  We study the $L^2$-gradient flows for functionals of the type
$\int_{\Omega}f(x,\mathbb{A}u)\,\mathrm{dx}$, where $f$ is a convex function of
linear growth and $\mathbb{A}$ is some first-order linear constant-coefficient
differential operator.
  To this end we identify the relaxation of the functional to the space
$\mathrm{BV}^{\mathbb{A}}\cap L^2$, identify its subdifferential, and show
pointwise representation formulas for the relaxation and the subdifferential,
both with and without Dirichlet boundary conditions. The existence and
uniqueness then follow from abstract semigroup theory.
  Another main novelty of our work is that we require no regularity or
continuity assumptions for $f$.
