Slow erosion with rough geological layers

In this paper, we study the slow erosion profile with rough geological layers. The mathematical model is a scalar conservation law which takes the form of an integro-differential equation with discontinuous flux functions. It is has been shown that, for a class of erosion functions, vertical jumps in the profile can occur in finite time even with smooth initial data. Three types of singularities can form in the solution, representing kinks, hyper-kinks, and jump discontinuities in the profile. The mathematical model studied in this paper is formulated in a transformed coordinate, where vertical jumps in the profile become an interval where the unknown is zero after applying a pointwise constraint. Front tracking approximate solutions are designed for both cases with or without jump discontinuities. Solutions to Riemann problems with discontinuous flux functions are derived, and suitable functionals that measure strengths of various wave types are introduced. Through the establishment of various a priori estimates, we achieve desired compactness which yields the existence of entropy weak solutions. Finally, a Kruzhkov-type entropy inequality is proved, leading to stability and uniqueness of the solutions.