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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. M. Dafermos, A system of hyperbolic conservation laws with frictional damping, Z. Angew. Math. Phys., 46 Special Issue (1995), 294-307.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[10]

L. Hsiao, "Quasilinear Hyperbolic Systems and Dissipative Mechanisms," World Scientific, Singapore, 1998. doi: 10.1142/9789812816917.  Google Scholar

[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.  Google Scholar

[12]

T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'Orsay, (1978), 46-53.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. L. Vazquez, "The Porous Medium Equation: Mathematical Theory," Oxford Science Publications, 2007.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. M. Dafermos, A system of hyperbolic conservation laws with frictional damping, Z. Angew. Math. Phys., 46 Special Issue (1995), 294-307.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[10]

L. Hsiao, "Quasilinear Hyperbolic Systems and Dissipative Mechanisms," World Scientific, Singapore, 1998. doi: 10.1142/9789812816917.  Google Scholar

[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.  Google Scholar

[12]

T. Nishida, Nonlinear hyperbolic equations and related topics in fluid dynamics, Publ. Math. D'Orsay, (1978), 46-53.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. L. Vazquez, "The Porous Medium Equation: Mathematical Theory," Oxford Science Publications, 2007.  Google Scholar

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