Chemotaxis compressible Navier-Stokes equations with density-dependent viscosity modeling vascular network formation

TL;DR

This paper proves the existence of global weak solutions for a coupled chemotaxis and compressible Navier-Stokes system modeling early blood vessel formation, extending mathematical understanding of vascular network development.

Contribution

It establishes the existence of solutions for a complex, coupled PDE system with density-dependent viscosity, relevant to biological vascular formation, for the first time in three dimensions.

Findings

Existence of global weak solutions for $ ext{Adiabatic exponent} > 4/3$

Coupled PDE system models early blood vessel formation

Uses approximation methods and entropy inequalities for proof

Abstract

The existence of global weak solutions to the compressible Navier-Stokes equations for the density of endothelial cells and their velocity, coupled to a reaction-diffusion equation for the concentration of the chemoattractant, is established in a three-dimensional torus for energy-finite initial data. The coupling of the equations arises through the chemotaxis force, which contributes to the momentum balance equation, and the signal production due to the cells in the chemotaxis equation. The equations model the self-assembly of endothelial cells during the early stages of blood vessel formation. The existence result holds for adiabatic pressure exponents $\gamma>4/3$, matching the exponent found in the existence analysis for the degenerate Keller-Segel equations. The proof leverages an approximation via Korteweg and drag terms, the BD entropy inequality, and a construction of weak solutions that are renormalized in the velocity variable.