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A Kato Type Theorem for the Inviscid Limit of the Navier-Stokes Equations with a Moving Rigid Body

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Abstract

The issue of the inviscid limit for the incompressible Navier-Stokes equations when a no-slip condition is prescribed on the boundary is a famous open problem. A result by Kato (Math Sci Res Inst Publ 2:85–98, 1984) says that convergence to the Euler equations holds true in the energy space if and only if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes. Of course, if one considers the motion of a solid body in an incompressible fluid, with a no-slip condition at the interface, the issue of the inviscid limit is as least as difficult. However it is not clear if the additional difficulties linked to the body’s dynamic make this issue more difficult or not. In this paper we consider the motion of a rigid body in an incompressible fluid occupying the complementary set in the space and we prove that a Kato type condition implies the convergence of the fluid velocity and of the body velocity as well, which seems to indicate that an answer in the case of a fixed boundary could also bring an answer to the case where there is a moving body in the fluid.

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References

  1. Bardos, C., Titi, E.S.: Euler equations for an ideal incompressible fluid. (Russian) Usp. Mat. Nauk 62, no. 3(375), 5–46 (2007); translation in Russ. Math. Surv. 62(3), 409–451 (2007)

  2. Conca C., San Martin J.A., Tucsnak M.: Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid. Comm. Part. Diffl. Eqs. 25(5–6), 1019–1042 (2000)

    MathSciNet  MATH  Google Scholar 

  3. San Martin J.A., Starovoitov V., Tucsnak M.: Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Rat. Mech. Anal. 161(2), 113–147 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Constantin P.: On the Euler equations of incompressible fluids. Bull. Amer. Math. Soc. (N.S.) 44(4), 603–621 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Desjardins B., Esteban M.J.: On weak solutions for fluid-rigid structure interaction: compressible and incompressible models. Comm. Part. Diffl. Eqs. 25(7–8), 1399–1413 (2000)

    MathSciNet  MATH  Google Scholar 

  6. Desjardins B., Esteban M.J.: Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Rat. Mech. Anal. 146(1), 59–71 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gérard-Varet D., Dormy E.: On the ill-posedness of the Prandtl equation. J. Amer. Math. Soc. 23(2), 591–609 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gérard-Varet D., Hillairet M.: Regularity issues in the problem of fluid structure interaction. Arch. Rat. Mech. Anal. 195(2), 375–407 (2010)

    Article  MATH  Google Scholar 

  9. W.E.: Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation. Acta Math. Sin. (Engl. Ser.) 16(2), 207–218 (2000)

  10. Gamblin P., Saint Raymond X.: On three-dimensional vortex patches. Bull. Soc. Math. France 123(3), 375–424 (1995)

    MathSciNet  MATH  Google Scholar 

  11. Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Classics in Mathematics. Berlin-Heidelberg-New York: Springer-Verlag, 2001

  12. Glass, O., Lacave, C., Sueur F.: On the motion of a small body immersed in a two dimensional incompressible perfect fluid. Preprint, 2011, http://arxiv.org/abs/1104.5404v1 [math.AP] 2011

  13. Glass, O., Sueur, F.: On the motion of a rigid body in a two-dimensional irregular ideal flow. Preprint 2011, http://arxiv.org/abs/1107.0575v2 [math.AP], 2011

  14. Glass O., Sueur F., Takahashi T.: Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid. Ann. Sci. École Norm. Sup. 45(1), 1–51 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Grenier, E.: Boundary layers. Handbook of mathematical fluid dynamics. Vol. III, Amsterdam: Elsevier, 2004, pp. 245–309

  16. Guo Y., Nguyen T.T.: A note on the Prandtl boundary layers. Comm. Pure Appl. Math. 64(10), 1416–1438 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Houot J.-G., Munnier A.: On the motion and collisions of rigid bodies in an ideal fluid. Asymptot. Anal. 56(3–4), 125–158 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Houot J.-G., San Martin J., Tucsnak M.: Existence and uniqueness of solutions for the equations modelling the motion of rigid bodies in a perfect fluid. J. Funct. Anal. 259(11), 2856–2885 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Iftimie D., Lopes Filho M.C., Nussenzveig Lopes H.J.: Incompressible flow around a small obstacle and the vanishing viscosity limit. Commun. Math. Phys. 287(1), 99–115 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Kato T.: On classical solutions of the two-dimensional nonstationary Euler equation. Arch. Rat. Mech. Anal. 25(1), 188–200 (1967)

    MATH  Google Scholar 

  21. Kato T.: Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. Seminar on nonlinear partial differential equations, Math. Sci. Res. Inst. Publ. 2, 85–98 (1984)

    Article  Google Scholar 

  22. Kelliher J.P.: On Kato’s conditions for vanishing viscosity. Indiana Univ. Math. J. 56(4), 1711–1721 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kikuchi K.: The existence and uniqueness of nonstationary ideal incompressible flow in exterior domains in R3. J. Math. Soc. Japan 38(4), 575–598 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lions, P.-L.: Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications 3, Oxford: Oxford Univ. Press, 1996

  25. Masmoudi N.: The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary. Arch. Rat. Mech. Anal. 142(4), 375–394 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ortega J.H., Rosier L., Takahashi T.: Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid. M2AN Math. Model. Numer. Anal. 39(1), 79–108 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ortega J.H., Rosier L., Takahashi T.: On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid, Ann. Inst. H. Poincaré Anal. Non Linéaire 24(1), 139–165 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Rosier C., Rosier L.: Smooth solutions for the motion of a ball in an incompressible perfect fluid. J. Funct. Anal. 256(5), 1618–1641 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Serre D.: Chute libre d’un solide dans un fluide visqueux incompressible. Existence. Japan J. Appl. Math. 4(1), 99–110 (1987)

    Article  MATH  Google Scholar 

  30. Takahashi T., Tucsnak M.: Global strong solutions for the two-dimensional motion of an infinite cylinder in a viscous fluid. J. Math. Fluid Mech. 6(1), 53–77 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Temam, R.: Problèmes mathématiques en plasticité. Méthodes Mathématiques de l’Informatique, 12. Paris:Gauthier-Villars, 1983

  32. Temam R., Wang X.: On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(3–4), 807–828 (1997)

    MathSciNet  MATH  Google Scholar 

  33. Wang X.: A Kato type theorem on zero viscosity limit of Navier-Stokes flows. Indiana Univ. Math. J. 50(Special Issue), 223–241 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang Y., Zang A.: Smooth solutions for motion of a rigid body of general form in an incompressible perfect fluid. J. Diff. Eqs. 252(7), 4259–4288 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wolibner W.: Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long. Math. Z. 37(1), 698–726 (1933)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France

    Franck Sueur

  2. UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005, Paris, France

    Franck Sueur

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  1. Franck Sueur
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Communicated by P. Constantin

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Sueur, F. A Kato Type Theorem for the Inviscid Limit of the Navier-Stokes Equations with a Moving Rigid Body. Commun. Math. Phys. 316, 783–808 (2012). https://doi.org/10.1007/s00220-012-1516-x

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