Multiphase cross-diffusion models for tissue structures: modeling, analysis, numerics

TL;DR

This paper derives and analyzes multiphase cross-diffusion models for tissue structures, establishing existence, uniqueness, and long-term behavior of solutions, supported by numerical simulations.

Contribution

It introduces a unified derivation of volume-filling cross-diffusion equations from physical principles and proves key mathematical properties of these models.

Findings

Proved global-in-time existence of bounded weak solutions for equal drag coefficients.

Established long-time behavior and weak-strong uniqueness using entropy methods.

Numerical simulations illustrate solution dynamics beyond entropy regimes.

Abstract

Volume-filling cross-diffusion equations for the components of a tissue structure are formally derived from mass conservation laws and force balances for the interphase pressures and viscous drag forces in a multiphase approach. The equations include Maxwell-Stefan, tumor-growth, thin-film solar cell models as well as novel volume-filling population systems. The Boltzmann and Rao entropy structures are explored. If the drag coefficients are all equal to one, the global-in-time existence of bounded weak solutions, their long-time behavior, and the weak-strong uniqueness of solutions to a regularized system are proved using entropy methods. In the general case, the resulting diffusion matrix is positively stable, ensuring local-in-time existence of solutions. Global-in-time existence of weak solutions is proved if the drag coefficients are sufficiently close to each other. This restriction is explained by the fact that the pressure forces are of degenerate type, while the drag forces are nondegenerate in the volume fractions. Numerical simulations are presented in one space dimension to illustrate the solution behavior beyond the entropy regime.