The variational approach to $s$-fractional heat flows and the limit cases $s\to 0^{+}$ and $s\to 1^{-}$

Abstract

This paper deals with the limit cases for $s$-fractional heat flows in a cylindrical domain, with homogeneous Dirichlet boundary conditions, as $s\to 0^{+}$ and $s\to 1^{-}$\,. To this purpose, we describe the fractional heat flows as minimizing movements of the corresponding Gagliardo seminorms, with respect to the $L^2$ metric. First, we provide an abstract stability result for minimizing movements in Hilbert spaces, with respect to a sequence of $\Gamma$-converging uniformly $\lambda$-convex energy functionals. Then, we provide the $\Gamma$-convergence analysis of the $s$-Gagliardo seminorms as $s\to 0^{+}$ and $s\to 1^{-}$\,, and apply the general stability result to such specific cases. As a consequence, we prove that $s$-fractional heat flows (suitably scaled in time) converge to the standard heat flow as $s\to 1^{-}$, and to a degenerate ODE type flow as $s\to 0^{+}$\,. 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 $s\to 0^{+}$\,, 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.