American Institute of Mathematical Sciences

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July  2011, 16(1): 333-344. doi: 10.3934/dcdsb.2011.16.333

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, United States
2. 

Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210, United States

Received  April 2010 Revised  September 2010 Published  April 2011

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: initial-boundary value problem, global existence, long-time behavior., Hyperbolic balance laws, blood flow, classical solution.
Mathematics Subject Classification: Primary: 35L50, 35L65; Secondary: 92C3.
Citation: Tong Li, Kun Zhao. On a quasilinear hyperbolic system in blood flow modeling. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 333-344. doi: 10.3934/dcdsb.2011.16.333
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.

show all references

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.

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