Second-order asymptotics of fractional Gagliardo seminorms as $s\to 1^{-}$ and convergence of the associated gradient flows

We study the second-order asymptotic expansion of the $s$-fractional Gagliardo seminorm as $s\to 1^{-}$ in terms of a higher order nonlocal functional. We prove a Mosco-convergence result for the energy functionals and that the $L^2$-gradient flows of the energies converge to the $L^2$-gradient flows of the Mosco-limit.