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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), 217-246.
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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