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On the Motion of Several Rigid Bodies in an Incompressible Viscous Fluid under the Influence of Selfgravitating Forces

Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 80)

Abstract

The global existence of weak solutions is proved for the problem of the motion of several rigid bodies in non-newtonian fluid of power-law with selfgraviting forces.

Keywords

Existence of weak solutions motion of several rigid bodies non-Newtonian fluid selfgravitating forces. 

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References

  1. 1.
    W. Borchers. Zur Stabilität und Faktorisierungsmethode für die Navier-Stokes-Gleichungen inkompressibler viskoser Flüssigkeiten. Habilitation Thesis, Univ. of Paderborn, 1992.Google Scholar
  2. 2.
    C. Bost, G.H. Cottet and E. Maitre. Numerical analysis of a penalization method for the three-dimensional motion of a rigid body in an incompressible viscous fluid. Preprint, 2009.Google Scholar
  3. 3.
    H. Brenner. The Stokes resistance of an arbitrary particle II. Chem. Engng. Sci. 19, 599–624, 1959.CrossRefGoogle Scholar
  4. 4.
    M. Bulïček, E. Feireisl and J. Mälek. Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients. Nonlinear Analysis: Real World Applications, 10, 2, 992–1015, 2009.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    C. Conca, J. San Martin and M. Tucsnak. Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid. Commun. Partial Differential Equation, 25, 1019–1042, 2000.MATHMathSciNetGoogle Scholar
  6. 6.
    B. Desjardins and M.J. Esteban. Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Rational Mech. Anal., 146, 59–71, 1999.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    B. Desjardins and M.J. Esteban. On weak solutions for fluid-rigid structure interaction: Compressible and incompressible models. Commun. Partial Differential Equations, 25, 1399–1413, 2000.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    B. Ducomet, E. Feireisl, H. Petzeltova, I. Straskraba. Global in time weak solutions for compressible barotropic self-graviting fluids, DCDS, A, 11, 1, 113–130.Google Scholar
  9. 9.
    B. Ducomet, E. Feireisl, H. Petzeltova, I. Straskraba: Existence global pour un fluide barotropique autogravitant, C. R. Acad. Paris, 332, 627–632, 2001.MATHMathSciNetGoogle Scholar
  10. 10.
    R.J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98, 511–547, 1989.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    R. Farwig. An Lq-analysis of viscous fluid flow past a rotating obstacle. Tˆohoku Math. J., 58, 129–147, 2005CrossRefMathSciNetGoogle Scholar
  12. 12.
    R. Farwig. Estimates of lower order derivatives of viscous fluid flow past a rotating obstacle. Banach Center Publications Warsaw, 70, 73–82, 2005.CrossRefMathSciNetGoogle Scholar
  13. 13.
    R. Farwig and T. Hishida. Stationary Navier-Stokes flow around a rotating obstacle. Funkcialaj Ekvacioj, 3, 371–403, 2007.CrossRefMathSciNetGoogle Scholar
  14. 14.
    R. Farwig, T. Hishida, and D. Müller. L q-theory of a singular “winding” integral operator arising from fluid dynamics. Pacific J. Math., 215, 297–312, 2004.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    E. Feireisl and J. Málek. On the Navier- Stokes equations with temperature dependent transport coefficients. Differ. Equ. Nonl. Mech., Art. Id 90616, 2006.Google Scholar
  16. 16.
    E. Feireisl. On the motion of rigid bodies in a viscous compressible fluid. Arch. Rational Mech. Anal., 167, 281–308, 2003.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    E. Feireisl, M. Hillairet and Š. Nečasová. On the motion of several rigid bodies in an incompressible non-Newtonian fluid. Nonlinearity, 21, 1349–1366, 2008.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    E. Feireisl and A. Novotný. Singular limits in thermodynamics of viscous fluids. Birkhäuser, Basel, 2008.Google Scholar
  19. 19.
    E. Feireisl, J. Neustupa, J. Stebel. Convergence of a Brinkman-type penalization for compressible fluid flows. Preprint of the Necas Centrum, 2010.Google Scholar
  20. 20.
    J. Frehse, J. Málek and M. Růźička. Large data existence result for unsteady flows of inhomogeneous heat-conducting incompressible fluids. Preprint of the Necas Centrum, 2008.Google Scholar
  21. 21.
    J. Frehse and M. Růźička. Non-homogeneous generalized Newtonian fluids. Math. Z., 260, 35–375, 2008.CrossRefGoogle Scholar
  22. 22.
    G.P. Galdi. On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch. Rat. Mech. Anal., 148, 53–88, 1999.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    G.P. Galdi. On the motion of a rigid body in a viscous fluid: A mathematical analysis with applications. Handbook of Mathematical Fluid Dynamics, Vol. I, Elsevier Sci., Amsterdam, 2002.Google Scholar
  24. 24.
    D. Gérard-Varet and M. Hillairet. Regularity issues in the problem of fluid structure interaction. Archive for Rational Mechanical Analysis. In press.Google Scholar
  25. 25.
    M.D. Gunzburger, H.C. Lee, and A. Seregin. Global existence of weak solutions for viscous incompressible flow around a moving rigid body in three dimensions. J. Math. Fluid Mech., 2, 219–266, 2000.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    T.I. Hesla. Collision of smooth bodies in a viscous fluid: A mathematical investigation. 2005. PhD Thesis – Minnesota.Google Scholar
  27. 27.
    M. Hillairet. Lack of collision between solid bodies in a 2D incompressible viscous flow. Comm. Partial Differential Equations, 32, 7–9, 1345–1371, 2007. Preprint – ENS Lyon.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    M. Hillairet and T. Takahashi. Collisions in three dimensional fluid structure interaction problems SIAM J. Math. Anal., 40, 6, 2451–2477, 2009MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    K.-H. Hoffmann and V.N. Starovoitov. On a motion of a solid body in a viscous fluid. Two dimensional case. Adv. Math. Sci. Appl., 9, 633–648, 1999.MATHMathSciNetGoogle Scholar
  30. 30.
    K.H. Hoffmann and V.N. Starovoitov. Zur Bewegung einer Kugel in einer zähen Flüssigkeit. (German) [On the motion of a sphere in a viscous fluid] Doc. Math., 5, 15–21, 2000.MATHMathSciNetGoogle Scholar
  31. 31.
    G. Kirchhoff. Über die Bewegung eines Rotationskörpers in einer Flüssigkeit. (German) [On the motion of a rotating body in a fluid] Crelle J. 71, 237–281, 1869.Google Scholar
  32. 32.
    H. Koch and V.A. Solonnikov. L P estimates for a solutions to the nonstationary Stokes equations. J. Math. Sci., 106, 3042–3072, 2001.CrossRefMathSciNetGoogle Scholar
  33. 33.
    J. Máalek, J. Neˇcas, M. Rokyta, and M. Růźička. Weak and measure-valued solutions to evolutionary PDE's. Chapman and Hall, London, 1996.Google Scholar
  34. 34.
    J.A. San Martin, V. Starovoitov, and M. Tucsnak. Global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Rational Mech. Anal., 161, 93–112, 2002.CrossRefMathSciNetGoogle Scholar
  35. 35.
    D. Serre. Chute libre d’un solide dans un fluide visqueux incompressible. Existence. Jap. J. Appl. Math., 4, 99–110, 1987.MATHCrossRefGoogle Scholar
  36. 36.
    H. Sohr. The Navier-Stokes equations: An elementary functional analytic approach. Birkhäuser Verlag, Basel, 2001.MATHGoogle Scholar
  37. 37.
    V.N. Starovoitov. Nonuniqueness of a solution to the problem on motion of a rigid body in a viscous incompressible fluid. J. Math. Sci., 130, 4893–4898, 2005.MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    V.N. Starovoitov. Behavior of a rigid body in an incompressible viscous fluid near boundary. In International Series of Numerical Mathematics, 147, 313–327, 2003.MathSciNetGoogle Scholar
  39. 39.
    N.V. Judakov. The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid. (Russian) Dinamika Splošn. Sredy Vyp., 18, 249–253, 1974.MathSciNetGoogle Scholar
  40. 40.
    L. Tartar. Compensated compactness and applications to partial differential equations. Heriot-Watt Symposium, Vol. IV, Res. Notes in Math. 39, Pitman, Boston, Mass., 136–212, 1979.MathSciNetGoogle Scholar
  41. 41.
    W. Thomson (Lord Kelvin). Mathematical and Physical Papers, Vol. 4. Cambridge University Press, 1982.Google Scholar
  42. 42.
    H.F. Weinberger. On the steady fall of a body in a Navier-Stokes fluid. Proc. Symp. Pure Mathematics 23, 421–440, 1973.Google Scholar
  43. 43.
    H.F. Weinberger. Variational properties of steady fall in Stokes flow. J. Fluid Mech. 52, 321–344, 1972.MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    J. Wolf. Existence of weak solutions to the equations of non-stationary motion of non-newtonian fluids with shear rate dependent viscosity. J. Math. Fluid Dynamics, 8, 1–35, 2006.Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.CEA, DAM, DIFArpajonFrance
  2. 2.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPraha 1Czech Republic

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