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

Abstract

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 this 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.

Keywords:

• Brinkman’s law
• Cahn–Hilliard equation
• Chemotaxis
• Darcy’s law
• Limit of vanishing viscosities
• Multiphase model
• Tumor growth

Mathematics Subject Classification:

• 35D30
• 35K35
• 35K86
• 35Q92
• 76D07
• 92C17
• 92C50

Acknowledgment

The authors want to thank Harald Garcke for helpful discussions. In addition, Andrea Signori wants to acknowledge the affiliation to the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica).

Additional information

Funding

Patrik Knopf was partially supported by the RTG 2339 ‘‘Interfaces, Complex Structures, and Singular Limits’’ of the German Science Foundation (DFG). Their support is gratefully acknowledged. Deutsche Forschungsgemeinschaft10.13039/501100001659.