Communications in Mathematical Physics

, Volume 270, Issue 3, pp 789–811 | Cite as

Regularity of Coupled Two-Dimensional Nonlinear Fokker-Planck and Navier-Stokes Systems

  • P. Constantin
  • C. Fefferman
  • E. S. Titi
  • A. Zarnescu
Article

Abstract

We consider systems of particles coupled with fluids. The particles are described by the evolution of their density, and the fluid is described by the Navier-Stokes equations. The particles add stress to the fluid and the fluid carries and deforms the particles. Because the particles perform rapid random motion, we assume that the density of particles is carried by a time average of the fluid velocity. The resulting coupled system is shown to have smooth solutions at all values of parameters, in two spatial dimensions.

References

  1. 1.
    Brézis H. and Gallouet T. (1980). Nonlinear Schrödinger evolution equations. Nonlinear Anal. 4(4): 677–681 CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Chemin, J.Y.: Perfect Incompressible Fluids. Oxford Lecture Series in Mathematics and its Applications. 14, New York: Clarendon Press Oxford University Press, 1998Google Scholar
  3. 3.
    Chemin J.Y. and Masmoudi N. (2001). About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal. 33(1): 84–112 CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Constantin P. (2005). Nonlinear Fokker-Planck Navier-Stokes systems. Commun. Math. Sciences 3(4): 531–544 MathSciNetMATHGoogle Scholar
  5. 5.
    Constantin P. and Foias C. (1988). Navier-Stokes Equations. U. Chicago Press, Chicago, IL MATHGoogle Scholar
  6. 6.
    Doi M. and Edwards S.F. (1988). The Theory of Polymer Dynamics. Oxford University Press, Oxford Google Scholar
  7. 7.
    E W., Li T.J. and Zhang P-W. (2004). Well-posedness for the dumbell model of polymeric fluids. Commun. Math. Phys. 248: 409–427 CrossRefADSMathSciNetMATHGoogle Scholar
  8. 8.
    Helgason, S.: Differential Geometry, Lie Groups and Symmetric spaces. London Academic Press, 1978Google Scholar
  9. 9.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators. Vol. 3, Berlin-Heidelberg- New York-Tokyo: Springer-Verlag 1985Google Scholar
  10. 10.
    Jourdain B., Lelievre T. and Le Bris C. (2002). Numerical analysis of micro-macro simulations of polymeric flows: a simple case. Math. Models in Appl. Science 12: 1205–1243 CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Jourdain B., Lelievre T. and Le Bris C. (2004). Esistence of solutions for a micro-macro model of polymeric fluid: the FENE model. J. Funct. Anal. 209: 162–193 CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Lemarié-Rieusset, P.G.: Recent Developments in the Navier-Stokes Problem. Chapmann and Hall/CRC Research Notes in Mathematics 431, Boca Raton, FL: CRC, 2002Google Scholar
  13. 13.
    Lin F-H., Liu C. and Zhang P. (2005). On hydrodynamics of viscoelastic fluids. CPAM 58: 1437–1471 MathSciNetMATHGoogle Scholar
  14. 14.
    Li T., Zhang H. and Zhang P-W. (2004). Local existence for the dumbell model of polymeric fluids. Commun. PDE 29: 903–923 CrossRefMATHGoogle Scholar
  15. 15.
    Onsager L. (1949). The effects of shape on the interaction of colloidal particles. Ann. N.Y. Acad. Sci 51: 627–659 CrossRefADSGoogle Scholar
  16. 16.
    Otto, F., Tzavaras, A.: Continuity of velocity gradients in suspensions of rod-like molecules. SFB Preprint 147 (2004), available at http://www.math.Umd.edu/Ntzavaras/reprints/existence.pdfGoogle Scholar
  17. 17.
    Renardy M. (1991). An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Analysis 23: 313–327 CrossRefMathSciNetGoogle Scholar
  18. 18.
    Sideris T. and Thomasses B. (2005). Global existence for 3D incompressible isotropic elastodynamics via the incompressible limit. CPAM 58: 750–788 MATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • P. Constantin
    • 1
  • C. Fefferman
    • 2
  • E. S. Titi
    • 3
    • 4
  • A. Zarnescu
    • 1
  1. 1.Department of MathematicsThe University of ChicagoChicagoUSA
  2. 2.Department of Mathematics, Princeton UniversityPrincetonUSA
  3. 3.Department of Mathematics and Department of Mechanics and Aerospace EngineeringUniversity of CaliforniaIrvineUSA
  4. 4.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

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