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July  2010, 9(4): 839-865. doi: 10.3934/cpaa.2010.9.839

Existence and uniqueness of a solution to a three-dimensional axially symmetric Biot problem arising in modeling blood flow

Taebeom Kim 1, , Sunčica Čanić 1, and  Giovanna Guidoboni 1,

1. 

Department of Mathematics, University of Houston, Houston, Texas 77204-3476, United States, United States

Received  June 2009 Revised  January 2010 Published  April 2010

We prove the existence of a unique weak solution to a problem associated with studying blood flow in compliant, viscoelastic arteries. The model problem is a linearization of the leading-order approximation of a viscous, incompressible, Newtonian fluid flow in a long and slender viscoelastic tube with small aspect ratio. The resulting model is of Biot type. The linearized model equations form a hyperbolic-parabolic system of partial differential equations with degenerate diffusion. The degenerate diffusion is a consequence of the fact that the effects of the fluid viscosity in the axial direction of a long and slender tube are small in comparison with the effects of the fluid viscosity in the radial direction. Degenerate fluid diffusion and hyperbolicity of the hyperbolic-parabolic system cause lower regularity of a weak solution and are a source of the main difficulties associated with the existence proof. Crucial for the existence proof is the viscoelasticity of vessel walls which provides the main smoothing mechanisms in the energy estimates which, via the compactness arguments, leads to the proof of the existence of a solution of this problem. This has interesting consequences for the understanding of the underlying hemodynamics application. Our analysis shows that the viscoelasticity of the vessel walls is crucial in smoothing sharp wave fronts that might be generated by the steep pressure pulses emanating from the heart, which are known to occur in, for example, patients with aortic insufficiency.
Keywords: Existence of a unique solution, fluid-structure interaction., blood flow, mixed hyperbolic-parabolic problem.
Mathematics Subject Classification: Primary: 35M10, 35G25, 35K65, 35Q99; Secondary: 76D03, 76D08, 76D9.
Citation: Taebeom Kim, Sunčica Čanić, Giovanna Guidoboni. Existence and uniqueness of a solution to a three-dimensional axially symmetric Biot problem arising in modeling blood flow. Communications on Pure & Applied Analysis, 2010, 9 (4) : 839-865. doi: 10.3934/cpaa.2010.9.839
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