  Free boundaries of biofilms advancing on surfaces evolve according to
conservation laws coupled with systems of partial differential equations for
velocities, pressures and chemicals affecting cell behavior. Thin film
approximations lead to complicated quasi-stationary systems coupling stationary
transport equations and compressible Stokes systems with
convection-reaction-diffusion equations.We establish existence, uniqueness and
stability of solutions of the different submodels involved and then obtain well
posedness results for the full system. Our analysis relies on the construction
of weak solutions for the steady transport equations under sign assumptions and
the reformulation of the compressible Stokes problem as an elliptic system with
enhanced regularity properties on the pressure. We need to consider velocity
fields whose divergence and normal boundary components satisfy sign conditions,
instead of vanishing as classical results require. Applications include the
study of cells, biofilms and tissues, where one phase is a liquid solution,
whereas the other one is assorted biomass.
