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 ?p ,?d (proliferating and necrotic cells, respectively), u (cell velocity) and n (nutrient concentration). The variables ?p,?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 (?p,?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 ?p and ?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.