American Institute of Mathematical Sciences

Advanced
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

[1]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks & Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625
[2]

Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59
[3]

Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure & Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759
[4]

Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917
[5]

Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015
[6]

Nataliia V. Gorban, Olha V. Khomenko, Liliia S. Paliichuk, Alla M. Tkachuk. Long-time behavior of state functions for climate energy balance model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1887-1897. doi: 10.3934/dcdsb.2017112
[7]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381
[8]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
[9]

Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212
[10]

Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917
[11]

Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319
[12]

Elena Rossi. Well-posedness of general 1D initial boundary value problems for scalar balance laws. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3577-3608. doi: 10.3934/dcds.2019147
[13]

Martn P. Árciga Alejandre, Elena I. Kaikina. Mixed initial-boundary value problem for Ott-Sudan-Ostrovskiy equation. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 381-409. doi: 10.3934/dcds.2012.32.381
[14]

Türker Özsarı, Nermin Yolcu. The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3285-3316. doi: 10.3934/cpaa.2019148
[15]

Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135
[16]

Haifeng Hu, Kaijun Zhang. Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1601-1626. doi: 10.3934/dcdsb.2014.19.1601
[17]

Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021205
[18]

Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683
[19]

Constantine M. Dafermos. Hyperbolic balance laws with relaxation. Discrete & Continuous Dynamical Systems, 2016, 36 (8) : 4271-4285. doi: 10.3934/dcds.2016.36.4271
[20]

Igor Chueshov, Irena Lasiecka, Justin Webster. Flow-plate interactions: Well-posedness and long-time behavior. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 925-965. doi: 10.3934/dcdss.2014.7.925

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (87)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy