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Abstract
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.
Mathematics Subject Classification: Primary: 35M10, 35G25, 35K65, 35Q99; Secondary: 76D03, 76D08, 76D99.
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