On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials

We consider a model describing the evolution of a tumor inside a host tissue in terms of the parameters ϕ_p, ϕ_d (proliferating and necrotic cells, respectively), u (cell velocity) and n (nutrient concentration). The variables ϕ_p, ϕ_d satisfy a vectorial Cahn-Hilliard-type system with nonzero forcing term (implying that their spatial means are not conserved in time), whereas u obeys a variant of Darcy's law and n satisfies a quasi-static diffusion equation. The main novelty of the present work stands in the fact that we are able to consider a configuration potential of singular type implying that the concentration vector (ϕ_p,ϕ_d) is constrained to remain in the range of physically admissible values. On the other hand, in the presence of nonzero forcing terms, this choice gives rise to a number of mathematical difficulties, especially related to the control of the mean values of ϕ_p and ϕ_d. For the resulting mathematical problem, by imposing suitable initial-boundary conditions, our main result concerns the existence of weak solutions in a proper regularity class.