  We investigate a mathematical theory for the erosion of sediment which begins
with the study of a non-linear, parabolic, weighted 4-Laplace equation on a
rectangular domain corresponding to a base segment of an extended landscape.
Imposing natural boundary conditions, we show that the equation admits entropy
solutions and prove regularity and uniqueness of weak solutions when they
exist. We then investigate a particular class of weak solutions studied in
previous work of the first author and produce numerical simulations of these
solutions. After introducing an optimal transportation problem for the sediment
flow, we show that this class of weak solutions implements the optimal
transportation of the sediment.
