A Semigroup Approach to an Integro-Differential Equation Modeling Slow Erosion

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.