Parabolic $\alpha$-Riesz flows and limit cases $\alpha \to 0^{+}$, $\alpha \to d^{-}$

Abstract

In this paper we introduce the notion of parabolic $\alpha$-Riesz flow, for $\alpha \in (0,d)$, extending the notion of $s$-fractional heat flows to negative values of the parameter $s=-\frac{\alpha }{2}$. Then, we determine the limit behaviour of these gradient flows as $\alpha \to 0^{+}$ and $\alpha \to d^{-}$. To this end we provide a preliminary $\Gamma$-convergence expansion for the Riesz interaction energy functionals. Then we apply abstract stability results for uniformly $\lambda$-convex functionals which guarantee that $\Gamma$-convergence commutes with the gradient flow structure.