Parabolic α-Riesz flows and limit cases α → 0+, α → d−

In this paper we introduce the notion of parabolic α-Riesz flow, for α ∈ (0, d), extending the notion of s-fractional heat flows to negative values of the parameter s=−α2. Then, we determine the limit behaviour of these gradient flows as α → 0+ and α → d−. To this end we provide a preliminary Γ-convergence expansion for the Riesz interaction energy functionals. Then we apply abstract stability results for uniformly λ-convex functionals which guarantee that Γ-convergence commutes with the gradient flow structure.