Journal of Mathematical Fluid Mechanics

, Volume 14, Issue 1, pp 1–32 | Cite as

Continuous Dependence on Initial Data in Fluid–Structure Motions

  • Giovanna Guidoboni
  • Marcello Guidorzi
  • Mariarosaria Padula


We prove a continuous dependence theorem for weak solutions of equations governing a fluid–structure interaction problem in two spatial dimensions. The proof is based on a priori estimates which, in particular, convey uniqueness of weak solutions. The estimates are obtained using Eulerian coordinates, without remapping the problem into a fixed domain.

Mathematics Subject Classification (2010)

74F10 76D03 76D05 


Uniqueness incompressible fluids fluid–structure interaction a priori estimates 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abels H.: On a diffusive interface model for two-phase flows of viscous, incompressible fluids with matched densities. Max Planck Inst. Math. Sci. 15, 307–352 (2007) (Preprint)Google Scholar
  2. 2.
    Abels H.: The intial value problem for the Navier–Stokes equationswith a free surface in L q-Sobolev spaces. Adv. Diff. Eq. 10, 45–64 (2005)MathSciNetMATHGoogle Scholar
  3. 3.
    Abels H.: On generalized solutions of two-phase flows for viscous incompressible fluis. Interfaces Free Bound 9, 31–65 (2007)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Abels H.: On the notion of generalized solutions of two-phase flows for viscous incompressible fluis. RISM Kokyuroku Bessatsu B1, 1–15 (2007)MathSciNetMATHGoogle Scholar
  5. 5.
    Adams R.: Sobolev Spaces. Academic Press, London (1975)MATHGoogle Scholar
  6. 6.
    Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge Univ. Press, (1967)Google Scholar
  7. 7.
    Beale J.T.: Large time regularity of viscous surface waves. Arch. Ratl. Mech. Anal. 84, 307–352 (1983)MathSciNetGoogle Scholar
  8. 8.
    Baiocchi C., Evans L.C., Frank L., Friedman A.: Uniqueness for two immiscible fluids in a one-dimensional porous medium. Arch. Ratl. Mech. Anal. 84, 307–352 (1983)Google Scholar
  9. 9.
    Bernstein B., Toupin R.A.: Korn’s inequalities for the sphere and circle. Arch. Ratl. Mech. Anal. 6, 51–64 (1960)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Beirao H., Veiga D.: On the existence of strong solutions to a coupled fluid–structure evolution problem. J. Math. Fluid Mech. 6(1), 21–52 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Cahn J., Hilliard J.E.: Free energy of non uniform system I. Interfacial energy. J. Chem. Phys. 28, 258–267 (1958)ADSCrossRefGoogle Scholar
  12. 12.
    Canic S., Hartley C.J., Rosenstrauch D., Tambaca J., Guidoboni G., Mikelic A.: Blood flow in compliant arteries: an effective viscoelastic reduced model, numerics and experimental validation. Ann. Biomed. Eng. 34(4), 575–592 (2006)CrossRefGoogle Scholar
  13. 13.
    Canic S., Tambaca J., Guidoboni G., Mikelic A., Hartley C.J., Rosenstrauch D.: Modeling viscoelastic behaviour of arterial walls and their interaction with pulsatile blood flow. SIAM J. Appl. Math. 67(1), 164–193 (2006)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Chambolle A., Desjardin B., Esteban M.J., Grandmont C.: Existence of weak solutions for unsteady fluid-plate interaction problem. J. Math. Fluid Mech. 7(3), 368–404 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Coutand D., Shkoller S.: Motion of an elastic solid inside an incompressible viscous fluid. Arch. Rat. Mech. Anal. 176(1), 25–102 (2005)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Coutand, D., Shkoller, S.: Unique solvability of the free boundary Navier–Stokes equations with surface tension. (Preprint) (2003)
  17. 17.
    Debussche A., Dettori L.: On Cahn–Hilliard equation with a logarithmic free energy. Nonlinear Anal. 24, 1491–1514 (1995)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Denisova, V.I.: Evolution of a closed interface between two liquids of different types. In: European Congress of Mathematics, vol. II, Barcelona, 2000, pp. 263–272. Progr. Math., vol. 202. Birkhäuser, BaselGoogle Scholar
  19. 19.
    Formaggia L., Gerbeau J.-F., Nobile F., Quarteroni A.: On the coupling of 3-D and 1-D NavierStokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Eng. 191, 561–582 (2001)MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    Friedrichs K.O.: On the boundary value problems of theory of elasticity and Korn’s inequatity. Ann Math Ser II 48, 441–471 (1947)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Kondrat’ev, V.A., Oleinik, O.A.: On the dependence of the constant in Korn’s inequality on parameters characterizing the geometry of the region. Commun. Moscow Math. Soc. Uspehi Mat. Nauk 44, 153–160 (1989) [Russian Math. Surv. 44, 187–195 (1989)]Google Scholar
  22. 22.
    Korn A.: Eigenschwingungen eins elastischen Körpers mit ruhender Oberflache. Akad. der Wissensch., Munich, Math-phys. Kl., Berichte 36, 351–401 (1906)Google Scholar
  23. 23.
    Glowinski, R.: Finite element methods for incompressible viscous flow. In: Ciarlet, P.G., Lions, J.-L. (eds.) Handbook of Numerical Analysis, vol. IX. North-Holland, Amsterdam (2003)Google Scholar
  24. 24.
    Gobert J.: Une inegalité fondamentale de la theorie de l’élasticité. Bull. Soc. Roy. Sci. Liege 31, 182–191 (1962)MathSciNetMATHGoogle Scholar
  25. 25.
    Grandmont C.: Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. SIAM J. Math. Anal. 40(2), 716–737 (2008)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Guidorzi, M., Padula, M.: Approximate solutions to the 2-D unsteady Navier–Stokes system with free surface. In: Hyperbolic Problems and Regularity questions, pp. 109–119. Trends in mathematics Birkhäuser Verlag, Basel (2007)Google Scholar
  27. 27.
    Guidorzi M., Padula M., Plotnikov P.: Hopf solutions to a fluid–elastic interaction model. Math. Models Methods Appl. Sci. 18(2), 215–269 (2008)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Hoenberg P.C., Halperin B.I.: Theory of critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977)ADSCrossRefGoogle Scholar
  29. 29.
    Jin B.J., Padula M.: On existence of non steady compressible viscous flows in a horizontal layer with free upper surface. Comm. Pure Appl. Anal. 1(3), 370–415 (2002)MathSciNetGoogle Scholar
  30. 30.
    Landau L.D., Lifshitz E.M.: Fluid Mechanics. Pergamon press, NY (1959)Google Scholar
  31. 31.
    Ladyzhenskaja O.A.: The Mathematical Theory of Viscous Incompressible Flow, vol. 2. Gordon & Breach, NY (1969)Google Scholar
  32. 32.
    Ladyzhenskaja O.A., Solonnikov V.A.: Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier–Stokes equations. Zap. Nauc. Sem. Leningrad Otd. Math. Inst. 59, 81–116 (1976)Google Scholar
  33. 33.
    Lowergrub J., Truskinosky L.: Quasi incompressible Cahn–Hiliard Fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454, 2617–2654 (1978)CrossRefGoogle Scholar
  34. 34.
    Nuori A., Poupad F.: An existence theorem for the multifluid Navier–Stokes problem. J. Differ. Equ. 122(1), 71–88 (1995)CrossRefGoogle Scholar
  35. 35.
    Padula M.: Free work and control of equilibrium configurations. Ann. Univ. Ferrara Sez. VII (N.S.) 49, 375–396 (2003)MathSciNetMATHGoogle Scholar
  36. 36.
    Padula, M.: Free work and control of equilibrium configurations. In: Rodrigues, J.F., Seregin, G., Urbano, J.M. (eds.) Trends in Partial Differential Equations on Mathematical Physics. Obidos 7–10 june, 2003. Progress in nonlinear differential equations and their applications, vol. 61. Birkäuser, Basel, pp. 213–223 (2005)Google Scholar
  37. 37.
    Padula, M.: Free Work identity and nonlinear instability in fluids with free boundaries. Recent advances in elliptic and parabolic problems. In: Chen, C.-C., Chipot, M., Lin, C.S. (eds.) Proceedings of the International Conference, Hsinchu, Taiwan University, 16–20 february 2004. World Scientific, Singapore, pp. 203–214 (2005)Google Scholar
  38. 38.
    Prodi, G.: Résultats récents et problèmes anciens dans la théorie des equations de Navier–Stokes. Colloquies internationaux du centre nationale de la richerche scientifique, n. 117, (1962)Google Scholar
  39. 39.
    Rowlinson, J.S.: Translation of J.D. van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, J. Stat. Phys. 20, 197–244 (1979)Google Scholar
  40. 40.
    Quarteroni, A., Sacco, R., Saleri, F.: Numerical mathematics. In: Texts in Applied Mathematics (TAM) vol. 37. Springer, New York (2000) (Second Edition (2007))Google Scholar
  41. 41.
    Quarteroni A., Tuveri M., Veneziani A.: Computational vascular fluid dynamics: problems, models and methods. Comput. Visual. Sci. 2, 163–197 (2000)MATHCrossRefGoogle Scholar
  42. 42.
    Shih S.M., Shen M.C.: Uniform asymptotic approximation for viscous fluid flow down an inclined plane. SIAM J. Math. Anal. 6, 560–582 (1975)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Sohr H.: The Navier–Stokes equations. An elementary functional analytic approach. Birkhäuser Verlag, Basel (2001)MATHGoogle Scholar
  44. 44.
    Solonnikov V.A.: Solvability of the problem of the motion of a viscous incompressible liquid buonded by a free surface. Math. USSR Izvestia 41, 1388–1427 (1977)MathSciNetMATHGoogle Scholar
  45. 45.
    Solonnikov V.A.: Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval. St. Petersburg Math. J. 3, 189–220 (1992)MathSciNetGoogle Scholar
  46. 46.
    Solonnikov V.A.: On the transient motion of an isolated volume of viscous incompressible fluid. Math. USSR Izvestia 31, 381–405 (1988)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Giovanna Guidoboni
    • 1
  • Marcello Guidorzi
    • 2
  • Mariarosaria Padula
    • 2
  1. 1.Department of Mathematical SciencesIndiana University and Purdue University at IndianapolisIndianapolisUSA
  2. 2.Dipartimento di MatematicaUniversità di FerraraFerraraItaly

Personalised recommendations