  We review the theory of Gradient Flows in the framework of convex and lower
semicontinuous functionals on ${\sf CAT}(\kappa)$-spaces and prove that they
can be characterized by the same differential inclusion $y_t'\in-\partial^-{\sf
E}(y_t)$ one uses in the smooth setting and more precisely that $y_t'$ selects
the element of minimal norm in $-\partial^-{\sf E}(y_t)$. This generalizes
previous results in this direction where the energy was also assumed to be
Lipschitz.
  We then apply such result to the Korevaar-Schoen energy functional on the
space of $L^2$ and ${\sf CAT}(0)$ valued maps: we define the Laplacian of such
$L^2$ map as the element of minimal norm in $-\partial^-{\sf E}(u)$, provided
it is not empty. The theory of gradient flows ensures that the set of maps
admitting a Laplacian is $L^2$-dense. Basic properties of this Laplacian are
then studied.
