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July  2012, 11(4): 1563-1576. doi: 10.3934/cpaa.2012.11.1563

The 2-component dispersionless Burgers equation arising in the modelling of blood flow

Tony Lyons 1,

1. 

School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

Received  May 2011 Revised  June 2011 Published  January 2012

This article investigates the properties of the solutions of the dispersionless two-component Burgers (B2) equation, derived as a model for blood-flow in arteries with elastic walls. The phenomenon of wave breaking is investigated as well as applications of the model to clinical conditions.
Keywords: breaking waves, dispersionless limit., Blood flow, nonlinear waves.
Mathematics Subject Classification: Primary: 35Q53, 37K65, 35B30; Secondary: 35B35, 35Q80, 35B6.
Citation: Tony Lyons. The 2-component dispersionless Burgers equation arising in the modelling of blood flow. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1563-1576. doi: 10.3934/cpaa.2012.11.1563
References:
[1]

D. J. Acheson, "Elementary Fluid Dynamics," Oxford University Press, Oxford, 1990. Google Scholar

[2]

M. Aniker, R. L. Rockwell and E. Ogden, Nonlinear analysis of flow pulses and shock waves in arteries. Part I: derivation and properties of mathematical model,, Zeitschrift f\, 22 (): 217.   Google Scholar

[3]

M. Aniker, R. L. Rockwell and E. Ogden, Nonlinear analysis of flow pulses and shock waves in arteries. Part II: parametric study related to clinical problems,, Zeitschrift f\, 22 (): 563.   Google Scholar

[4]

F. Bauer and J. A. Nohel, "The Qualitative Theory of Ordinary Differential Equations: An Introduction," W.A Benjamin, New York, 1969. Google Scholar

[5]

R. Beals, D. Sattinger and J. Szmigielski, Accoustic scattering and the extended KdV hierachy, Adv. Math., 140 (1998), 190-206. doi: doi: 10.1005/aima.1998.1768.  Google Scholar

[6]

A Boutet de Monvel, A. Kostenko, D. Shepelsky and G. Teschi, Long-time asymptotics for teh Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588. doi: doi: 10.1137/090748500.  Google Scholar

[7]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[8]

M. Chen, S.-Q. Liu and Y. Zhang, A two-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15; nlin.SI/0501028. Google Scholar

[9]

R. M. Chen and Y. Liu, Wave breaking and global existence for a generalized two-component Camassa-Holm system, International Mathematics Research Notices, Article ID rnq118 (2010), 36 pages. doi: doi10.1093/imrn/rnq118.  Google Scholar

[10]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shalow water equation, Acta. Math., 181 (1998), 229-243. doi: doi: 10.1007/BF02392586.  Google Scholar

[11]

A. Constantin, V. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa -Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: doi: 10.1088/0266-5611/22/6/017.  Google Scholar

[12]

A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132. doi: doi: 10.1016/j.physleta.2008.10.050.  Google Scholar

[13]

A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam Res., 40 (2008), 175-211. Google Scholar

[14]

A. Constantin and D. Lannes, The hydro-dynamical relevance of teh Camassa-Holm and Degasperis-Proceisi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. Google Scholar

[15]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Commun. Pure Appl. Math., 52 (1999), 949-982. Google Scholar

[16]

A. Constantin and W. Strauss, Stability of a class of solitary waves in compressible elastic rods,, Phys. Lett. A, 270 (): 140.   Google Scholar

[17]

H. H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta. Math., 127 (1998), 193-207. Google Scholar

[18]

J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513. Google Scholar

[19]

J. Escher and Z. Yin, Well-posedness, blow-up phenomena and global solutions for the b-equation, J. Reine Angew. Math., 624 (2008), 51-80. doi: doi: 10.1515/CRELLE.2008.080.  Google Scholar

[20]

G. Falqui, On a Camassa-Holm type equation with two dependent variables, J. Phys. A, 39 (2006), 327-342. doi: doi: 10.1088/0305-4470/39/2/004.  Google Scholar

[21]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278. doi: doi: 10.1016/j.jfa.2010.02.008.  Google Scholar

[22]

D. Henry, Infinite propagation speed for a two component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009) 597-606. doi: doi: 10.3934/dcdsb.2009.12.597.  Google Scholar

[23]

D. D. Holm and C. Tronci, Geodesic Vlasov equations and their integrable moment closures, J. Geom. Mech., 1 (2009), 181-208. Google Scholar

[24]

D. D. Holm and R. I. Ivanov, Two-component CH system: inverse scattering, peakons and geometry, Inverse Problems, 27 (2011), 045013. doi: doi: 10.1088/0266-5611/27/4/045013.  Google Scholar

[25]

R. I. Ivanov, Extended Camassa-Holm hierarchy and conserved quantities, Z. Naturforsch., 61a (2006), 133-138; nlin.SI/0601066. Google Scholar

[26]

R. I. Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46 (2009), 389-396. Google Scholar

[27]

R. S. Johnson, Camassa-Holm, Kortweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: doi: 10.1017/S0022112001007224.  Google Scholar

[28]

J. Keener and J. Sneyd, "Mathematical Physiology 1: Cellular Physiology," Springer, 2009. Google Scholar

[29]

Y. Kodama and B. Konopelchenko, Singular sector of the Burgers-Hopf hierarchy and deformations of hyperelliptic curves, J. Phys. A: Math. Gen., 35 (2002), L489-L500. Google Scholar

[30]

J. Lighthill, "Mathematical Biofluiddynamics," SIAM, Philadelphia, PA. 1975. Google Scholar

[31]

S.-Q. Liu and Y. Zhang, Deformations of semisimple bi-Hamiltonian structures of hydrodynamic type, J. Geom. Phys., 54 (2005), 427-53. doi: 10.1088/0305-4470/35/31/104.  Google Scholar

[32]

P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900.  Google Scholar

[33]

T. J. Pedley, Blood flow in arteries and veins, in "Perspectives in Fluid Dynamics," Cambridge University Press, 2003. Google Scholar

[34]

W. A. Strauss, "Partial Differential Equations: An Introduction," John Wiley & Sons Inc., 1990. Google Scholar

[35]

G. B Whitham, "Linear and Nonlinear Waves," John Wiley & Sons, New York, 1980. Google Scholar

show all references

References:
[1]

D. J. Acheson, "Elementary Fluid Dynamics," Oxford University Press, Oxford, 1990. Google Scholar

[2]

M. Aniker, R. L. Rockwell and E. Ogden, Nonlinear analysis of flow pulses and shock waves in arteries. Part I: derivation and properties of mathematical model,, Zeitschrift f\, 22 (): 217.   Google Scholar

[3]

M. Aniker, R. L. Rockwell and E. Ogden, Nonlinear analysis of flow pulses and shock waves in arteries. Part II: parametric study related to clinical problems,, Zeitschrift f\, 22 (): 563.   Google Scholar

[4]

F. Bauer and J. A. Nohel, "The Qualitative Theory of Ordinary Differential Equations: An Introduction," W.A Benjamin, New York, 1969. Google Scholar

[5]

R. Beals, D. Sattinger and J. Szmigielski, Accoustic scattering and the extended KdV hierachy, Adv. Math., 140 (1998), 190-206. doi: doi: 10.1005/aima.1998.1768.  Google Scholar

[6]

A Boutet de Monvel, A. Kostenko, D. Shepelsky and G. Teschi, Long-time asymptotics for teh Camassa-Holm equation, SIAM J. Math. Anal., 41 (2009), 1559-1588. doi: doi: 10.1137/090748500.  Google Scholar

[7]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[8]

M. Chen, S.-Q. Liu and Y. Zhang, A two-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys., 75 (2006), 1-15; nlin.SI/0501028. Google Scholar

[9]

R. M. Chen and Y. Liu, Wave breaking and global existence for a generalized two-component Camassa-Holm system, International Mathematics Research Notices, Article ID rnq118 (2010), 36 pages. doi: doi10.1093/imrn/rnq118.  Google Scholar

[10]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shalow water equation, Acta. Math., 181 (1998), 229-243. doi: doi: 10.1007/BF02392586.  Google Scholar

[11]

A. Constantin, V. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa -Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: doi: 10.1088/0266-5611/22/6/017.  Google Scholar

[12]

A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, 372 (2008), 7129-7132. doi: doi: 10.1016/j.physleta.2008.10.050.  Google Scholar

[13]

A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam Res., 40 (2008), 175-211. Google Scholar

[14]

A. Constantin and D. Lannes, The hydro-dynamical relevance of teh Camassa-Holm and Degasperis-Proceisi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. Google Scholar

[15]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Commun. Pure Appl. Math., 52 (1999), 949-982. Google Scholar

[16]

A. Constantin and W. Strauss, Stability of a class of solitary waves in compressible elastic rods,, Phys. Lett. A, 270 (): 140.   Google Scholar

[17]

H. H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta. Math., 127 (1998), 193-207. Google Scholar

[18]

J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst., 19 (2007), 493-513. Google Scholar

[19]

J. Escher and Z. Yin, Well-posedness, blow-up phenomena and global solutions for the b-equation, J. Reine Angew. Math., 624 (2008), 51-80. doi: doi: 10.1515/CRELLE.2008.080.  Google Scholar

[20]

G. Falqui, On a Camassa-Holm type equation with two dependent variables, J. Phys. A, 39 (2006), 327-342. doi: doi: 10.1088/0305-4470/39/2/004.  Google Scholar

[21]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal., 258 (2010), 4251-4278. doi: doi: 10.1016/j.jfa.2010.02.008.  Google Scholar

[22]

D. Henry, Infinite propagation speed for a two component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009) 597-606. doi: doi: 10.3934/dcdsb.2009.12.597.  Google Scholar

[23]

D. D. Holm and C. Tronci, Geodesic Vlasov equations and their integrable moment closures, J. Geom. Mech., 1 (2009), 181-208. Google Scholar

[24]

D. D. Holm and R. I. Ivanov, Two-component CH system: inverse scattering, peakons and geometry, Inverse Problems, 27 (2011), 045013. doi: doi: 10.1088/0266-5611/27/4/045013.  Google Scholar

[25]

R. I. Ivanov, Extended Camassa-Holm hierarchy and conserved quantities, Z. Naturforsch., 61a (2006), 133-138; nlin.SI/0601066. Google Scholar

[26]

R. I. Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion, 46 (2009), 389-396. Google Scholar

[27]

R. S. Johnson, Camassa-Holm, Kortweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: doi: 10.1017/S0022112001007224.  Google Scholar

[28]

J. Keener and J. Sneyd, "Mathematical Physiology 1: Cellular Physiology," Springer, 2009. Google Scholar

[29]

Y. Kodama and B. Konopelchenko, Singular sector of the Burgers-Hopf hierarchy and deformations of hyperelliptic curves, J. Phys. A: Math. Gen., 35 (2002), L489-L500. Google Scholar

[30]

J. Lighthill, "Mathematical Biofluiddynamics," SIAM, Philadelphia, PA. 1975. Google Scholar

[31]

S.-Q. Liu and Y. Zhang, Deformations of semisimple bi-Hamiltonian structures of hydrodynamic type, J. Geom. Phys., 54 (2005), 427-53. doi: 10.1088/0305-4470/35/31/104.  Google Scholar

[32]

P. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900-1906. doi: 10.1103/PhysRevE.53.1900.  Google Scholar

[33]

T. J. Pedley, Blood flow in arteries and veins, in "Perspectives in Fluid Dynamics," Cambridge University Press, 2003. Google Scholar

[34]

W. A. Strauss, "Partial Differential Equations: An Introduction," John Wiley & Sons Inc., 1990. Google Scholar

[35]

G. B Whitham, "Linear and Nonlinear Waves," John Wiley & Sons, New York, 1980. Google Scholar

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