Global existence and weak-strong uniqueness for chemotaxis compressible Navier-Stokes equations modeling vascular network formation. (English) Zbl 1534.35296
Summary: 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 \(\gamma > 8/5\). The solutions satisfy a relative energy inequality, which allows for the proof of the weak-strong uniqueness property.
MSC:
| 35Q30 | Navier-Stokes equations |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 76N06 | Compressible Navier-Stokes equations |
| 76Z05 | Physiological flows |
| 92C17 | Cell movement (chemotaxis, etc.) |
| 92C35 | Physiological flow |
| 35D30 | Weak solutions to PDEs |
| 35D35 | Strong solutions to PDEs |
| 35K57 | Reaction-diffusion equations |
| 35K65 | Degenerate parabolic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
Keywords:
compressible Navier-Stokes equations; chemotaxis force; global existence of solutions; weak-strong uniqueness; relative energyReferences:
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