Diffuse interface models for tumour growth within a non-isothermal Cahn-Hilliard theory for phase separation: thermodynamics, chemotaxis and stability

In this thesis we provide a scheme for phase separation by accounting for diffusion, dynamic equations and consistency with thermodynamics. The constituents are two compressible fluids and, for the non-simple character of the mixture, an extra energy flux is allowed to occur. Since also thermal effects are included, the result is a whole set of evolution equations for the concentration, the velocity and the temperature which describes a non-isothermal Navier-Stokes-Cahn-Hilliard model for phase separation in a binary mixture with extra fluxes and within the Fourier heat theory. Alternative heat theories may be proposed for this Navier-Stokes-Cahn-Hilliard theory. Meanwhile the mixing problem is described graphically. Moreover the model may be generalized including a source term, and this doesn't affect the thermodynamic scheme. Then we describe and then compare two mathematical models for chemotactic processes: the pioneeristic Keller-Segel model and the hydrodynamic model by Chavanis and Sire. The first one is able to describe clusters or peaks, the second one involves inertial effects together with a friction force and leads to network patterns or filaments that are in good agreement with the experimental results. We analyze the linear stability of an infinite, stationary and homogeneous distribution of cells for determining the critical thresholds above which chemotactic collapse is allowed and cellular aggregation is reproduced. Then we discuss the differences between the two models, moreover we show the analogy between the instability criterion for biological populations and the Jeans instability criterion in an astrophysical setting. Finally we propose a different approach for the derivation of new diffuse interface models for tumour growth (with chemotaxis and active transport) based on the Cahn-Hilliard theory, combined with the (stationary) Darcy momentum equation.