American Institute of Mathematical Sciences

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December  2011, 6(4): 625-646. doi: 10.3934/nhm.2011.6.625

Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model

Tong Li 1, and  Kun Zhao 2,

1. 

Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419
2. 

Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States

Received  April 2010 Revised  May 2011 Published  December 2011

This paper is concerned with 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 $L^\infty$ entropy weak solutions exist globally in time when the initial data are large, rough and contains vacuum states. Furthermore, based on entropy principle and the theory of divergence measure field, it is shown that any $L^\infty$ entropy weak solution converges to a constant equilibrium state exponentially fast as time goes to infinity. The physiological relevance of the theoretical results obtained in this paper is demonstrated.
Keywords: Large data, Large-time behavior., Initial-boundary value problem, Blood flow, Entropy weak solution, Hyperbolic balance laws, Global existence.
Mathematics Subject Classification: Primary: 35L50, 35L65; Secondary: 92C3.
Citation: 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
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, Computing and Visualization in Science, 4 (2002), 147-155. doi: 10.1007/s007910100066.  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.  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., 3 (2005), 559-596.  Google Scholar

[6]

S. Čanić and D. Mirković, A hyperbolic system of conservation laws in modeling endovascular treatment of abdominal aortic aneurysm, in "Hyperbolic Problems: Theory, Numerics, Applications'' (eds. H. Freistühler and G. Warnecke), Vol. I, II, International Series of Numerical Mathematics, 140, 141, Birkhäuser, Basel, (2001), 227-236.  Google Scholar

[7]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Rat. Mech. Anal., 147 (1999), 89-118. doi: 10.1007/s002050050146.  Google Scholar

[8]

G.-Q. Chen and H. Frid, Vanishing viscosity limit for initial-boundary value problems for conservation laws, in "Nonlinear Partial Differntial Equations" (Evanston, IL, 1998), Contemp. Math., 238, Amer. Math. Soc., Providence, RI, (1999), 35-51.  Google Scholar

[9]

K. N. Chueh, C. C. Conley and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., 26 (1977), 373-392. doi: 10.1512/iumj.1977.26.26029.  Google Scholar

[10]

X. X. Ding, G. Q. Chen and P. Z. Luo, Convergence of the fraction step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Comm. Math. Phys., 121 (1989), 63-84. doi: 10.1007/BF01218624.  Google Scholar

[11]

R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys., 91 (1983), 1-30. doi: 10.1007/BF01206047.  Google Scholar

[12]

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" (Yamaguchi, 2000), Lecture Notes in Computational Science and Engineering, 19, Springer, Berlin, (2002).  Google Scholar

[13]

L. Formaggia, F. Nobile, A. Quarteroni and A. Veneziani, Multiscale modeling of the circulatory system: A preliminary analysis, Computing and Visualization in Science, 2 (1999), 75-83. doi: 10.1007/s007910050030.  Google Scholar

[14]

A. Heidrich, Global weak solutions to initial-boundary-value problems for the one-dimensional quasilinear wave equation with large data, Arch. Rat. Mech. Anal., 126 (1994), 333-368. doi: 10.1007/BF00380896.  Google Scholar

[15]

L. Hsiao and K. J. Zhang, The global weak solution and relaxation limits of the initial-boundary value problem to the bipolar hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci., 10 (2000), 1333-1361. doi: 10.1142/S0218202500000653.  Google Scholar

[16]

F. M. Huang, P. Marcati and R. H. Pan, Convergence to Barenblatt solution for the compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 176 (2005), 1-24. doi: 10.1007/s00205-004-0349-y.  Google Scholar

[17]

F. M. Huang and R. H. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 166 (2003), 359-376. doi: 10.1007/s00205-002-0234-5.  Google Scholar

[18]

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

[19]

T. Li and K. Zhao, On a quasilinear hyperbolic system in blood flow modeling, Discrete & Continuous Dynamical Systems-B, 16 (2011), 333-344. Google Scholar

[20]

P.-L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math., 49 (1996), 599-638. doi: 10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5.  Google Scholar

[21]

P.-L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems, Comm. Math. Phys., 163 (1994), 415-431. doi: 10.1007/BF02102014.  Google Scholar

[22]

T.-P. Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math, 13 (1996), 25-32. doi: 10.1007/BF03167296.  Google Scholar

[23]

T.-P. Liu and T. Yang, Compressible Euler equations with vacuum, J. Differential Equations, 140 (1997), 223-237. doi: 10.1006/jdeq.1997.3281.  Google Scholar

[24]

T.-P. Liu and T. Yang, Compressible flow with vacuum and physical singularity. Cathleen Morawetz: A great mathematician, Methods Appl. Anal., 7 (2000), 495-509.  Google Scholar

[25]

P. Marcati and B. Rubino, Hyperbolic to parabolic relaxation theory for quasilinear first order systems, J. Differential Equations, 162 (2000), 359-399.  Google Scholar

[26]

F. Murat, Compacité par compensation, Ann. Scoula Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 489-507.  Google Scholar

[27]

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

[28]

R. H. Pan and K. Zhao, Initial boundary value problem for compressible Euler equations with damping, Indiana University Mathematics Journal, 57 (2008), 2257-2282. doi: 10.1512/iumj.2008.57.3366.  Google Scholar

[29]

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 (): 990.  doi: 10.1137/S0036139999355199.  Google Scholar

[30]

Y. C. Qiu and K. J. Zhang, On the relaxation limits of the hydrodynamic model for semiconductor devices, Math. Models and Methods in Appl. Sciences, 12 (2002), 333-363. doi: 10.1142/S0218202502001684.  Google Scholar

[31]

J. L. Vazquez, "The Porous Medium Equation: Mathematical Theory,'' Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 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, Computing and Visualization in Science, 4 (2002), 147-155. doi: 10.1007/s007910100066.  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.  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., 3 (2005), 559-596.  Google Scholar

[6]

S. Čanić and D. Mirković, A hyperbolic system of conservation laws in modeling endovascular treatment of abdominal aortic aneurysm, in "Hyperbolic Problems: Theory, Numerics, Applications'' (eds. H. Freistühler and G. Warnecke), Vol. I, II, International Series of Numerical Mathematics, 140, 141, Birkhäuser, Basel, (2001), 227-236.  Google Scholar

[7]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Rat. Mech. Anal., 147 (1999), 89-118. doi: 10.1007/s002050050146.  Google Scholar

[8]

G.-Q. Chen and H. Frid, Vanishing viscosity limit for initial-boundary value problems for conservation laws, in "Nonlinear Partial Differntial Equations" (Evanston, IL, 1998), Contemp. Math., 238, Amer. Math. Soc., Providence, RI, (1999), 35-51.  Google Scholar

[9]

K. N. Chueh, C. C. Conley and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., 26 (1977), 373-392. doi: 10.1512/iumj.1977.26.26029.  Google Scholar

[10]

X. X. Ding, G. Q. Chen and P. Z. Luo, Convergence of the fraction step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Comm. Math. Phys., 121 (1989), 63-84. doi: 10.1007/BF01218624.  Google Scholar

[11]

R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys., 91 (1983), 1-30. doi: 10.1007/BF01206047.  Google Scholar

[12]

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" (Yamaguchi, 2000), Lecture Notes in Computational Science and Engineering, 19, Springer, Berlin, (2002).  Google Scholar

[13]

L. Formaggia, F. Nobile, A. Quarteroni and A. Veneziani, Multiscale modeling of the circulatory system: A preliminary analysis, Computing and Visualization in Science, 2 (1999), 75-83. doi: 10.1007/s007910050030.  Google Scholar

[14]

A. Heidrich, Global weak solutions to initial-boundary-value problems for the one-dimensional quasilinear wave equation with large data, Arch. Rat. Mech. Anal., 126 (1994), 333-368. doi: 10.1007/BF00380896.  Google Scholar

[15]

L. Hsiao and K. J. Zhang, The global weak solution and relaxation limits of the initial-boundary value problem to the bipolar hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci., 10 (2000), 1333-1361. doi: 10.1142/S0218202500000653.  Google Scholar

[16]

F. M. Huang, P. Marcati and R. H. Pan, Convergence to Barenblatt solution for the compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 176 (2005), 1-24. doi: 10.1007/s00205-004-0349-y.  Google Scholar

[17]

F. M. Huang and R. H. Pan, Convergence rate for compressible Euler equations with damping and vacuum, Arch. Ration. Mech. Anal., 166 (2003), 359-376. doi: 10.1007/s00205-002-0234-5.  Google Scholar

[18]

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

[19]

T. Li and K. Zhao, On a quasilinear hyperbolic system in blood flow modeling, Discrete & Continuous Dynamical Systems-B, 16 (2011), 333-344. Google Scholar

[20]

P.-L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math., 49 (1996), 599-638. doi: 10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5.  Google Scholar

[21]

P.-L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems, Comm. Math. Phys., 163 (1994), 415-431. doi: 10.1007/BF02102014.  Google Scholar

[22]

T.-P. Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math, 13 (1996), 25-32. doi: 10.1007/BF03167296.  Google Scholar

[23]

T.-P. Liu and T. Yang, Compressible Euler equations with vacuum, J. Differential Equations, 140 (1997), 223-237. doi: 10.1006/jdeq.1997.3281.  Google Scholar

[24]

T.-P. Liu and T. Yang, Compressible flow with vacuum and physical singularity. Cathleen Morawetz: A great mathematician, Methods Appl. Anal., 7 (2000), 495-509.  Google Scholar

[25]

P. Marcati and B. Rubino, Hyperbolic to parabolic relaxation theory for quasilinear first order systems, J. Differential Equations, 162 (2000), 359-399.  Google Scholar

[26]

F. Murat, Compacité par compensation, Ann. Scoula Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), 489-507.  Google Scholar

[27]

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

[28]

R. H. Pan and K. Zhao, Initial boundary value problem for compressible Euler equations with damping, Indiana University Mathematics Journal, 57 (2008), 2257-2282. doi: 10.1512/iumj.2008.57.3366.  Google Scholar

[29]

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 (): 990.  doi: 10.1137/S0036139999355199.  Google Scholar

[30]

Y. C. Qiu and K. J. Zhang, On the relaxation limits of the hydrodynamic model for semiconductor devices, Math. Models and Methods in Appl. Sciences, 12 (2002), 333-363. doi: 10.1142/S0218202502001684.  Google Scholar

[31]

J. L. Vazquez, "The Porous Medium Equation: Mathematical Theory,'' Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.  Google Scholar

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