Existence of weak solutions to multiphase Cahn-Hilliard-Darcy and Cahn-Hilliard-Brinkman models for stratified tumor growth with chemotaxis and general source terms

We investigate a multiphase Cahn-Hilliard model for tumor growth with general source terms. The multiphase approach allows us to consider multiple cell types and multiple chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared to classical two-phase tumor growth models, the multiphase model can be used to describe a stratified tumor exhibiting several layers of tissue (e.g., proliferating, quiescent and necrotic tissue) more precisely. Our model consists of a convective Cahn-Hilliard type equation to describe the tumor evolution, a velocity equation for the associated volume-averaged velocity field, and a convective reaction-diffusion type equation to describe the density of the chemical species. The velocity equation is either represented by Darcy's law or by the Brinkman equation. We first construct a global weak solution of the multiphase Cahn-Hilliard-Brinkman model. After that, we show that such weak solutions of the system converge to a weak solution of the multiphase Cahn-Hilliard-Darcy system as the viscosities tend to zero in some suitable sense. This means that the existence of a global weak solution to the Cahn-Hilliard-Darcy system is also established.