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

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.