Mathematical modeling of landscape erosion: Convergence, singularities, and shocks in a continuous, transport-limited model

This dissertation will examine continuous mathematical models of transport-limited landscape erosion. First, we will develop a coupled pair of partial differential equations describing landscape erosion: one equation will describe water evolution, and the second will describe evolution of the landscape. In subsequent chapters, we will investigate convergence, steady-state, and numerical approximation of solutions to the water equation, as well as stability of the coupled system. In the final chapter, we will find a class of nonphysical, unbounded solutions, suggest a remedy, and present two types of shocks found in this improved system. We will conclude by showing how these shocks first form a convex hillslope, and then transition to a mature, concave steady-state.