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## Cahn–Hilliard–Brinkman systems for tumour growth

 1 Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany 2 Department of Mathematics, University of Trento, Trento, Italy

* Corresponding author: Harald Garcke

Received  March 2020 Revised  January 2021 Published  March 2021

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.

Citation: Matthias Ebenbeck, Harald Garcke, Robert Nürnberg. Cahn–Hilliard–Brinkman systems for tumour growth. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021034
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Typical situation for the formal asymptotic analysis
The tumour and healthy regions $\Omega_T$ and $\Omega_H$
Schematic sketch of the inner region close to $\Sigma(0)$
Initial tumour size for initial data $r$: A slightly perturbed sphere
Comparison of Cahn--Hilliard--Darcy and Cahn--Hilliard--Brinkman models: Tumour at time $t = 12$ for $\beta = 0.1$, left side for the CHD model, right side for the CHB model with $\eta = 10^{-5}$
Influence of different mobilities: Tumour at time $t = 9$ for $\eta = 10^{-5}$, $\beta = 0.1$ and $\alpha = \rho_S = 2$, but with different mobilities, left $m(\varphi) = \frac{1}{2}(1+\varphi)^2$, middle $m(\varphi) = \epsilon$, right $m(\varphi) = 10^{-3}\epsilon$
Influence of the adhesion parameter $\beta$: Evolution of the tumour with $m(\varphi) = \tfrac{1}{2}(1+\varphi)^2$ and $\eta = 0.1$, above for $\beta = 0.1$ at time $t = 1, 3, 6, 10$, below for $\beta = 0.01$ at time $t = 1, 1.5, 2, 2.5$
Influence of viscosiy I: Tumour and velocity for $\beta = 0.01$ at time $t = 2.5$, left for $\eta = 0.1$, right for $\eta = 100$, on top the tumour and below the velocity magnitude
Influence of viscosity II: Evolution of the tumour at time $t = 1, 3, 6, 10$ with $\beta = 0.1$, $\nu = 0$ and a no-slip boundary condition on the left boundary, on top for $\eta = 0.1$ and below for $\eta = 10$
Velocity profiles for different viscosities: The velocity magnitude at time $t = 10$ with $\beta = 0.1$, $\nu = 0$ and a no-slip boundary condition on the left boundary, left for $\eta = 0.1$, right for $\eta = 10$
Influence of viscosity contrast: Tumour at time $t = 10$ with $\beta = 0.1$, $\nu = 0$ and a no-slip b. c. on the left boundary, with $\eta_- = 0.01$, $\eta_+ = 1$; $\eta_- = 1$, $\eta_+ = 0.01$; $\eta_- = 0.01$, $\eta_+ = 10$; $\eta_- = 10$, $\eta_+ = 0.01$
Influence of initial profile I: Tumour at time $t = 0, 0.3, 1, 1.6$ with $\eta = 100$ and with the initial profile corresponding to $r_1$
Influence of initial profile II: Tumour at time $t = 0, 0.7, 1.2, 2.6$ with $\eta = 100$ and with the initial profile corresponding to $r_2$
Influence of initial profile III: Tumour at time $t = 0, 0.5, 1.2, 2.4$ with $\eta = 100$ and with the initial profile corresponding to $r_3$
Influence of initial profile IV: Tumour at time $t = 0, 0.3, 1.3, 2.3$ with $\eta = 0.01$ and with the initial profile corresponding to $r_4$
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