  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.
