zbMATHCrossRefGoogle Scholar
  82. 82.
    Zhang, H., Zhang, P.: Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ration. Mech. Anal. 181(2), 373–400 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  83. 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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Authors and Affiliations

  1. 1.Courant InstituteNew York UniversityNew YorkUSA

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