Existence and uniqueness of a solutionto a three-dimensional axially symmetric Biot problem arising inmodeling blood flow
Tae‐Beom Kim, Sunčica Čanić, Giovanna Guidoboni · Communications on Pure & Applied Analysis · 2010
We prove the existence of a unique weak solution to a problem associated with studying blood flowin compliant, viscoelastic arteries. The model problem is alinearization of the leading-order approximation of a viscous,incompressible, Newtonian fluid flow in a long and slenderviscoelastic tube with small aspect ratio. The resulting model is ofBiot type. The linearized model equations form ahyperbolic-parabolic system of partial differential equations withdegenerate diffusion. The degenerate diffusion is a consequence ofthe fact that the effects of the fluid viscosity in the axialdirection of a long and slender tube are small in comparison withthe effects of the fluid viscosity in the radial direction.Degenerate fluid diffusion and hyperbolicity of thehyperbolic-parabolic system cause lower regularity of a weaksolution and are a source of the main difficulties associated withthe existence proof. Crucial for the existence proof is theviscoelasticity of vessel walls which provides the main smoothingmechanisms in the energy estimates which, via the compactnessarguments, leads to the proof of the existence of a solution ofthis problem. This has interesting consequences for theunderstanding of the underlying hemodynamics application. Ouranalysis shows that the viscoelasticity of the vessel walls iscrucial in smoothing sharp wave fronts that might be generated bythe steep pressure pulses emanating from the heart, which are knownto occur in, for example, patients with aortic insufficiency.