A semigroup approach to an integro-differential equation modeling slow erosion
Alberto Bressan and Wen Shen
Abstract: The paper is concerned with a scalar conservation law with nonlocal flux, providing a model for granular flow with slow erosion and deposition. While the solution $u = u(t,x)$ can have jumps, the inverse function $x = x(t,u)$ is always Lipschitz continuous; its derivative has bounded variation and satisfies a balance law with measure-valued sources. Using a backward Euler approximation scheme combined with a nonlinear projection operator, we construct a continuous semigroup whose trajectories are the unique entropy weak solutions to this balance law. Going back to the original variables, this yields the global well-posedness of the Cauchy problem for the granular flow model.