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.

中文翻译：

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用于肿瘤生长的 Cahn-Hilliard-Brinkman 系统

介绍了一种肿瘤生长的相场模型，该模型基于对流速度场的 Brinkman 定律。该模型将用于肿瘤演化的对流 Cahn-Hilliard 方程与用于营养物的反应-扩散-平流方程以及用于流体速度的 Brinkman-Stokes 型定律相结合。该模型是从基本热力学原理推导出来的，通过匹配渐近法推导出锐界面极限，并针对在一个相中退化导致四阶退化抛物线方程的迁移率的情况提出了存在理论。最后，数值结果描述了解决方案的定性特征并说明了某些情况下的不稳定性。