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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 |
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.
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.
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.
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.
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.
|
[6] |
S. Čanić and D. Mirković, A hyperbolic system of conservation laws in modeling endovascular treatment of abdoninal aortic aneurysm,, in, 141 (2000), 227.
|
[7] |
C. M. Dafermos, A system of hyperbolic conservation laws with frictional damping,, Z. Angew. Math. Phys., 46 Special Issue (1995), 294.
|
[8] |
L. Formaggia, F. Nobile and A. Quarteroni, A one-dimensional model for blood flow: application to vascular prosthesis,, in, 19 (2002), 137.
|
[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. Google Scholar |
[10] |
L. Hsiao, "Quasilinear Hyperbolic Systems and Dissipative Mechanisms,", World Scientific, (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.
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.
|
[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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
|
[6] |
S. Čanić and D. Mirković, A hyperbolic system of conservation laws in modeling endovascular treatment of abdoninal aortic aneurysm,, in, 141 (2000), 227.
|
[7] |
C. M. Dafermos, A system of hyperbolic conservation laws with frictional damping,, Z. Angew. Math. Phys., 46 Special Issue (1995), 294.
|
[8] |
L. Formaggia, F. Nobile and A. Quarteroni, A one-dimensional model for blood flow: application to vascular prosthesis,, in, 19 (2002), 137.
|
[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. Google Scholar |
[10] |
L. Hsiao, "Quasilinear Hyperbolic Systems and Dissipative Mechanisms,", World Scientific, (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.
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.
|
[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.
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.
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.
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.
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.
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.
doi: 10.1081/PDE-120020497. |
[19] |
J. L. Vazquez, "The Porous Medium Equation: Mathematical Theory,", Oxford Science Publications, (2007).
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