A “Lakes and Rivers” Heuristic Metaphor for the Singular Limit of a Nonlinear Diffusion PDE

We introduce a new PDE free boundary problem that in a singular limit admits an interesting metaphoric interpretation, describing how "water" originating from "rainfall" flows along a landscape of varying height, forming "lakes" and also "rivers" and "surface flows," the latter described by transport equations along downhill gradient flows. We stress that this interpretation is a metaphor only, and we make absolutely no claim that the nonlinear PDE we study in any way models actual physical processes.