Mechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn–Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell–cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. In the overall model an equation of Cahn–Hilliard type is coupled to the system of linear elasticity and a reaction–diffusion equation for a nutrient concentration. The highly non-linear coupling between a fourth-order Cahn–Hilliard equation and the quasi-static elasticity system lead to new challenges which cannot be dealt within a gradient flow setting which was the method of choice for other elastic Cahn–Hilliard systems. We show existence, uniqueness and regularity results. In addition, several continuous dependence results with respect to different topologies are shown. Some of these results give uniqueness for weak solutions and other results will be helpful for optimal control problems.

到目前为止，在基于Cahn-Hilliard方法的相场肿瘤模型中，机械效应大多被忽略。在本文中，我们研究了肿瘤生长的宏观力学模型，其中借助Ginzburg-Landau型能量考虑了细胞间粘附作用。在整个模型中，Cahn–Hilliard型方程式与线性弹性系统和营养物浓度的反应扩散方程式耦合。四阶Cahn-Hilliard方程与准静态弹性系统之间的高度非线性耦合导致了新的挑战，而梯度流设置是其他弹性Cahn-Hilliard系统选择的方法，因此无法应对这些新挑战。我们显示存在性，唯一性和规律性结果。此外，显示了针对不同拓扑的几个连续依赖结果。这些结果中的一些为弱解提供了唯一性，而其他结果对于优化控制问题将有所帮助。