Multicomponent tumor models using Maxwell-Stefan and Cahn-Hilliard equations

Abstract

Cross-diffusion systems for multicomponent fluids with phase separation are analyzed. The equations consist of the Maxwell-Stefan equations for the mass fractions together with Cahn-Hilliard-type terms, solved in a bounded domain with no-flux boundary conditions. Such equations arise as simplifed models for tumor growth. The mass fractions describe healthy and tumor tissues and the interstitial fluid. Compared to the literature, we allow for degenerate diffusion matrices. The key of the analysis is the thermodynamical structure, with a free energy that consists of the Boltzmann energies of the mass fractions, a nonconvex potential energy, and a gradient term penalizing phase changes. The global existence of bounded weak solutions and the weak-strong uniqueness property are proved by combining estimates for the free energy and entropy.