  We consider a (degenerate) cross-diffusion model of tumor growth structured
by phenotypic trait. We prove the existence of weak solutions and the
incompressible limit as the pressure becomes stiff extending methods recently
introduced in the context of two-species cross-diffusion systems. In the
stiff-pressure limit, the compressible model generates a free boundary problem
of Hele-Shaw type. Moreover, we prove a new L4-bound on the pressure gradient.
