We consider the continuity equation for a population density subject to (i) a density upper-bound that depends on space and time and (ii) a velocity that minimizes the kinetic energy. A solution is constructed via the Wasserstein minimizing movement scheme for a corresponding time-dependent energy. Motion of the solution is driven by a decreasing density constraint.

With a few assumptions, we prove this solution moves according to a free boundary problem of modified Hele-Shaw type that depends on the density constraint. In order to do this, we utilize a modified porous medium equation as an approximation to the original problem. Viscosity solution arguments are used to prove that given a decreasing density constraint, the porous medium equation solutions converge to the Hele-Shaw free boundary problem solution. By analyzing the Wasserstein gradient flow structure of the time-dependent energies involved, we next show that the porous medium equation solutions also converge to a solution of the original problem, thus identifying it with the Hele-Shaw description.

In addition, we consider complications of the analysis without each assumption, perform numerical simulations supporting the results, and explore some limiting situations of the dynamics.