On a quasilinear hyperbolic system in blood flow modeling

Tong Li; Kun Zhao

doi:10.3934/dcdsb.2011.16.333

Article Contents

2011, Volume 16, Issue 1: 333-344. Doi: 10.3934/dcdsb.2011.16.333

This issue Previous Article Vanishing singularity in hard impacting systems Next Article Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system

On a quasilinear hyperbolic system in blood flow modeling

Tong Li^1, and
Kun Zhao^2,

1.
Department of Mathematics, University of Iowa, Iowa City, IA 52242
2.
Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210

Received: March 31, 2010

Revised: August 31, 2010

Published: June 2011

Abstract Related Papers Cited by

Abstract

This paper aims at an initial-boundary value problem on bounded domains for a one-dimensional quasilinear hyperbolic model of blood flow with viscous damping. It is shown that, for given smooth initial data close to a constant equilibrium state, there exists a unique global smooth solution to the model. Time asymptotically, it is shown that the solution converges to the constant equilibrium state exponentially fast as time goes to infinity due to viscous damping and boundary effects.

Keywords:

Mathematics Subject Classification: Primary: 35L50, 35L65; Secondary: 92C35.

Citation:

Related Papers

Cited by

References

[1]	M. Anliker, R. L. Rockwell and E. Ogden, Nonlinear analysis of flow pulses and shock waves in arteries, Part I: Derivation and properties of mathematical model, Z. Angew. Math. Phys., 22 (1971), 217-246.doi: 10.1007/BF01591407.
[2]	A. C. L. Barnard, W. A. Hunt, W. P. Timlake and E. Varley, A theory of fluid flow in compliant tubes, Biophysical J., 6 (1966), 717-724.doi: 10.1016/S0006-3495(66)86690-0.
[3]	S. Čanić, Blood flow through compliant vessels after endovascular repair: Wall deformations induced by the discontinuous wall properties, Comput. Visual. Sci., 4 (2002), 147-155.doi: 10.1126/science.4.83.147.
[4]	S. Čanić and E. H. Kim, Mathematical analysis of the quasilinear effects in a hyperbolic model blood flow through compliant axi-symmetric vessels, Math. Methods Appl. Sci., 26 (2003), 1161-1186.doi: 10.1002/mma.407.
[5]	S. Čanić, D. Lamponi, A. Mikelić and J. Tambača, Self-consistent effective equations modeling blood flow in medium-to-large compliant arteries, Multiscale Model. Simul., 5 (2005), 559-596.
[6]	S. Čanić and D. Mirković, A hyperbolic system of conservation laws in modeling endovascular treatment of abdoninal aortic aneurysm, in "Hyperbolic Problems: Theory, Numerics, Applications: Eighth International Conference In Magdeburg, 2000," 227-236, International series of numerical mathematics (eds. H. Freist黨ler and G. Warnecke), 141, Birkh鋟ser, Basel, 2001.
[7]	C. M. Dafermos, A system of hyperbolic conservation laws with frictional damping, Z. Angew. Math. Phys., 46 Special Issue (1995), 294-307.
[8]	L. Formaggia, F. Nobile and A. Quarteroni, A one-dimensional model for blood flow: application to vascular prosthesis, in "Mathematical Modelling and Numerical Simulation in Continuum Mechanics," 137-153, Lecture Notes in Computational Science and Engineering, 19 (eds. I. Babuska, P.G. Ciarbet and T. Miyoshi), Springer-Verlag, Berlin, 2002.
[9]	L. Formaggia, F. Nobile, A. Quarteroni and A. Veneziani, Multiscale modeling of the circulatory system: A preliminary analysis, Comput. Visual. Sci., 2 (1999), 75-83.
[10]	L. Hsiao, "Quasilinear Hyperbolic Systems and Dissipative Mechanisms," World Scientific, Singapore, 1998.doi: 10.1142/9789812816917.
[11]	T. Li and S. Čanić, Critical thresholds in a quasilinear hyperbolic model of blood flow, Netw. Heterog. Media, 4 (2009), 527-536.doi: 10.3934/nhm.2009.4.527.
[12]	T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'Orsay, (1978), 46-53.
[13]	K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations, 131 (1996), 171-188.doi: 10.1006/jdeq.1996.0159.
[14]	K. Nishihara and T. Yang, Boundary effect on asymptotic behavior of solutions to the $p$-system with damping, J. Differentail Equations, 156 (1999), 439-458.doi: 10.1006/jdeq.1998.3598.
[15]	M. Olufsen, C. Peskin, W. Kim, E. Pedersen, A. Nadim and J. Larsen, Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions, Annals of Biomedical Engineering, 28 (2000), 1281-1299.doi: 10.1114/1.1326031.
[16]	R. H. Pan and K. Zhao, 3D compressible Euler equations with damping in bounded domains, J. Differential Equations, 246 (2009), 581-596.doi: 10.1016/j.jde.2008.06.007.
[17]	A. J. Pullan, N. P. Smith and P. J. Hunter, An anatomically based model of transient coronary blood flow in the heart, SIAM J. Appl. Math., 62 (2002), 990-1018.doi: 10.1137/S0036139999355199.
[18]	T. C. Sideris, B. Thomases and D. H. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial Differential Equations, 28 (2003), 795-816.doi: 10.1081/PDE-120020497.
[19]	J. L. Vazquez, "The Porous Medium Equation: Mathematical Theory," Oxford Science Publications, 2007.