The variational approach to s-fractional heat flows and the limit cases s → 0+ and s → 1−

This paper deals with the limit cases for s-fractional heat flows in a cylindrical domain, with homogeneous Dirichlet boundary conditions, as s→0+ and s→1−. We describe the fractional heat flows as minimizing movements of the corresponding Gagliardo seminorms, with respect to the L2 metric. To this end, we first provide a Γ-convergence analysis for the s-Gagliardo seminorms as s→0+ and s→1−; then, we exploit an abstract stability result for minimizing movements in Hilbert spaces, with respect to a sequence of Γ-converging uniformly λ-convex energy functionals. We prove that s-fractional heat flows (suitably scaled in time) converge to the standard heat flow as s→1−, and to a degenerate ODE type flow as s→0+. Moreover, looking at the next order term in the asymptotic expansion of the s-fractional Gagliardo seminorm, we show that suitably forced s-fractional heat flows converge, as s→0+, to the parabolic flow of an energy functional that can be seen as a sort of renormalized 0-Gagliardo seminorm: the resulting parabolic equation involves the first variation of such an energy, that can be understood as a zero (or logarithmic) Laplacian.