Existence of a Unique Solution to a Nonlinear Moving-Boundary Problem of Mixed Type Arising in Modeling Blood Flow

  • Sunčica Čanić
  • Andro Mikelić
  • Tae-Beom Kim
  • Giovanna Guidoboni
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 153)


We prove the (local) existence of a unique mild solution to a nonlinear moving-boundary problem of a mixed hyperbolic-degenerate parabolic type arising in modeling blood flow through compliant (viscoelastic) arteries.

Key words

Moving boundary problem PDE of mixed type Hyperbolicdegenerate parabolic problem Blood flow Compliant arteries 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Adams. Sobolev. Spaces. Second Edition. Academic Press, New York 2008.Google Scholar
  2. 2.
    R.L. Armentano, J.G. Barra, J. Levenson, A. Simon, and R.H. weak Pichel. Arterial wall mechanics in conscious dogs: assessment of viscous, inertial, and elastic moduli to characterize aortic wall behavior. Circ. Res. 76 (1995), pp. 468–478.Google Scholar
  3. 3.
    R.L. Armentano, J.L. Megnien, A. Simon, F. Bellenfant, J.G. Barra, and J. Levenson. Effects of hypertension on viscoelasticity of carotid and femoral arteries in humans. Hypertension 26 (1995), pp. 48–54.Google Scholar
  4. 4.
    R.D. Bauer, R. Busse, A. Shabert, Y. Summa, and E. Wetterer. Separate de-termination of the pulsatile elastic and viscous forces developed in the arterial wall in vivo. Pflugers Arch. 380 (1979), pp. 221–226.CrossRefGoogle Scholar
  5. 5.
    S. Čanić and E.-H. Kim. Mathematical analysis of the quasilinear effects in a hyperbolic model of blood flow through compliant axi-symmetric vessels, Mathematical Methods in the Applied Sciences, 26(14) (2003), pp. 1161–1186.CrossRefMathSciNetGoogle Scholar
  6. 6.
    S. Čanić, J. Tambača, G. Guidobini, A. Mikelić, C.J. Hartley, and D. Rosen- strauch. Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow SIAM J. App. Math., V 67-1 (2006), 164–193.Google Scholar
  7. 7.
    S. Čanić, A. Mikelić, D. Lamponi, and J. Tambača. Self-Consistent Effec-tive Equations Modeling Blood Flow in Medium-to-Large Compliant Arteries. SIAM J. Multiscale Analysis and Simulation 3(3) (2005), pp. 559–596.CrossRefGoogle Scholar
  8. 8.
    S. Čanić, C.J. Hartley, D. Rosenstrauch, J. Tambača, G. Guidoboni, and A. Mikelić. Blood Flow in Compliant Arteries: An Effective Viscoelastic Reduced Model, Numerics and Experimental Validation. Annals of Biomedical Engineering. 34 (2006), pp. 575–592.CrossRefGoogle Scholar
  9. 9.
    D. Coutand and S. Shkoller. On the motion of an elastic solid inside of an incompressible viscous fluid. Archive for rational mechanics and analysis, Vol. 176, pp. 25–102, 2005.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    D. Coutand and S. Shkoller. On the interaction between quasilinear elastodynamics and the Navier-Stokes equations Archive for Rational Mechanics and Analysis, Vol. 179, pp. 303–352, 2006.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    A. Chambolle, B. Desjardins, M. Esteban, and C. Grandmont. Existence of weak solutions for an unsteady fluid-plate interaction problem. J Math. Fluid Mech. 7 (2005), pp. 368–404.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    B. Desjardin, M.J. Esteban, C. Grandmont, and P. Le Tallec. Weak solutions for a fluid-elastic structure interaction model. Revista Mathem’atica complutense, Vol. XIV, num. 2, 523–538, 2001.Google Scholar
  13. 13.
    L.C. Evans, Parrial differential equations. Graduate Studies in Mathematics, (19), American Mathematical Society, RI 2002.Google Scholar
  14. 14.
    M. Guidorzi, M. Padula, and P. Plotnikov. Galerkin method for fuids in domains with elastic walls. University of Ferrara, Preprint.Google Scholar
  15. 15.
    A. Mikelić. Bijectivity of the Frechét derivative for a Biot problem in blood flow. Private communication. Dec 19, 2009.Google Scholar
  16. 16.
    F. Nobile, Numerical Approximation of Fluid-Structure Interaction Problems with Application to Haemodynamics, Ph.D. Thesis, EPFL, Lausanne, 2001.Google Scholar
  17. 17.
    G. Pontrelli. A mathematical model of flow through a viscoelastic tube. Med. Biol. Eng. Comput, 2002.Google Scholar
  18. 18.
    A. Quarteroni, M. Tuveri, and A. Veneziani. Computational vascular fluid dynamics: problems, models and methods. Survey article, Comput. Visual. Sci. 2 (2000), pp. 163–197.CrossRefMATHGoogle Scholar
  19. 19.
    T.-B. Kim. Some mathematical issues in blood flow problems. Ph.D. Thesis. University of Houston 2009.Google Scholar
  20. 20.
    T.-B. Kim, S. čanić, G. Guidoboni. Existence and uniqueness of a solution to a three-dimensional axially symmetric Biot problem arising in modeling blood flow. Communications on Pure and Applied Analysis. To appear (2009).Google Scholar
  21. 21.
    B. da Veiga. On the existence of strong solutions to a coupled fluid-structure evolution problem. Journal of Mathematical Fluid Mechanics, Vol. 6, pp. 1422–6928 (Print), pp. 1422–6952 (Online), 2004.Google Scholar
  22. 22.
    E. Zeidler. Nonlinear Functional Analysis and its Applications I. (Fixed Point Theorems) 1986, Springer-Verlag New York, Inc.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Sunčica Čanić
    • 1
  • Andro Mikelić
    • 2
    • 3
  • Tae-Beom Kim
    • 1
  • Giovanna Guidoboni
    • 1
  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA
  2. 2.Université de LyonLyonFrance
  3. 3.Institut Camille Jordan, UMR 5208, Département de MathéematiquesUniversité Lyon 1Villeurbanne CedexFrance

Personalised recommendations