  Models of tumor growth, now commonly used, present several levels of
complexity, both in terms of the biomedical ingredients and the mathematical
description. The simplest ones contain competition for space using purely fluid
mechanical concepts. Another possible ingredient is the supply of nutrients
through vasculature. The models can describe the tissue either at the level of
cell densities, or at the scale of the solid tumor, in this latter case by
means of a free boundary problem.
  Our first goal here is to formulate a free boundary model of Hele-Shaw type,
a variant including growth terms, starting from the description at the cell
level and passing to a certain limit. A detailed mathematical analysis of this
purely mechanical model is performed. Indeed, we are able to prove strong
convergence in passing to the limit, with various uniform gradient estimates;
we also prove uniqueness for the asymptotic Hele-Shaw type problem. The main
tools are nonlinear regularizing effects for certain porous medium type
equations, regularization techniques \`a la Steklov, and a Hilbert duality
method for uniqueness. At variance with the classical Hele-Shaw problem, here
the geometric motion governed by the pressure is not sufficient to completely
describe the dynamics. A complete description requires the equation on the cell
number density.
  Using this theory as a basis, we go on to consider the more complex model
including nutrients. We obtain the equation for the limit of the coupled
system; the method relies on some BV bounds and space/time a priori estimates.
Here, new technical difficulties appear, and they reduce the generality of the
results in terms of the initial data. Finally, we prove uniqueness for the
system, a main mathematical difficulty.
