Circular, stationary profiles emerging in unidirectional abrasion

We describe a PDE model of bedrock abrasion by impact of moving particles and show that by assuming unidirectional impacts the modification of a geometrical PDE due to Bloore exhibits circular arcs as solitary profiles. We demonstrate the existence and stability of these stationary, travelling shapes by numerical experiments based on finite difference approximations. Our simulations show that, depending on initial profile shape and other parameters, these circular profiles may evolve via long transients which, in a geological setting, may appear as non-circular stationary profiles.