Existence and uniqueness of maximal strong solution of a 1D blood flow in a network of vessels

Abstract

We study the well-posedness of a system of one-dimensional partial differential equations modeling blood flows in a network of vessels with viscoelastic walls. We prove the existence and uniqueness of maximal strong solution for this type of hyperbolic/parabolic model. We also prove a stability estimate under suitable nonlinear Robin boundary conditions.

Asymptotic stability of rarefaction wave for a blood flow model

“…In this case, the diffusive effect induced by the viscous term makes the system of hyperbolic/parabolic nature. In fact, as pointed out by the authors in [23], for the Kelvin-Voigt blood flow model, even if the hyperbolic nature of this system is dominant, because the viscous term is small compared to other terms, this additional viscous term plays an important role in numerical simulations [29], in estimation problems [6], and when data coming from numerical models are compared with in vivo data [2]. The authors in [1] also observed this phenomenon.…”

“…In this paper, we consider the following one-dimensional blood flow model in a network of vessels with viscoelastic walls (see [5,23]):…”

“…In particular, when the coefficients α 1 and α 2 in (1.2) take different values, the system (1.1) occurs different forms. For example, in the Kelvin-Voigt blood flow model, the pressure is given by (see [23,24])…”

“…Motivated by the papers [23] and [5], we will consider the model (1.1) with a more general pressure law than that in (1.3), namely, the pressure P is expressed by (1.2) where α 1 > 0 and α 2 > 0. On the other hand, the results in [5] obtained using numerical simulations are in accordance with the non-dimensional analysis which reveals that the viscous damping term is of one order of magnitude smaller than the remaining terms of the system.…”

Optimal Decay Rates of Solutions to a Blood Flow Model

“…Even so, the hyperbolic nature of the system (1.1) is still dominant in the blood flow model. Recently, Maity [14] showed that the existence and uniqueness of maximal strong solutions of the system. For other results concerning numerical simulations of this model, we refer to the interesting works [1,2,5,17,22,23].…”