Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth

We study the existence of weak solutions to a Cahn-Hilliard-Darcy system coupled witha convection-reaction-diffusion equation through the fluxes, through the source terms and in Darcy’slaw. The system of equations arises from a mixture model for tumour growth accounting for transportmechanisms such as chemotaxis and active transport. We prove, via a Galerkin approximation, theexistence of global weak solutions in two and three dimensions, along with new regularity results forthe velocity field and for the pressure. Due to the coupling with the Darcy system, the time derivativeshave lower regularity compared to systems without Darcy flow, but in the two dimensional case weemploy a new regularity result for the velocity to obtain better integrability and temporal regularityfor the time derivatives. Then, we deduce the global existence of weak solutions for two variantsof the model; one where the velocity is zero and another where the chemotaxis and active transportmechanisms are absent