  Two phase solid-fluid mixture models are ubiquitous in biological
applications. For instance, models for growth of tissues and biofilms combine
time dependent and quasi-stationary boundary value problems set in domains
whose boundary moves in response to variations in the mechano-chemical
variables. For a model of biofilm spread, we show how to obtain better posed
models by characterizing the time derivatives of relevant quasi-stationary
magnitudes in terms of additional boundary value problems. We also give
conditions for well posedness of time dependent submodels set in moving domains
depending on the motion of the boundary. After constructing solutions for
transport, diffusion and elliptic submodels for volume fractions,
displacements, velocities, pressures and concentrations with the required
regularity, we are able to handle the full model of biofilm spread in moving
domains assuming we know the dynamics of the boundary. These techniques are
general and can be applied in models with a similar structure arising in
biological and chemical engineering applications.
