  In some models involving nonlinear conservation laws, physical mechanisms
exist which prevent the formation of shocks. This gives rise to conservation
laws with a constraint on the gradient of the solution. We approach this
problem by studying a related conservation law with a spatial nonlocal term. We
prove existence, uniqueness and stability of solution of the Cauchy problem for
this nonlocal conservation law. In turn, this allows us to provide a notion of
solution to the conservation law with a gradient constraint. The proof of
existence is based on a time-stepping technique, and an $L^1$-contraction
estimate follows from stability results of Karlsen and Risebro.
