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

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  • 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.
    Mathematics Subject Classification: Primary: 35Q53, 37K65, 35B30; Secondary: 35B35, 35Q80, 35B65.


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  • [1]

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


    M. Aniker, R. L. Rockwell and E. OgdenNonlinear analysis of flow pulses and shock waves in arteries. Part I: derivation and properties of mathematical model, Zeitschrift für Angewandte Mathematik und Physik., 22 (1971a), 217-246.


    M. Aniker, R. L. Rockwell and E. OgdenNonlinear analysis of flow pulses and shock waves in arteries. Part II: parametric study related to clinical problems, Zeitschrift für Angewandte Mathematik und Physik., 22 (1971b), 563-581.


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


    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.


    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.


    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.


    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.


    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.


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


    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.


    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.


    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.


    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.


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


    A. Constantin and W. StraussStability of a class of solitary waves in compressible elastic rods, Phys. Lett. A, 270 (200), 140-148.


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


    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.


    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.


    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.


    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.


    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.


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


    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.


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


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


    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.


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


    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.


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


    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.


    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.


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


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


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

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