Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport
Harald Garcke, Kei Fong Lam · European Journal of Applied Mathematics · 2016
We consider a diffuse interface model for tumour growth consisting of a Cahn–Hilliard equation with source terms coupled to a reaction–diffusion equation. The coupled system of partial differential equations models a tumour growing in the presence of a nutrient species and surrounded by healthy tissue. The model also takes into account transport mechanisms such as chemotaxis and active transport. We establish well-posedness results for the tumour model and a variant with a quasi-static nutrient. It will turn out that the presence of the source terms in the Cahn–Hilliard equation leads to new difficulties when one aims to derivea prioriestimates. However, we are able to prove continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms.