Global existence and weak-strong uniqueness for chemotaxis compressible Navier-Stokes equations modeling vascular network formation

A model of vascular network formation is analyzed in a bounded domain, consisting of the compressible Navier-Stokes equations for the density of the endothelial cells and their velocity, coupled to a reaction-diffusion equation for the concentration of the chemoattractant, which triggers the migration of the endothelial cells and the blood vessel formation. The coupling of the equations is realized by the chemotaxis force in the momentum balance equation. The global existence of finite energy weak solutions is shown for adiabatic pressure coefficients $\gamma >8\mathrm{/}5$. The solutions satisfy a relative energy inequality, which allows for the proof of the weak–strong uniqueness property.