Motivated by the mathematical modeling of tumor invasion in healthy tissues, we propose a generalized compressible diphasic Navier-Stokes Cahn-Hilliard model that we name G-NSCH. We assume that the two phases of the fluid represent two different populations of cells: cancer cells and healthy tissue. We include in our model possible friction and proliferation effects. The model aims to be as general as possible to study the possible mechanical effects playing a role in the invasive growth of a tumor. In the present work, we focus on the analysis and numerical simulation of the G-NSCH model. Our G-NSCH system is derived rigorously and satisfies the basic mechanics of fluids and the thermodynamics of particles. Under simplifying assumptions, we prove the existence of global weak solutions. We also propose a structure-preserving numerical scheme based on the scalar auxiliary variable method to simulate our system and present some numerical simulations validating the properties of the numerical scheme and illustrating the solutions of the G-NSCH model.

翻译：受健康组织中肿瘤侵袭的数学建模启发，我们提出了一个广义可压缩两相Navier-Stokes Cahn-Hilliard模型，并将其命名为G-NSCH。我们假设流体的两相分别代表两类不同的细胞群体：癌细胞和健康组织。在模型中，我们考虑了可能的摩擦效应和增殖效应。该模型旨在尽可能具有普适性，以研究在肿瘤侵袭生长中可能起作用的力学效应。本研究聚焦于G-NSCH模型的分析与数值模拟。我们的G-NSCH系统经过严格推导，满足流体基本力学和粒子热力学。在简化假设下，我们证明了全局弱解的存在性。我们还基于标量辅助变量方法提出了一种保结构的数值格式来模拟该系统，并通过数值模拟验证了该数值格式的性质，展示了G-NSCH模型的解的形态。