# Cahn–Hilliard–Brinkman systems for tumour growth

@article{Ebenbeck2020CahnHilliardBrinkmanSF, title={Cahn–Hilliard–Brinkman systems for tumour growth}, author={Matthias Ebenbeck and Harald Garcke and Robert Nurnberg}, journal={Discrete \& Continuous Dynamical Systems - S}, year={2020} }

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…

## 8 Citations

### Cahn-Hilliard-Brinkman models for tumour growth: Modelling, analysis and optimal control

- Mathematics
- 2020

Phase field models recently gained a lot of interest in the context of tumour growth models. In this work we study several diffuse interface models for tumour growth in a bounded domain with…

### Numerical analysis for a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport

- Mathematics, Computer ScienceJ. Num. Math.
- 2022

A fully-discrete finite element approximation of the model and stability bounds for the discrete scheme are introduced and it is shown that discrete solutions exist and depend continuously on the initial and boundary data.

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

- MathematicsCommunications in Partial Differential Equations
- 2021

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…

### Pressure jump and radial stationary solutions of the degenerate Cahn-Hilliard equation

- Mathematics
- 2022

The Cahn-Hilliard equation with degenerate mobility is used in several areas including the modeling of living tissues. We are interested in quantifying the pressure jump at the interface in the case…

### Degenerate Cahn–Hilliard and incompressible limit of a Keller–Segel model

- MathematicsCommunications in Mathematical Sciences
- 2022

The Keller-Segel model is a well-known system representing chemotaxis in living organisms. We study the convergence of a generalized nonlinear variant of the Keller-Segel to the degenerate…

### From Vlasov equation to degenerate nonlocal Cahn-Hilliard equation

- Mathematics
- 2022

We provide a rigorous mathematical framework to establish the hydrodynamic limit of the Vlasov model introduced in [31] by Noguchi and Takata in order to describe phase transition of ﬂuids by kinetic…

### Sharp-interface limits of the diffuse interface model for two-phase inductionless magnetohydrodynamic fluids

- MathematicsArXiv
- 2021

It is shown that the sharp interface limit of the models are the standard incompressible inductionless magnetohydrodynamic equations coupled with several different interface conditions for different choice of the mobilities.

### Strong well-posedness and inverse identification problem of a non-local phase field tumour model with degenerate mobilities

- MathematicsEuropean Journal of Applied Mathematics
- 2021

We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol. 22, Springer, Cham, pp.…

## References

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### Cahn-Hilliard-Brinkman models for tumour growth: Modelling, analysis and optimal control

- Mathematics
- 2020

Phase field models recently gained a lot of interest in the context of tumour growth models. In this work we study several diffuse interface models for tumour growth in a bounded domain with…

### Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis

- MathematicsJournal of Differential Equations
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### A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport

- Biology
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Using basic thermodynamic principles, a Cahn–Hilliard–Darcy model for tumour growth including nutrient diffusion, chemotaxis, active transport, adhesion, apoptosis and proliferation is derived and several sharp interface models are developed.

### On a Cahn-Hilliard-Brinkman Model for Tumor Growth and Its Singular Limits

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In the singular limit of vanishing viscosity, a Darcy-type system related to Cahn-Hilliard-Darcy type models for tumour growth is recovered and well-posedness of the model as well as existence of strong solutions will be established for a broad class of potentials.

### On a Cahn–Hilliard–Darcy System for Tumour Growth with Solution Dependent Source Terms

- Mathematics
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We study the existence of weak solutions to a mixture model for tumour growth that consists of a Cahn–Hilliard–Darcy system coupled with an elliptic reaction-diffusion equation. The Darcy law gives…

### A multiphase Cahn--Hilliard--Darcy model for tumour growth with necrosis

- Biology
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We derive a Cahn–Hilliard–Darcy model to describe multiphase tumour growth taking interactions with multiple chemical species into account as well as the simultaneous occurrence of proliferating,…

### Global weak solutions and asymptotic limits of a Cahn--Hilliard--Darcy system modelling tumour growth

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It is proved, via a Galerkin approximation, the existence of global weak solutions in two and three dimensions, along with new regularity results for the velocity field and for the pressure.

### Analysis of Cahn‐Hilliard‐Brinkman models for tumour growth

- MathematicsPAMM
- 2019

We introduce and mathematically analyse a new Cahn–Hilliard–Brinkman model for tumour growth allowing for chemotaxis. Outflow boundary conditions are considered in order not to influence tumour…

### On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects

- Mathematics
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### A Cahn‐Hilliard–type equation with application to tumor growth dynamics

- Mathematics
- 2017

We consider a Cahn‐Hilliard–type equation with degenerate mobility and single‐well potential of Lennard‐Jones type. This equation models the evolution and growth of biological cells such as solid…