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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 and 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.

[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. 

[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. 

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[25]

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

[26]

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

[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.

[28]

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

[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.

[30]

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

[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.

[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.

[33]

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

[34]

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

[35]

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

show all references

References:
[1]

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

[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. 

[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. 

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[25]

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

[26]

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

[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.

[28]

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

[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.

[30]

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

[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.

[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.

[33]

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

[34]

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

[35]

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

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