Abstract
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.
AMS(MOS) subject classifications. 35M10, 35G25, 35K65, 35Q99, 76D03, 76D08, 76D99.
Research supported by NSF under grant DMS-0806941, by NSF/NIH grant number DMS-0443826, and by the Texas Higher Education Board under ARP grant 003652- 0051-2006.
Research supported by NSF/NIH under grant DMS-0443826.
Graduate Student Support by the NSF/NIGMS under grant 0443826, and by the Texas Higher Education Board under ARP grant 003652-0051-2006.
Research supported by the NSF under grant DMS-0811138, and by the Texas Higher Education Board under ARP grant 003652-0051-2006.
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References
R. Adams. Sobolev. Spaces. Second Edition. Academic Press, New York 2008.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
L.C. Evans, Parrial differential equations. Graduate Studies in Mathematics, (19), American Mathematical Society, RI 2002.
M. Guidorzi, M. Padula, and P. Plotnikov. Galerkin method for fuids in domains with elastic walls. University of Ferrara, Preprint.
A. Mikelić. Bijectivity of the Frechét derivative for a Biot problem in blood flow. Private communication. Dec 19, 2009.
F. Nobile, Numerical Approximation of Fluid-Structure Interaction Problems with Application to Haemodynamics, Ph.D. Thesis, EPFL, Lausanne, 2001.
G. Pontrelli. A mathematical model of flow through a viscoelastic tube. Med. Biol. Eng. Comput, 2002.
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.
T.-B. Kim. Some mathematical issues in blood flow problems. Ph.D. Thesis. University of Houston 2009.
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).
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.
E. Zeidler. Nonlinear Functional Analysis and its Applications I. (Fixed Point Theorems) 1986, Springer-Verlag New York, Inc.
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Čanić, S., Mikelić, A., Kim, TB., Guidoboni, G. (2011). Existence of a Unique Solution to a Nonlinear Moving-Boundary Problem of Mixed Type Arising in Modeling Blood Flow. In: Bressan, A., Chen, GQ., Lewicka, M., Wang, D. (eds) Nonlinear Conservation Laws and Applications. The IMA Volumes in Mathematics and its Applications, vol 153. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9554-4_11
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DOI: https://doi.org/10.1007/978-1-4419-9554-4_11
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