Mathematical derivation and analysis of a mixture model of tumor growth

We derive, through the periodic homogenization theory in thin heterogeneous domains, a 2 D model consisting of Hele-Shaw equation coupled with the convective Cahn-Hilliard equation with non-constant mobility. The upscaled set of equations, which models in particular tumor growth, is then analyzed and we prove some regularity results. We heavily rely on the two-scale convergence concept in thin heterogeneous media associated to some Sobolev inequalities such as the Gagliardo-Nirenberg and Agmon inequalities to achieve our goal.