Center for Complex and Nonlinear Science

We consider the theory of erosion and investigate connections to the theory of optimal transport. The mathematical theory of erosion is based on two coupled nonlinear partial differential equations. The first one describing the water flow can be considered to be an averaged Navier-Stokes equation, and the second one describing the sediment flow is a very degenerate nonlinear parabolic equation. In the first half of this work, we prove existence, uniqueness, and regularity properties of weak solutions to the second model equation describing the sediment flow. This forms the basis to define an optimal transport problem for the movement of sediment; the second half of this work is devoted to this optimal transport problem for the sediment. We solve the optimal transport problem. Furthermore, we demonstrate that the optimal transport problem distinguishes a particular class of solutions to the model equation. The movement of sediment according to the solution of the optimal transport problem is identical to the movement of sediment according to these solutions of the model equation. The physical interpretation is that if the sediment flows according to the model equation on the surface of separable solutions consisting of valleys separated by mountain ridges, that are observed in simulation and in nature, then the sediment is ``optimally transported.''