Abstract We study a phase field model proposed recently in the context of tumour growth. The model couples a Cahn–Hilliard–Brinkman (CHB) system with an elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenoidal, as its divergence is a function of the nutrient and the phase field variable, i.e., solution-dependent, and frictionless boundary conditions are prescribed for the velocity to avoid imposing unrealistic constraints on the divergence relation. In this paper we give a first result on the existence of weak and stationary solutions to the CHB model for tumour growth with singular potentials, specifically the double obstacle potential and the logarithmic potential, which ensures that the phase field variable stays in the physically relevant interval. New difficulties arise from the interplay between the singular potentials and the solution-dependent source terms, but can be overcome with several key estimates for the approximations of the singular potentials, which maybe of independent interest. As a consequence, included in our analysis is an existence result for a Darcy variant, and our work serves to generalise recent results on weak and stationary solutions to the Cahn–Hilliard inpainting model with singular potentials.

中文翻译：

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具有奇异势和源项的 Cahn-Hilliard-Brinkman 模型的弱平稳解

摘要 我们研究了最近在肿瘤生长背景下提出的相场模型。该模型将 Cahn-Hilliard-Brinkman (CHB) 系统与营养物质的椭圆反应扩散方程相结合。由布林克曼定律控制的流体速度不是螺线管的，因为它的散度是养分和相场变量的函数，即为速度规定了与溶液相关的无摩擦边界条件，以避免对流体施加不切实际的约束散度关系。在本文中，我们首先给出了关于具有奇异电位的肿瘤生长的 CHB 模型的弱解和平稳解的存在性的第一个结果，特别是双障碍电位和对数电位，这确保了相场变量保持在物理相关的区间. 奇异势和依赖解的源项之间的相互作用产生了新的困难，但可以通过对奇异势的近似值的几个关键估计来克服，这些估计可能是独立的。因此，我们的分析中包含了 Darcy 变体的存在结果，我们的工作有助于将最近关于具有奇异势的 Cahn-Hilliard 修复模型的弱解和平稳解的结果推广。