Global Existence and Weak-Strong Uniqueness for Chemotaxis Compressible Navier-Stokes Equations Modeling Vascular...

Area of Science:

• Mathematical modeling
• Biophysics
• Computational fluid dynamics

Background:

• Vascular network formation is crucial for tissue development and repair.
• Understanding the mathematical principles governing angiogenesis is essential for therapeutic interventions.
• Existing models often simplify the complex interplay of cellular behavior and chemical signaling.

Purpose of the Study:

• To develop and analyze a mathematical model for vascular network formation.
• To investigate the role of chemotaxis in endothelial cell migration and blood vessel development.
• To establish the existence and properties of weak solutions for the proposed model.

Main Methods:

• Utilizing compressible Navier-Stokes equations to model endothelial cell density and velocity.
• Employing a reaction-diffusion equation to describe chemoattractant concentration.
• Coupling these equations through a chemotaxis force term in the momentum balance.
• Proving the global existence of finite energy weak solutions for adiabatic pressure coefficients γ > 8/5.

Main Results:

• Demonstrated the global existence of finite energy weak solutions for the coupled system.
• Established a relative energy inequality for these solutions.
• Utilized the inequality to prove the weak-strong uniqueness property of the solutions.

Conclusions:

• The mathematical model provides a framework for understanding vascular network formation.
• The existence of weak solutions under specific conditions supports the model's validity.
• The weak-strong uniqueness property enhances the reliability of simulation results.