Global existence of weak solutions to an angiogenesis model

Abstract

The aim of this paper is to prove global existence of weak solutions to the angiogenesis model proposed by Tosin et al. (Bull Math Biol 68(7):1819–1836, 2006). The model consists of compressible Navier–Stokes equations coupled with a reaction–diffusion equation describing the concentration of a chemical solution responsible of endothelial cells migration and blood vessels formation. Proofs are based on the control of the entropy term associated with the entropy solution of the hyperbolic mass conservation equation and the adaptation of the results of Feireisl (Dynamics of viscous compressible fluids, Oxford University Press, 2003), Lions (Mathematical topics in fluid mechanics, vol 2, compressible models. Oxford Lecture Series in Mathematics and its Applications, 1998) and Novotny and Straskhaba (Introduction to the theory of compressible flow, Oxford University Press, 2004), are inevitable for all models dealing with compressible Navier–Stokes equations. We use the vanishing artificial viscosity method to prove existence of a solution; the main difficulty for passing to the limit is the lack of compactness due to hyperbolic equation, which usually induces resonance phenomenon. This is overcome by using the concept of the compactness of effective viscous pressure combined with suitable renormalized solutions to the hyperbolic mass conservation equation.