Skip to main content
Open Access Publications from the University of California

Motion due to Dynamic Density Constraints

  • Author(s): Woodhouse, Brent Alan
  • Advisor(s): Kim, Christina
  • Biskup, Marek
  • et al.

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.

Main Content
Current View