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September  2009, 4(3): 527-536. doi: 10.3934/nhm.2009.4.527

Critical thresholds in a quasilinear hyperbolic model of blood flow

Tong Li 1, and  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  November 2008 Revised  May 2009 Published  July 2009

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.
Keywords: blood flow, quasi-linear hyperbolic system, global regularity., Critical thresholds, finite-time singularities.
Mathematics Subject Classification: Primary: 35L60, 35L67; Secondary: 35L45, 76L0.
Citation: Tong Li, Sunčica Čanić. Critical thresholds in a quasilinear hyperbolic model of blood flow. Networks and Heterogeneous Media, 2009, 4 (3) : 527-536. doi: 10.3934/nhm.2009.4.527
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