  We consider a model of congestion dynamics with chemotaxis: The density of
cells follows a chemical signal it generates, while subject to an
incompressibility constraint. The incompressibility constraint results in the
formation of patches, describing regions where the maximal density has been
reached. The dynamics of these patches can be described by either Hele-Shaw or
Richards equation type flow (depending on whether we consider the model with
diffusion or the model with pure advection). Our focus in this paper is on the
construction of weak solutions for this problem via a variational discrete time
scheme of JKO type. We also establish the uniqueness of these solutions. In
addition, we make more rigorous the connection between this incompressible
chemotaxis model and the free boundary problems describing the motion of the
patches in terms of the density and associated pressure variable. In
particular, we obtain new results characterizing the pressure variable as the
solution of an obstacle problem and prove that in the pure advection case the
dynamic preserves patches.
