  We consider a congested aggregation model that describes the evolution of a
density through the competing effects of nonlocal Newtonian attraction and a
hard height constraint. This provides a counterpoint to existing literature on
repulsive-attractive nonlocal interaction models, where the repulsive effects
instead arise from an interaction kernel or the addition of diffusion. We
formulate our model as the Wasserstein gradient flow of an interaction energy,
with a penalization to enforce the constraint on the height of the density.
From this perspective, the problem can be seen as a singular limit of the
Keller-Segel equation with degenerate diffusion. Two key properties distinguish
our problem from previous work on height constrained equations: nonconvexity of
the interaction kernel (which places the model outside the scope of classical
gradient flow theory) and nonlocal dependence of the velocity field on the
density (which causes the problem to lack a comparison principle). To overcome
these obstacles, we combine recent results on gradient flows of nonconvex
energies with viscosity solution theory. We characterize the dynamics of patch
solutions in terms of a Hele-Shaw type free boundary problem and, using this
characterization, show that in two dimensions patch solutions converge to a
characteristic function of a disk in the long-time limit, with explicit rate of
convergence. We believe that a key contribution of the present work is our
blended approach, combining energy methods with viscosity solution theory.
