Parabolic $α$-Riesz flows and limit cases $α\to 0^+$, $α\to d^-$

In this paper we introduce the notion of parabolic $α$ -Riesz flow, for $α \in (0, d)$ , extending the notion of $s$ -fractional heat flows to negative values of the parameter $s = - \frac{α}{2}$ . Then, we determine the limit behaviour of these gradient flows as $α \to 0^{+}$ and $α \to d^{-}$ . 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.

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

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

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

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

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

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

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