A Mass-Conserving Formulation of the Generalized Benjamin-Bona-Mahony-Burgers Equation on Star Networks

TL;DR

This paper introduces a mass-conserving formulation of the generalized Benjamin-Bona-Mahony-Burgers equation on arterial network models, ensuring mass conservation and analyzing wave behavior at network junctions with numerical simulations.

Contribution

It develops a new mass-conserving PDE formulation for gBBMB on star networks with variable coefficients, and proves well-posedness and energy-based extensions.

Findings

Mass conservation is maintained in the network formulation.

Local and global well-posedness are established under certain conditions.

Numerical simulations illustrate wave scattering at network junctions.

Abstract

The generalized Benjamin-Bona-Mahony-Burgers equation (gBBMB) describes the flow of blood through a long, viscoelastic artery. In this article we introduce a formulation of gBBMB valid on networks with semi-infinite edges joined at a single junction, with the network's edges corresponding to a segment of the arterial tree. To reflect sudden changes in the material properties of blood vessels, the coefficients of gBBMB are allowed to take different values on each edge of the network. Critically, our formulation ensures that the total mass of the solution to gBBMB is constant in time, even in the presence of dissipation. We also establish local-in-time well-posedness of this new formulation for sufficiently regular initial data. Then, we show how energy methods can be used to extend the local solution to a solution valid for all positive times, provided certain constraints are imposed on the parameters of the model PDE and the network. To build intuition for how waves scatter off the central junction of a network with two edges, we demonstrate the results of some numerical simulations.