Cahn-Hilliard-Brinkman systems for tumour growth

Abstract

A phase field model for tumour growth is introduced that is based on a Brinkman law for convective velocity fields. The model couples a convective Cahn-Hilliard equation for the evolution of the tumour to a reaction-diffusion-advection equation for a nutrient and to a Brinkman-Stokes type law for the fluid velocity. The model is derived from basic thermodynamical principles, sharp interface limits are derived by matched asymptotics and an existence theory is presented for the case of a mobility which degenerates in one phase leading to a degenerate parabolic equation of fourth order. Finally numerical results describe qualitative features of the solutions and illustrate instabilities in certain situations.

arXiv comment46 pages, 15 figures

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• Strong well-posedness and inverse identification problem of a non-local phase field tumor model with degenerate mobilities
• Analysis and discretization of the Ohta-Kawasaki equation with forcing and degenerate mobility
• Existence of weak solutions to multiphase Cahn-Hilliard-Darcy and Cahn-Hilliard-Brinkman models for stratified tumor growth with chemotaxis and general source terms
• Analysis and numerical simulation of a generalized compressible Cahn-Hilliard-Navier-Stokes model with friction effects