Existence of 3D Strong Solutions for a System Modeling a Deformable Solid Inside a Viscous Incompressible Fluid

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Abstract

We study a coupled system modeling the movement of a deformable solid inside a viscous incompressible fluid. For the solid we consider a given deformation which has to obey several physical constraints. The motion of the fluid is modeled by the incompressible Navier–Stokes equations in a time-dependent bounded domain of \(\mathbb {R}^3\), and the solid satisfies the Newton’s laws. Our contribution consists in adapting and completing in dimension 3, some existing results, in a framework where the regularity of the deformation of the solid is limited. We rewrite the main system in domains which do not depend on time, by using a new means of defining a change of variables, and a suitable change of unknowns. We study the corresponding linearized system before setting a local-in-time existence result. Global existence is obtained for small data, and in particular for deformations of the solid which are close to the identity.

Keywords

Navier–Stokes equations Incompressible fluid Fluid–structure interactions Deformable solid Strong solutions 

Mathematics Subject Classification

35Q30 76D03 76D05 74F10 
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Notes

Acknowledgments

This work was partially supported by the ANR-project CISIFS, 09-BLAN-0213-03. The author thanks in particular Professor Jean-Pierre Raymond for his advices and helpful remarks on this work.

References

  1. 1.
    Boulakia, M.: Existence of weak solutions for the motion of an elastic structure in an incompressible viscous fluid. C. R. Math. Acad. Sci. Paris 336, 985–990 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Boulakia, M., Schwindt, E.L., Takahashi, T.: Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid. Interfaces Free Boundaries 14, 273–306 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bourguignon, J.P., Brezis, H.: Remarks on the Euler equation. J. Funct. Anal. 15, 341–363 (1974)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chambolle, A., Desjardins, B., Esteban, M.J., Grandmont, C.: Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. J. Math. Fluid Mech. 7, 368–404 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ciarlet, P.G.: Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity. Amsterdam, North-Holland (1988)MATHGoogle Scholar
  6. 6.
    Coutand, D., Shkoller, S.: Motion of an elastic solid inside an incompressible viscous fluid. Arch. Rational Mech. Anal. 176, 25–102 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Coutand, D., Shkoller, S.: The interaction between quasilinear elastodynamics and the Navier–Stokes equations. Arch. Rational Mech. Anal. 179, 303–352 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cumsille, P., Takahashi, T.: Wellposedness for the system modeling the motion of a rigid body of arbitrary form in an incompressible viscous fluid. Czechoslov. Math. J. 58(133), 961–992 (2008)CrossRefMATHGoogle Scholar
  9. 9.
    Desjardins, B., Esteban, M.J., Grandmont, C., Le Tallec, P.: Weak solutions for a fluid–elastic structure interaction model. Rev. Math. Comput. 14, 523–538 (2001)MathSciNetMATHGoogle Scholar
  10. 10.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. 1. Springer, New York (1994)MATHGoogle Scholar
  11. 11.
    Grubb, G., Solonnikov, V.A.: Boundary value problems for nonstationary Navier–Stokes equations treated by pseudo-differential methods. Math. Scand. 69, 217–290 (1991)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press Inc., Harcourt Brace Jovanovich Publishers, New York (1981)MATHGoogle Scholar
  13. 13.
    Inoue, A., Wakimoto, M.: On existence of the Navier–Stokes equation in a time dependent domain. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24, 303–319 (1977)MathSciNetMATHGoogle Scholar
  14. 14.
    Nečasová, Š., Takahashi, T., Tucsnak, M.: Weak solutions for the motion of a self-propelled deformable structure in a viscous incompressible fluid. Acta Appl. Math. 116(3), 329–352 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Raymond, J.-P.: Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions. Ann. I. H. Poincaré - AN 24, 921–951 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    San Martín, J., Scheid, J.-F., Takahashi, T., Tucsnak, M.: An initial and boundary value problem modeling of fish-like swimming. Arch. Rational Mech. Anal. 188, 429–455 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    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, 1499–1532 (2003)MathSciNetMATHGoogle Scholar
  18. 18.
    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, 53–77 (2004)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Temam, R.: Problèmes mathématiques en plasticité. Gauthier-Villars, Paris (1983)MATHGoogle Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques UMR CNRS 6620Université Blaise PascalAubière CedexFrance

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