Critical thresholds in a quasilinear hyperbolic model of blood flow

Tong Li; Sunčica Čanić; Tong Li; Sunčica Čanić

doi:10.3934/nhm.2009.4.527

Networks and Heterogeneous Media

2009, Volume 4, Issue 3: 527-536. doi: 10.3934/nhm.2009.4.527

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Critical thresholds in a quasilinear hyperbolic model of blood flow

Tong Li ^{1
,},
Sunčica Čanić ^{2
,}

1.
Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419
2.
Department of Mathematics, University of Houston, Houston, TX 77204-3476

Received: 01 November 2008 Revised: 01 May 2009
Primary: 35L60, 35L67; Secondary: 35L45, 76L05.

Critical threshold phenomena in a one dimensional quasi-linear hyperbolic model of blood flow with viscous damping are investigated. We prove global in time regularity and finite time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the blood flow model. New results are obtained showing that the class of data that leads to global smooth solutions includes the data with negative initial Riemann invariant slopes and that the magnitude of the negative slope is not necessarily small, but it is determined by the magnitude of the viscous damping. For the data that leads to shock formation, we show that shock formation is delayed due to viscous damping.
- Critical thresholds,
- blood flow,
- finite-time singularities,
- quasi-linear hyperbolic system,
- global regularity
Citation: Tong Li, Sunčica Čanić. Critical thresholds in a quasilinear hyperbolic model of blood flow[J]. Networks and Heterogeneous Media, 2009, 4(3): 527-536. doi: 10.3934/nhm.2009.4.527

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Abstract

Critical threshold phenomena in a one dimensional quasi-linear hyperbolic model of blood flow with viscous damping are investigated. We prove global in time regularity and finite time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the blood flow model. New results are obtained showing that the class of data that leads to global smooth solutions includes the data with negative initial Riemann invariant slopes and that the magnitude of the negative slope is not necessarily small, but it is determined by the magnitude of the viscous damping. For the data that leads to shock formation, we show that shock formation is delayed due to viscous damping.
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