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Archive for Rational Mechanics and Analysis

, Volume 218, Issue 2, pp 907–944 | Cite as

Uniqueness Results for Weak Solutions of Two-Dimensional Fluid–Solid Systems

  • Olivier Glass
  • Franck Sueur
Article

Abstract

In this paper, we consider two systems modelling the evolution of a rigid body in an incompressible fluid in a bounded domain of the plane. The first system corresponds to an inviscid fluid driven by the Euler equation whereas the other one corresponds to a viscous fluid driven by the Navier–Stokes system. In both cases we investigate the uniqueness of weak solutions, à la Yudovich for the Euler case, à la Leray for the Navier–Stokes case, as long as no collision occurs.

Keywords

Weak Solution Rigid Body Euler Equation Strong Solution Uniqueness Result 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    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. Commun. Partial Differ. Equ. 25(5–6), 1019–1042 (2000)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Dashti M., Robinson J.C.: The motion of a fluid–rigid disc system at the zero limit of the rigid disc radius. Arch. Ration. Mech. Anal. 200(1), 285–312 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Desjardins B., Esteban M.: On weak solutions for fluid–rigid structure interaction: compressible and incompressible models. Commun. Partial Differ. Equ. 25(7–8), 1399–1413 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Desjardins B., Esteban M.J.: Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146(1), 59–71 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    DiPerna R.J., Lions P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)MathSciNetADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Feireisl E.: On the motion of rigid bodies in a viscous incompressible fluid. Dedicated to Philippe Bénilan. J. Evol. Equ. 3(3), 419–441 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Feireisl, E.: On the motion of rigid bodies in a viscous fluid. Mathematical theory in fluid mechanics (Paseky, 2001). Appl. Math. 47(6), 463–484 (2002)Google Scholar
  8. 8.
    Feireisl E.: On the motion of rigid bodies in a viscous compressible fluid. Arch. Ration. Mech. Anal. 167(4), 281–308 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Galdi, G.P.: On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications. Handbook of Mathematical Fluid Dynamics, Vol. I. North-Holland, Amsterdam, 653–791, 2002Google Scholar
  10. 10.
    Geissert M., Götze K., Hieber M.: Lp-theory for strong solutions to fluid–rigid body interaction in Newtonian and generalized Newtonian fluids. Trans. Am. Math. Soc. 365(3), 1393–1439 (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    Gérard-Varet D., Hillairet M.: Regularity issues in the problem of fluid–structure interaction. Arch. Ration. Mech. Anal. 195(2), 375–407 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gilbarg, D., Trudinger, N.S. Elliptic Partial Differential Equations of Second Order, 2nd edn. Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer, Berlin, 1983Google Scholar
  13. 13.
    Glass O., Lacave C., Sueur F.: On the motion of a small body immersed in a two dimensional incompressible perfect fluid. Bull. Soc. Math. Fr. 142(3), 489–536 (2014)MathSciNetGoogle Scholar
  14. 14.
    Glass O., Sueur F.: The movement of a solid in an incompressible perfect fluid as a geodesic flow. Proc. Am. Math. Soc. 140, 2155–2168 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Glass O., Sueur F., Takahashi T.: Smoothness of the motion of a rigid body immersed in an incompressible perfect fluid. Ann. Sci. E. N. S. 45(1), 1–51 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Glass O., Sueur F.: On the motion of a rigid body in a two-dimensional irregular ideal flow, SIAM J. Math. Anal. 44(5), 3101–3126 (2012)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Grandmont C., Maday Y.: Existence for an unsteady fluid–structure interaction problem. M2AN Math. Model. Numer. Anal. 34(3), 609–636 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gunzburger M., Lee H.-C., Seregin G.A.: Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions. J. Math. Fluid Mech. 2(3), 219–266 (2000)MathSciNetADSCrossRefzbMATHGoogle Scholar
  19. 19.
    Hesla, T.I.: Collisions of Smooth Bodies in Viscous Fluids: A Mathematical Investigation. PhD thesis, University of Minnesota, revised version (2005)Google Scholar
  20. 20.
    Hillairet M.: Lack of collision between solid bodies in a 2D incompressible viscous flow. Commun. Partial Differ. Equ. 32(7–9), 1345–1371 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hoffmann K.-H., Starovoitov V.N.: On a motion of a solid body in a viscous fluid. Two-dimensional case. Adv. Math. Sci. Appl. 9(2), 633–648 (1999)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Inoue A., Wakimoto M.: On existence of solutions of the Navier–Stokes equation in a time dependent domain. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24(2), 303–319 (1977)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Judakov, N.V.: The solvability of the problem of the motion of a rigid body in a viscous incompressible fluid (in Russian). Dinamika Splošn. Sredy 18, 249–253 (1974)Google Scholar
  24. 24.
    Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1933)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Leray J.: Étude de diverses équations intégrales non linéaires et de quelques problèmes de l’hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)MathSciNetGoogle Scholar
  26. 26.
    Lions, P.-L.: Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, Vol. 3, 1996Google Scholar
  27. 27.
    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. Ration. Mech. Anal. 161(2), 113–147 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Serre D.: Chute libre d’un solide dans un fluide visqueux incompressible. Existence. Jpn. J. Appl. Math. 4(1), 99–110 (1987)CrossRefzbMATHGoogle Scholar
  29. 29.
    Starovoitov V.N.: Nonuniqueness of a solution to the problem on motion of a rigid body in a viscous incompressible fluid. J. Math. Sci. 130(4), 4893–4898 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Takahashi T.: Analysis of strong solutions for the equations modeling the motion of a rigid–fluid system in a bounded domain. Adv. Differ. Equ. 8(12), 1499–1532 (2003)zbMATHGoogle Scholar
  31. 31.
    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)MathSciNetADSCrossRefzbMATHGoogle Scholar
  32. 32.
    Yudovich, V.I.: Non-stationary flows of an ideal incompressible fluid. Z̆. Vyčisl. Mat. i Mat. Fiz. 3, 1032–1066 (1963) (in Russian). [English translation in USSR Comput. Math. Math. Phys. 3, 1407–1456 (1963)]Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.CEREMADE, UMR CNRS 7534, Université Paris-DauphineParis Cedex 16France
  2. 2.Institut de Mathématiques de Bordeaux, UMR CNRS 5251Université de BordeauxTalence CedexFrance

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