  We consider a diffuse interface model for tumor growth recently proposed in
[Y. Chen, S.M. Wise, V.B. Shenoy, J.S. Lowengrub, A stable scheme for a
nonlinear, multiphase tumor growth model with an elastic membrane, Int. J.
Numer. Methods Biomed. Eng., 30 (2014), 726-754]. In this new approach sharp
interfaces are replaced by narrow transition layers arising due to adhesive
forces among the cell species. Hence, a continuum thermodynamically consistent
model is introduced. The resulting PDE system couples four different types of
equations: a Cahn-Hilliard type equation for the tumor cells (which include
proliferating and dead cells), a Darcy law for the tissue velocity field, whose
divergence may be different from 0 and depend on the other variables, a
transport equation for the proliferating (viable) tumor cells, and a
quasi-static reaction diffusion equation for the nutrient concentration. We
establish existence of weak solutions for the PDE system coupled with suitable
initial and boundary conditions. In particular, the proliferation function at
the boundary is supposed to be nonnegative on the set where the velocity ${\bf
u}$ satisfies ${\bf u}\cdot\nu>0$, where $\nu$ is the outer normal to the
boundary of the domain. We also study a singular limit as the diffuse interface
coefficient tends to zero.
