  An existence and uniqueness result, up to fattening, for crystalline mean
curvature flows with forcing and arbitrary (convex) mobilities, is proven. This
is achieved by introducing a new notion of solution to the corresponding level
set formulation. Such a solution satisfies the comparison principle and a
stability property with respect to the approximation by suitably regularized
problems. The results are valid in any dimension and for arbitrary, possibly
unbounded, initial closed sets. The approach accounts for the possible presence
of a time-dependent bounded forcing term, with spatial Lipschitz continuity. As
a by-product of the analysis, the problem of the convergence of the
Almgren-Taylor-Wang minimizing movements scheme to a unique (up to fattening)
"flat flow" in the case of general, possibly crystalline, anisotropies is
settled.
