# American Institute of Mathematical Sciences

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

## On a quasilinear hyperbolic system in blood flow modeling

 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.
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.
 [1] Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2022, 27 (7) : 3749-3778. doi: 10.3934/dcdsb.2021205 [18] Constantine M. Dafermos. Hyperbolic balance laws with relaxation. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4271-4285. doi: 10.3934/dcds.2016.36.4271 [19] Yue-Jun Peng, Yong-Fu Yang. Long-time behavior and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3683-3706. doi: 10.3934/dcds.2015.35.3683 [20] Igor Chueshov, Irena Lasiecka, Justin Webster. Flow-plate interactions: Well-posedness and long-time behavior. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 925-965. doi: 10.3934/dcdss.2014.7.925

2020 Impact Factor: 1.327