Mathematical Models for Erosion and the Optimal Transportation of Sediment

TL;DR

This paper develops a mathematical framework for sediment erosion using a nonlinear parabolic equation, proving solution properties and linking solutions to optimal transportation of sediment.

Contribution

It introduces a new mathematical model for sediment erosion, proves existence, regularity, and uniqueness of solutions, and connects these solutions to optimal transportation theory.

Findings

Existence of entropy solutions for the erosion model

Regularity and uniqueness of weak solutions

Numerical simulations demonstrating sediment transportation

Abstract

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.