  We define rigorously a solution to the fourth-order total variation flow
equation in $\mathbb{R}^n$. If $n\geq3$, it can be understood as a gradient
flow of the total variation energy in $D^{-1}$, the dual space of $D^1_0$,
which is the completion of the space of compactly supported smooth functions in
the Dirichlet norm. However, in the low dimensional case $n\leq2$, the space
$D^{-1}$ does not contain characteristic functions of sets of positive measure,
so we extend the notion of solution to a larger space. We characterize the
solution in terms of what is called the Cahn-Hoffman vector field, based on a
duality argument. This argument relies on an approximation lemma which itself
is interesting.
  We introduce a notion of calibrability of a set in our fourth-order setting.
This notion is related to whether a characteristic function preserves its form
throughout the evolution. It turns out that all balls are calibrable. However,
unlike in the second-order total variation flow, the outside of a ball is
calibrable if and only if $n\neq2$. If $n\neq2$, all annuli are calibrable,
while in the case $n=2$, if an annulus is too thick, it is not calibrable.
  We compute explicitly the solution emanating from the characteristic function
of a ball. We also provide a description of the solution emanating from any
piecewise constant, radially symmetric datum in terms of a system of ODEs.
