We investigate a non-local, non-linear conservation law, first introduced by A.C. Fowler to describe morphodynamics of dunes, see [6, 7]. A remarkable feature is the violation of the maximum principle, which allows for erosion phenomenon. We prove well posedness for initial data in $L^2$ and give an explicit counterexample for the maximum principle. We also provide numerical simulations corroborating our theoretical results.
"A non-monotone nonlocal conservation law for dune morphodynamics." Differential Integral Equations 23 (1/2) 155 - 188, January/February 2010. https://doi.org/10.57262/die/1356019392