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We review the theory of Gradient Flows in the framework of convex and lowersemicontinuous functionals on ${\sf CAT}(\kappa)$-spaces and prove that theycan be characterized by the same differential inclusion $y_t'\in-\partial^-{\sfE}(y_t)$ one uses in the smooth setting and more precisely that $y_t'$ selectsthe element of minimal norm in $-\partial^-{\sf E}(y_t)$. This generalizesprevious results in this direction where the energy was also assumed to beLipschitz.  We then apply such result to the Korevaar-Schoen energy functional on thespace 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 $-\partial^-{\sf E}(u)$, providedit is not empty. The theory of gradient flows ensures that the set of mapsadmitting a Laplacian is $L^2$-dense. Basic properties of this Laplacian arethen studied.

A differential perspective on Gradient Flows on ${\sf CAT}(\kappa)$-spaces and applications