Well-posedness, ill-posedness, and traveling waves for models of pulsatile flow in viscoelastic vessels

TL;DR

This paper investigates mathematical models of pulsatile blood flow in viscoelastic vessels, establishing well-posedness for some models, ill-posedness for others, and demonstrating the existence of periodic traveling wave solutions.

Contribution

It provides the first rigorous analysis of well-posedness and ill-posedness in these models, and proves existence of traveling waves in the context of blood flow modeling.

Findings

One model is well-posed for initial value problems.

A related model is ill-posed but admits solutions in analytic spaces.

Existence of periodic traveling wave solutions is established.

Abstract

We study dispersive models of fluid flow in viscoelastic vessels, derived in the study of blood flow. The unknowns in the models are the velocity of the fluid in the axial direction and the displacement of the vessel wall from rest. We prove that one such model has a well-posed initial value problem, while we argue that a related model instead has an ill-posed initial value problem; in the second case, we still prove the existence of solutions in analytic function spaces. Finally we prove the existence of some periodic traveling waves.