# On the Hilbert Method in the Kinetic Theory of Multicellular Systems: Hyperbolic Limits and Convergence Proof

## TL;DR

This paper proves the existence of solutions for a multicellular kinetic system and rigorously derives its hyperbolic macroscopic limit, advancing the mathematical understanding of biological cell and chemoattractant dynamics.

## Contribution

It provides a rigorous proof of solution existence and derives the hyperbolic limit of a multicellular kinetic model using compactness methods.

## Key findings

- Existence of global-in-time solutions established.
- Derivation of a macroscopic Cattaneo-type system from kinetic equations.
- Hyperbolic limit obtained through rigorous compactness arguments.

## Abstract

We consider a system of two kinetic equations modelling a multicellular system : The first equation governs the dynamics of cells, whereas the second kinetic equation governs the dynamics of the chemoattractant. For this system, we first prove the existence of global-in-time solution. The proof of existence relies on a fixed point procedure after establishing some a priori estimates. Then, we investigate the hyperbolic limit after rescaling of the kinetic system. It leads to a macroscopic system of Cattaneo type. The rigorous derivation is established thanks to a compactness method.