  We are interested in the gradient flow of a general first order convex
functional with respect to the $L^1$-topology. By means of an implicit
minimization scheme, we show existence of a global limit solution, which
satisfies an energy-dissipation estimate, and solves a non-linear and non-local
gradient flow equation, under the assumption of strong convexity of the energy.
Under a monotonicity assumption we can also prove uniqueness of the limit
solution, even though this remains an open question in full generality. We also
consider a geometric evolution corresponding to the $L^1$-gradient flow of the
anisotropic perimeter. When the initial set is convex, we show that the limit
solution is monotone for the inclusion, convex and unique until it reaches the
Cheeger set of the initial datum. Eventually, we show with some examples that
uniqueness cannot be expected in general in the geometric case.
