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

Critical thresholds in a quasilinear hyperbolic model of blood flow

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.
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
[1]

Wen-Rong Dai. Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3501-3524. doi: 10.3934/dcds.2012.32.3501

[2]

Manas Bhatnagar, Hailiang Liu. Sharp critical thresholds in a hyperbolic system with relaxation. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5851-5869. doi: 10.3934/dcds.2021098

[3]

Priyanjana M. N. Dharmawardane. Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity. Conference Publications, 2013, 2013 (special) : 197-206. doi: 10.3934/proc.2013.2013.197

[4]

Jaakko Kultima, Valery Serov. Reconstruction of singularities in two-dimensional quasi-linear biharmonic operator. Inverse Problems and Imaging, 2022, 16 (5) : 1047-1061. doi: 10.3934/ipi.2022011

[5]

Tong Li, Kun Zhao. On a quasilinear hyperbolic system in blood flow modeling. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 333-344. doi: 10.3934/dcdsb.2011.16.333

[6]

Tong Li, Kun Zhao. Global existence and long-time behavior of entropy weak solutions to a quasilinear hyperbolic blood flow model. Networks and Heterogeneous Media, 2011, 6 (4) : 625-646. doi: 10.3934/nhm.2011.6.625

[7]

Gang Tian. Finite-time singularity of Kähler-Ricci flow. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1137-1150. doi: 10.3934/dcds.2010.28.1137

[8]

Arno Berger. On finite-time hyperbolicity. Communications on Pure and Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963

[9]

Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure and Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243

[10]

Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463

[11]

Yongqin Liu, Shuichi Kawashima. Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1113-1139. doi: 10.3934/dcds.2011.29.1113

[12]

Tong Li, Hailiang Liu. Critical thresholds in a relaxation system with resonance of characteristic speeds. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 511-521. doi: 10.3934/dcds.2009.24.511

[13]

Zhe Xie, Jiangwei Zhang, Yongqin Xie. Asymptotic behavior of quasi-linear evolution equations on time-dependent product spaces. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022171

[14]

Young-Pil Choi, Seung-Yeal Ha, Jeongho Kim. Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication. Networks and Heterogeneous Media, 2018, 13 (3) : 379-407. doi: 10.3934/nhm.2018017

[15]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81

[16]

Kunio Hidano, Dongbing Zha. Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1735-1767. doi: 10.3934/cpaa.2019082

[17]

Xueqin Peng, Gao Jia. Existence and asymptotical behavior of positive solutions for the Schrödinger-Poisson system with double quasi-linear terms. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2325-2344. doi: 10.3934/dcdsb.2021134

[18]

Juanjuan Huang, Yan Zhou, Xuerong Shi, Zuolei Wang. A single finite-time synchronization scheme of time-delay chaotic system with external periodic disturbance. Mathematical Foundations of Computing, 2019, 2 (4) : 333-346. doi: 10.3934/mfc.2019021

[19]

Yuya Tanaka, Tomomi Yokota. Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production. Discrete and Continuous Dynamical Systems - B, 2023, 28 (1) : 262-286. doi: 10.3934/dcdsb.2022075

[20]

Sanjeeva Balasuriya. Uncertainty in finite-time Lyapunov exponent computations. Journal of Computational Dynamics, 2020, 7 (2) : 313-337. doi: 10.3934/jcd.2020013

2021 Impact Factor: 1.41

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]